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+{"text":"\\section*{Introduction}\n\nLet $X$ be a noetherian scheme and denote by $\\mathop{\\sf FEC}\\nolimits(X)$ the site of finite \\'etale coverings of $X$. Then we can define the assignment $\\Pi_1:Y\\mapsto \\Pi_1(Y)$, where $Y$ is an \\'etale scheme over $X$ and $\\Pi_1(Y)$ is the \\'etale fundamental groupoid of $Y$. In this paper, we will show that this association, which is a 2-functor, is in actuality a costack over $\\mathop{\\sf FEC}\\nolimits(X)$. The costack, the dual notion of a stack, is a rather overlooked construction. However, as we will see, for a covariant 2-functor to be a costack is actually a very natural property, as it merely means that the 2-categorical version of the Seifert-van Kampen theorem holds for every covering. Hence, by proving that the 2-functor $\\Pi_1$ is a costack, we have proven that a slightly reformulated version of the Seifert-van Kampen theorem holds for the \\'etale fundamental groupoid.\n\nBut more is true. As we will show in this paper, $\\Pi_1$ is not just a costack over $X$, but indeed the 2-terminal costack. This demonstrates that the \\'etale fundamental groupoid is indeed defined by the Seifert-van Kampen theorem. This result agrees with the one obtained in \\cite{top.groupoid}, where we showed that the topological fundamental groupoid was a 2-terminal costack. Hence we are able to talk about the fundamental groupoid of a site, which could be of independent interest. Indeed, although not discussed in this paper, another use of this axiomatisation is that we can now also talk about other fundamental objects. For example, we could define the fundamental groupoid scheme of a site $\\bf{T}$ to be the 2-terminal objects in the 2-category of costacks over $\\bf{T}$ with values in groupoid schemes.\n\\newline\n\nThis paper was written as part of my PhD thesis at the University of Leicester under the supervision of Dr. Frank Neumann. He introduced me to the fundamental groupoid and inspired much of this paper, for which I would like to thank him.\n\n\\section{Preliminaries}\nIn this section we will fix some basic terminology. Recall that a 2-category is a category enriched in categories. A category $\\mathfrak{G}$ is said to be a \\emph{groupoid}\\index{Groupoid!} if every morphism is an isomorphism. \n\n\\begin{De} Let $\\mathfrak{G}$ be a groupoid such that for every $a,b\\in \\mathfrak{G}$, $\\mathsf {Hom}(a,b)$ is endowed with the profinite topology and further the composition and inverse maps are continuous. We say that $\\mathfrak{G}$ is a finitely connected profinite groupoid, if it is equivalent to the 2-coproduct of finitely many connected groupoids coming from profinite groups.\n\\end{De}\n\nNote that this is not the best way to define a profinite groupoid in the general case, since it does not deal with the topology on its connected components. One should define it as simply the filtered 2-limit of finite groupoids endowed with the profinite topology. But for the purposes of this paper, this simpler definition is adequate.\n\n\n\\subsection{2-Limits and 2-Colimits}\\label{sec.2-lim.2-colim}\n\nJust like limits and colimits are of fundamental importance in category theory, so too are 2-limits and 2-colimits in 2-category theory. In this subsection, we will talk a little about the 2-limit as well as the filtered 2-colimit of 2-functors. \n\\newline\n\nLet $I$ be a category and $\\mathfrak{F}:I\\rightarrow \\mathfrak{Cat}$ be  a covariant 2-functor from the category $I$ to the 2-category $\\mathfrak{Cat}$ of categories. For an element $i\\in I$ we let $\\mathfrak{F}_i$ be the value of $\\mathfrak{F}$ at $i$. For a morphism $\\psi:i\\rightarrow j$ we let $\\psi_*:\\mathfrak{F}_i\\to \\mathfrak{F}_j$ be the induced functor. For any $i\\xrightarrow{\\psi} j\\xrightarrow{\\nu} k$, one has the natural transformation $\\mu_{\\psi,\\nu}:\\nu_*\\psi_*\\rightarrow (\\nu\\psi)_*$ satisfying the coherence condition. We can now give the definitions:\n\nObjects of the category 2-$\\mathop{\\sf lim}\\nolimits\\limits_i\\mathfrak{F}_i$ (or simply 2-$\\mathop{\\sf lim}\\nolimits \\mathfrak{F}$) are collections $(x_i,\\xi_\\psi)$, where $x_i$ is an object of $\\mathfrak{F}_i$, while $\\xi_\\psi:\\psi_*(x_j)\\to x_i$ for $i\\leq j$, is an isomorphism of the category $\\mathfrak{F}_j$ satisfying the 1-cocycle condition. That is, for any $i\\xrightarrow{\\psi} j\\xrightarrow{\\nu} k$, the diagram\n$$\\xymatrix{\\nu_*(\\psi_*(x_i)) \\  \\  \\ar[r]^{\\nu_*(\\xi_{\\psi})}\\ar[d]^{\\mu_{\\psi,\\nu}} & \\nu_*(x_j)\\ar[d]^{\\xi_{\\nu}} \\\\ \n\t          (\\nu\\psi)_*(x_i)\\ar[r]_{\\xi_{\\psi\\nu}} & x_k }$$ \ncommutes. A morphism from $(x_i,\\xi_{\\psi})$ to $(y_i,\\eta_{\\psi})$ is  a collection $(f_i)$, where $f_i:x_i\\rightarrow y_i$ is a morphism of $\\mathfrak{F}_i$ such that for any $\\psi:i\\rightarrow j$, the following \n$$\\xymatrix{\\psi_*(x_i)\\ar[r]^{\\xi_{\\psi}}\\ar[d]_{\\psi_*(f_i)}& x_j\\ar[d]^{f_j} \\\\ \n\\psi_*(y_i)\\ar[r]_{\\eta_{\\psi}}& y_j}$$\n is a commutative diagram. \n\\newline\n\nThe dual notion $2$-$\\mathop{\\sf colim}\\nolimits\\limits_i \\mathfrak{F}_i$ is a category together with a family of functors\n $$\\alpha_i:\\mathfrak{F}_i\\to 2\\text{-}\\mathop{\\sf colim}\\nolimits\\limits_i \\mathfrak{F}_i$$\n and natural transformations $\\lambda_{ij}:\\alpha_j\\psi_{ij}\\Longrightarrow \\alpha_i$, $i\\leq j$,  satisfying  the $1$-cocycle condition: For any $i\\leq j\\leq k$,\n $$\\lambda_{ik}=\\lambda_{ij}\\circ (\\lambda_{jk}\\star\\psi_{ij}).$$\nFurthermore, one requires that for any category ${\\sf C}$, the canonical functor\n$$\\kappa:\\mathsf {Hom}_{\\mathfrak{Cat}}(2\\text{-}\\mathop{\\sf colim}\\nolimits\\limits_i\\mathfrak{F}_i,\\mathsf{C})\\to 2\\text{-}\\mathop{\\sf lim}\\nolimits\\limits_i\\mathsf {Hom}_{\\mathfrak{Cat}}(\\mathfrak{F}_i,{\\sf C})$$\nis an equivalence of categories. Here the functor $\\kappa$ is given by $\\kappa(\\chi)=(\\chi\\circ  \\alpha_i, \\chi_i\\star \\lambda_{ij})$. It is well-known that $2$-$\\mathop{\\sf colim}\\nolimits$ exists and is unique up to a unique equivalence of categories (see \\cite[pp. 192-193] {br}).\n\n\\section{Stacks and Costacks}\\label{2-mathematics}\n\n\\subsection{Stacks}\\label{stacksection}\n\nThis subsection follows \\cite{I.Mo} closely, even on notation. Let $X$ be a site and $\\mathfrak{F}:X^{op}\\rightarrow\\mathfrak{Cat}$ a 2-functor where $\\mathfrak{Cat}$ is the 2-category of categories. This is called a \\emph{fibered category}\\index{Fibered! Category} over $X$. It should be noted to avoid any confusion that this is sometimes called a prestack. If we have two fibered categories over $X$, then a morphism between them is called a \\emph{fibered functor}\\index{Fibered! Functor}. \n\nLet $X$ be a site and $\\mathfrak{F}:X^{op}\\rightarrow\\mathfrak{Cat}$ a 2-functor. Let $U$ be an object in $X$ and $\\mathfrak{U}=\\{U_i\\rightarrow U\\}$ a covering of $U$. Then we can consider the following diagram: \n$$\\xymatrix{\\bigsqcap\\limits_{i\\in I}\\mathfrak{F}(U_i)\\ar@<-.5ex>[r] \\ar@<.5ex>[r]& \\bigsqcap\\limits_{i,j\\in I}\\mathfrak{F}(U_{ij})\\ar@<-.7ex>[r]\\ar[r]\\ar@<.7ex>[r] & \\bigsqcap\\limits_{i,j,k\\in I}\\mathfrak{F}(U_{ijk}).}$$\nHere $U_{ij}=U_i\\times_{U}U_j$ and $U_{ijk}=U_i\\times_{U}U_j\\times_{U}U_k$.\nWe denote its 2-limit by $2$-$\\mathop{\\sf lim}\\nolimits(\\mathfrak{U},\\mathfrak{F})$. Note that the 2-limit in this case is usually called the \\emph{descent data} and denoted by $\\mathsf{Des}(\\mathfrak{U},\\mathfrak{F})$. However, working with 2-limits makes it much clearer what a costack should be, hence we will keep using the above notation. \n\n\\begin{De}[Stack]\\label{stack} A fibered category $\\mathfrak{F}$ over $X$ is called a \\emph{stack}\\index{Stack} if for all objects $U$ of $X$ and for all coverings $\\mathfrak{U}$ of $U$, the functor $\\mathfrak{F}(U)\\rightarrow 2$-$\\mathop{\\sf lim}\\nolimits(\\mathfrak{U},\\mathfrak{F})$ is an equivalence of categories.\n\\end{De}\n\nJust like for sheaves, we can talk about the associated stack of a 2-functor.\n\n\\begin{De}\\label{uniquestack} Let $\\mathfrak{F}$ be a fibered category over a site $X$. Then $\\hat{\\mathfrak{F}}$ is the associated stack of $\\mathfrak{F}$ if for every stack $\\mathfrak{G}$ over $X$, we have an equivalence of categories \n$$\\mathsf {Hom}_{\\mathsf{Fb}}(\\mathfrak{F},\\mathfrak{G})\\cong\\mathsf {Hom}_{\\mathsf{St}}(\\hat{\\mathfrak{F}},\\mathfrak{G})$$\nwhere $\\mathsf{Fb}$ denotes the 2-category of fibered categories and $\\mathsf{St}$ the 2-category of stacks.\n\\end{De}\n\n\\begin{Pro}[Direct Stackification]\\label{direct} Let $\\mathfrak{F}:X^{op}\\rightarrow\\mathfrak{Cat}$ be a 2-functor. Define $\\mathfrak{F}'(U):=2$-$\\mathop{\\sf colim}\\nolimits_\\mathfrak{U}(2$-$\\mathop{\\sf lim}\\nolimits(\\mathfrak{U},\\mathfrak{F}))$. In other words, we take the filtered 2-colimits of the categories $2$-$\\mathop{\\sf lim}\\nolimits(\\mathfrak{U},\\mathfrak{F})$ over all coverings $\\mathfrak{U}$ of $U$. After iterating it 3 times we get the stackification. That is to say, we have \n$$\\hat{\\mathfrak{F}}=\\mathfrak{F}'''(U)=2\\text{-}\\mathop{\\sf colim}\\nolimits_\\mathfrak{U}(2\\text{-}\\mathop{\\sf lim}\\nolimits(\\mathfrak{U},\\mathfrak{F}'')).$$ \n\\end{Pro}\n\nFor the proof of this theorem, see \\cite[Theorem 3.8]{Street}.\n\n\\subsection{Costacks}\n\nLet $X$ be a site and $\\mathfrak{F}:X\\rightarrow\\mathfrak{Cat}$ a 2-functor where $\\mathfrak{Cat}$ is the 2-category of categories. Dually to the above subsection, we call $\\mathfrak{F}$ a \\emph{cofibered category}\\index{Cofibered category} over $X$. \n\nTake an object $U$ of $X$ and a covering $\\mathfrak{U}=\\{U_i\\rightarrow U\\}$. Then we can consider the following diagram: \n$$\\xymatrix{\\bigsqcap\\limits_{i\\in I}\\mathfrak{F}(U_i)& \\bigsqcap\\limits_{i,j\\in I}\\mathfrak{F}(U_{ij})\\ar@<-.5ex>[l] \\ar@<.5ex>[l]& \\bigsqcap\\limits_{i,j,k\\in I}\\mathfrak{F}(U_{ijk})\\ar@<-.7ex>[l]\\ar[l]\\ar@<.7ex>[l]}$$\nwhere $U_{ij}=U_i\\times_{U}U_j$ and $U_{ijk}=U_i\\times_{U}U_j\\times_{U}U_k$.\nWe denote its 2-colimit by $2$-$\\mathop{\\sf colim}\\nolimits(\\mathfrak{U},\\mathfrak{F})$. \n\n\\begin{De}[Costack]\\label{cost} A cofibered category $\\mathfrak{F}$ over $X$ is called a \\emph{costack}\\index{Costack} if for all objects $U$ of $X$ and for all coverings $\\mathfrak{U}$ of $U$, the functor $\\mathfrak{F}(U)\\leftarrow 2$-$\\mathop{\\sf colim}\\nolimits(\\mathfrak{U},\\mathfrak{F})$ is an equivalence of categories.\n\\end{De}\n\nAlternatively, we can define a costack using stacks. Namely, $\\mathfrak{F}$ is a costack if for every category $C$, the assignment $U\\mapsto \\mathsf {Hom}_{\\mathfrak{C}}(\\mathfrak{F}(U),C)$ is a stack. It is clear that this is equivalent to the above since $\\mathsf {Hom}(A,-)$ is left exact. However, it should be noted that since it is not (in general) right exact, the duality between stacks and costacks breaks down here. Namely, it would not be sufficient to check that $U\\mapsto\\mathsf {Hom}_\\mathfrak{Cat}(\\mathfrak{F}(U),-)$ is a costack for $\\mathfrak{F}$ to be a stack.\n\n\\begin{De}[Uniqueness Property]\\label{uniquecostack} Let $\\mathfrak{F}$ be a cofibered category over a site $X$. Then we will say that $\\mathfrak{F}$ has an associated costack $\\hat{\\mathfrak{F}}$ if for every costack $\\mathfrak{G}$ over $X$, we have an equivalence of categories \n$$\\mathsf {Hom}_{\\mathsf{Cofb}}(\\mathfrak{G},\\mathfrak{F})\\cong\\mathsf {Hom}_{\\mathsf{Cost}}(\\mathfrak{G},\\hat{\\mathfrak{F}})$$\nwhere $\\mathsf{Cofb}$ denotes the 2-category of cofibered categories and $\\mathsf{Cost}$ the 2-category of costacks.\n\\end{De}\n\nNote that if our category took values in groupoids, then it would be enough to check it for every groupoid. Unfortunately, unlike for stacks, it is not known whether every cofibered category has an associated costack. However, if it exists, it is clearly unique in the 2-categorical sense.\n\n\n\\section{Galois Categories} \\label{galoischapter\n\nIn this section we will generalise the classical (connected) Galois categories to the finitely connected case. Our main interest in them is to study the \\'etale fundamental groupoid of a finitely connected, noetherian scheme. \n\nIn the first subsection, we will show that the analogue of Grothendieck's classical result holds, and that a finitely connected Galois category is equivalent to the category of $\\mathfrak{G}$-$\\mathop{\\sf FSets}\\nolimits$, where $\\mathfrak{G}$ is a finitely connected, profinite groupoid and $\\mathop{\\sf FSets}\\nolimits$ denotes the category of finite sets.\n\nIn the next subsection, we will talk about the whole 2-category of such Galois categories, which includes the morphisms and 2-morphisms. We sharpen the result above, by proving that there is a 2-equivalence between the 2-category of finitely connected categories and the 2-category of finitely connected profinite groupoids. While there are many generalisations of Galois categories, some of which are no doubt more general than ours, this reformulation of the classical theorem seems to be new. \n\nIndeed, it will follow that an even more general formulation is true, namely Corollary \\ref{Galoisstackequivalence}.\n\n\n\n\\subsection{Finitely Connected Galois Categories}\n\nIn order to give the main definition, recall that a morphism $u:A\\to B$ of a category $\\mathcal{C}$ is an \\emph{epimorphism}\\index{Morphism! Epimorphism} (resp. \\emph{monomorphism}\\index{Morphism! Monomorphism}) if for any object $X$ the induced map $\\mathsf {Hom}_{\\mathcal{C}}(B,X)\\to \\mathsf {Hom}_{\\mathcal{C}}(A,X)$ (resp. $\\mathsf {Hom}_{\\mathcal{C}}(X,A)\\to \\mathsf {Hom}_{\\mathcal{C}}(X,B))$)  is injective. Moreover, an epimorphism $u$ is called a \\emph{strict epimorphism}\\index{Morphism! Strict epimorphism} if the pull-back \n$$\\xymatrix{A\\times _B A\\ar@{-->}[r]^{p_1} \\ar@{-->}[d]^{p_2} & A\\ar[d]^u\\\\A\\ar[r]^u & B}$$ \nexists and $B$ is the coequaliser of the diagram: \n$$\\xymatrix{ A\\times_B A \\ar@<0.5ex>[r]^{\\ \\ \\ p_1} \\ar@<-0.5ex>[r]_{\\ \\ \\ p_2} & A\\ar[r]^{u} & B. }$$\n\nWe give the following definition of a finitely connected Galois category which differs from the standard definition of a Galois category (see for example \\cite{AGGT}).\n\n\\begin{De}\\label{f-c G c} A {\\sf (finitely-connected) Galois category}\\index{Galois Category!} is a category $\\mathcal{C}$ together with a set of covariant functors $\\{\\mathcal{F}_j:\\mathcal{C}\\rightarrow \\mathop{\\sf FSets}\\nolimits\\}_{j\\in J}$, satisfying the following axioms: \n\\begin{enumerate}\n\\item Finite limits exist in $\\mathcal{C}$. \n\\item Finite colimits exist in $\\mathcal{C}$. \n\\item Any morphism $u:Y\\rightarrow X$ in $\\mathcal{C}$ factors as $Y\\xrightarrow{u'}X'\\xrightarrow{u''}X$, where $u'$ is a strict epimorphism and $u''$ is a monomorphism and there is an isomorphism $v:X'\\coprod X''\\to X$ such that $u''=vi_1$, where $i_1:X'\\to X'\\coprod X''$ is the standard inclusion.\n\\item Every $\\mathcal{F}_j$ is right exact, i.e. $\\mathcal{F}_j$ respects finite colimits.\n\\item Every $\\mathcal{F}_j$ is left exact,  i.e. $\\mathcal{F}_j$ respects finite limits.\n\\item There exists a finite subset $I\\subset J$ such that any $\\{u:Y\\rightarrow X\\}$ in $\\mathcal{C}$ is an isomorphism if and only if $\\mathcal{F}_i(u)$ is an isomorphism for all $i\\in I$.\n\\end{enumerate}\n\\end{De} \n\nIf $I$ can be chosen to be a one element set, then $\\mathcal{C}$ is called \\emph{connected}\\index{Galois Category! Connected}. This is clearly equivalent to the standard definition of a Galois category. Several facts (Lemma 2.6 i), ii); Prop 3.1; Prop. 3.2 (1), (3) i), iii)) proven in \\cite{AGGT} have immediate generalisations in our situation. To state these statements, recall that an object $X$ is called \\emph{connected}\\index{Object! Connected} if $X\\not =0$ and  for any decomposition $X=Y\\coprod Z$ one has $X=0$ or $Y=0$. Here $0$ denotes the initial object. \n\n\\begin{Pro} \\label{2612} Let $\\mathcal{C}$ be a finitely connected Galois category. Then the following properties hold:\n\\begin{itemize} \n\\item[i)] A morphism $u$ is a monomorphism (resp.  strong epimorphism) if and only if for all $i\\in I$, the map  $\\mathcal{F}_i(u)$ is injective (resp. surjective). A morphism is an isomorphism if and only if it is a monomorphism and a strong epimorphism.\n\\item[ii)] An object $X$ is initial (resp. terminal) provided  for all $i\\in I$, the set $\\mathcal{F}_i(X)=\\emptyset$ (resp. $\\mathcal{F}(X)=*$). Here $*$ denotes the singleton.\n\\item[iii)] Any object $X$ has a unique decomposition $X=U_1\\coprod \\cdots \\coprod U_k$, where the $U_i$ are connected. \n\\item[iv)] If $U$ and $V$ are connected, then any morphism $U\\to V$ is a strong epimorphism. In particular, any endomorphism $U\\to U$ is an automorphism.\n\\item[v)] If $U$ is connected, then for any objects $A_1\\cdots A_m$ the natural map\n$$\\coprod_{i=1}^m \\mathsf {Hom}(U, A_i)\\to \\mathsf {Hom}(U, \\coprod_{i=1}^m A_i)$$\nis a bijection.\n\\end{itemize}\n\\end{Pro}\n\nThe proof is the same as for the connected case. (see \\cite{AGGT}).\n\n\\begin{Le}\\label{2712}  Let $\\mathcal{C}$ be a finitely-connected Galois category and $t=e_1\\coprod \\cdots \\coprod  e_d$ be a decomposition of the terminal object as a coproduct of connected objects. Then for each $1\\leq i\\leq d$  one has (after re-indexing)  $\\mathcal{F}_i(e_i)=*$ and $\\mathcal{F}_i(e_j)=\\emptyset$, $j\\not= i, \\  1\\leq i,j\\leq d$.\n\\end{Le}\n\n\\begin{proof}  Let $1\\leq i\\leq d$. Since $0\\to e_i$ is not an isomorphism, there exists at least one $\\mathcal{F}$ such that $\\mathcal{F}(e_i)\\not =\\emptyset$. After reindexing we can assume that $\\mathcal{F}=\\mathcal{F}_i$. Since $\\mathcal{F}_i(t)=*$ and $\\mathcal{F}_i$ respects coproducts we see that \n$$\\mathcal{F}_i(e_1)\\coprod \\cdots \\coprod \\mathcal{F}_i(e_d)=*.$$\nSo all terms except $\\mathcal{F}_i(e_i)$ are empty sets and the result follows.\n\\end{proof}\n\nNow we are in the position to prove the following result.\n\n\\begin{Le}[Main Lemma]\\label{G.MainLemma} Let $\\mathcal{C}$ be a finitely-connected Galois category. If\n$$t=e_1\\coprod \\cdots \\coprod  e_d$$\nis the decomposition of the terminal object $t$ as a coproduct of connected objects, then there is an equivalence of categories\n$$\\mathcal{C}\\cong \\mathcal{C}_1\\times \\cdots \\times \\mathcal{C}_d$$\nwhere $C_i$ , $1\\leq i\\leq d$ is the following full subcategory of $C$:\n$$\\mathcal{C}_i=\\{X\\in \\mathcal{C}| \\ \\mathcal{F}_j(X)=\\emptyset, j\\not =i, 1\\leq j\\leq d\\}.$$\nFurthermore, the pair $(\\mathcal{C}_i,\\mathcal{F}_i)$ is a connected Galois category and for any element $k\\in I$ the functor $\\mathcal{F}_k$ is isomorphic to exactly one of the functors $\\mathcal{F}_1,\\cdots , \\mathcal{F}_d$.\n\\end{Le}\n\n\\begin{proof} We proceed by induction on $d$. Assume $d=1$. Thus $t$ is connected. In this case $\\mathcal{C}_1=\\mathcal{C}$. Take any of $\\mathcal{F}_i$ and call it $\\mathcal{F}$. First we show that $\\mathcal{F}$ reflects isomorphisms, meaning that if $v$ is a morphism, such that $\\mathcal{F}(v)$ is an isomorphism, then $v$ is an isomorphism. If $U$ is connected, then $U\\to t$ is a strict epimorphism thanks to Proposition \\ref{2612} iv). It follows that $\\mathcal{F}(U)\\to \\mathcal{F}(t)=*$ is a strict epimorphism. Thus $\\mathcal{F}(U)\\not=\\emptyset$. Since any object is a coproduct of connected ones and $\\mathcal{F}$ respects coproducts, it follows that if $A$ is not an initial object, then $\\mathcal{F}(A)\\not =\\emptyset.$\nAssume $u:A\\to B$ is a monomorphism, such that $\\mathcal{F}(u)$ is an isomorphism. Then $B\\cong A\\coprod \\mathcal{C}$. Hence $\\mathcal{F}(\\mathcal{C})=\\emptyset$, so $\\mathcal{C}=0$ and $u$ is an isomorphism. \n\nNow  let $v:A\\to B$ be a general morphism, such that $\\mathcal{F}(v)$ is an isomorphism. Consider the following commutative diagram\n$$\\xymatrix{A\\ar[drr]^{id}\\ar[dr]_{\\delta}\\ar[ddr]_{id}&&\\\\ & A\\times_BA\\ar[r]\\ar[d]&A\\ \\ar[d]_v\\\\&A\\ar[r]^v& B .}$$\nHere $\\delta$ is the diagonal map and hence a monomorphism. Apply $\\mathcal{F}$ to this diagram and use the fact that $\\mathcal{F}$ preserves pulbacks and $\\mathcal{F}(v)$ is an isomorphism. We obtain that $\\mathcal{F}(\\delta)$ is an isomorphism. Thus $\\delta$ is an isomorphism. It follows that $v$ is a monomorphism (thanks to \\cite[Lemma 2.4]{AGGT}) and hence an isomorphism. \nThus $\\mathcal{F}$ reflects isomorphisms and $(\\mathcal{C},\\mathcal{F})$ is a connected Galois category. By \\cite[Theorem 2.8]{AGGT} any other $\\mathcal{F}_i$ is isomorphic to $\\mathcal{F}$. Hence we have proven the lemma for $d=1$.\n\nAssume now that $d>1$. Thanks to Lemma \\ref{2712}, we have $\\mathcal{F}_i(e_j)=\\emptyset$ for all $j\\not =i$ and $\\mathcal{F}_i(e_i)=*$, $1\\leq i,j\\leq d$. One easily sees that for each $1\\leq i\\leq d$ the subcategory $\\mathcal{C}_i$ is closed under finite limits and colimits. Assume $X=\\coprod_{j=1}^k U_j$ is a decomposition as a coproduct of connected objects. \n\n\\underline{Claim 1}: We have $X\\in \\mathcal{C}_i$ if and only if $U_1,\\cdots, U_k\\in \\mathcal{C}_i$. In fact if  $U_1,\\cdots, U_k\\in \\mathcal{C}_i$, then for any $j\\not =i$, $1\\leq j\\leq d$ one has $\\mathcal{F}_j(U_1)=\\cdots =\\mathcal{F}_j(U_k)=\\emptyset$. Thus\n$\\mathcal{F}_j(X)=\\mathcal{F}_j(U_1)\\coprod\\cdots\\coprod \\mathcal{F}_j(U_k)=\\emptyset.$ Hence $X\\in \\mathcal{C}_i$. Conversely, if $X\\in \\mathcal{C}_i$, then $$\\emptyset =\\mathcal{F}_j(X)=\\mathcal{F}_j(U_1)\\coprod\\cdots\\coprod \\mathcal{F}_j(U_k).$$\nThus $\\mathcal{F}_j(U_1)=\\cdots= \\mathcal{F}_j(U_k)=\\emptyset$ and $U_1,\\cdots U_k\\in \\mathcal{C}_i$.\n\n\\underline{Claim 2}: The object $e_i$ is a terminal object in the category $\\mathcal{C}_i$. So, we have to prove that the set $\\mathsf {Hom}(X,e_i)$ is a singleton provided $X\\in \\mathcal{C}_i$. By the first claim it is enough to assume that $X$ is connected. According to Proposition \\ref{2612} v) \n $$*=\\mathsf {Hom}(X,t)=\\mathsf {Hom}(X,e_1)\\coprod \\cdots \\coprod \\mathsf {Hom}(X,e_d).$$\n So the set $\\mathsf {Hom}(X,e_i)$ has at most one element. To show that it has exactly one element, we need to show that $\\mathsf {Hom}(X,e_j)=\\emptyset$ for $j\\not =i$. In fact, assume there is a morphism $X\\to e_j$. Since both objects are connected, this map must be a strict epimorphism. This implies that $\\emptyset =\\mathcal{F}_j (X)\\to \\mathcal{F}_j(e_j)=*$ is surjective and hence a contradiction. Thus our second claim is proven. It follows from the case $d=1$, that the pair $(\\mathcal{C}_i,\\mathcal{F}_i)$ is a connected Galois category.\n\n\\underline{Claim 3}: Our third claim is that if $i\\not =j$, then for any objects $0\\not =X\\in \\mathcal{C}_i$ and $Y\\in \\mathcal{C}_j$ one has $\\mathsf {Hom}(X,Y)=\\emptyset$. In fact, since $e_j$ is terminal in $\\mathcal{C}_j$ there exists a unique morphism $Y\\to e_j$. Thus it suffices to show that $\\mathsf {Hom}(X,e_j)=\\emptyset$, but this was shown in the proof of Claim 2. \n\\newline \n\nDefine the functor\n$$\\xi:\\mathcal{C}_1\\times \\cdots \\times \\mathcal{C}_d\\to \\mathcal{C}$$\nby $\\xi(X_1,\\cdots,X_d)=X_1\\coprod \\cdots \\coprod X_d$.\nWe will show that the functor $\\xi$ is an equivalence of categories. Take an object $X\\in \\mathcal{C}$ and consider the pull-back\n$$\\xymatrix{X_i\\ar[d]\\ar[r]&X\\ar[d]\\\\e_i\\ar[r]&t.}$$\n\\underline{Claim 4}: We want to show that $X_i\\in \\mathcal{C}_i$. In fact, take any $j\\not =i$. Since the functor $\\mathcal{F}_j$ respects pullbacks we obtain a diagram of sets\n$$\\xymatrix{\\mathcal{F}_j(X_i)\\ar[d]\\ar[r]&\\mathcal{F}_j(X)\\ar[d] \\ \\\\ \n\\emptyset \\ar[r]& \\mathcal{F}_j(t).}$$\nIt follows that $\\mathcal{F}_j(X_i)=\\emptyset$ and thus $X_i\\in \\mathcal{C}_i$. Moreover, the natural morphism $X_1\\coprod X_d\\to X$ is an isomorphism, because every $\\mathcal{F}_i$ takes it to an isomorphism. It follows that the functor $\\xi$ is essentially surjective. It remains to show that the functor $\\xi$ is full and faithful. Take the objects $X_i,Y_i\\in \\mathcal{C}_i$, $i=1,\\cdots,d$. We have \n$$\\mathsf {Hom}(\\coprod_{i=1}^d,\\coprod_{i=1}^dY_i)=\\prod_{i=1}^d\\mathsf {Hom}(X_i, Y_1\\coprod \\cdots \\coprod Y_d)$$\nThus it remains to show that for any object $Z\\in \\mathcal{C}_i$ one has \n$$\\mathsf {Hom}(Z, Y_1\\coprod \\cdots \\coprod Y_d)=\\mathsf {Hom}(Z,Y_i)$$\nIf $Z$ is connected this follows from Proposition \\ref{2612} v) and Claim 3. For the general case we decompose $Z=Z_1\\coprod \\cdots \\coprod Z_k$. \n\nThen  we  have\n\\begin{align*}\n\\mathsf {Hom}(Z, Y_1\\coprod \\cdots \\coprod Y_d)&=\\mathsf {Hom}(\\coprod_{j=1}^kZ_j, Y_1\\coprod \\cdots \\coprod Y_d)\\\\ \n&=\\prod_{j=1}^k\\mathsf {Hom}(Z_j, Y_1\\coprod \\cdots \\coprod Y_d)\\\\ &=\\prod_j\\mathsf {Hom}(Z_j,Y_i)\\\\\n&=\\mathsf {Hom}(\\coprod Z_j,Y_i)\\\\ \n&=\\mathsf {Hom}(Z,Y_i)\n\\end{align*}\nHence $\\xi$ is an equivalence of categories.\n\\end{proof}\n\nRecall the following easy but important fact:\n\n\\begin{Pro}\\label{homrep} Let $\\mathfrak{G}$ be a connected groupoid and let $x\\in\\mathfrak{G}$. Then we have an equivalence of categories \n$$\\mathsf {Hom}_{\\mathfrak{Cat}}(\\mathfrak{G},\\mathsf{Sets})\\cong \\mathsf {Aut}(x)\\text{-}\\mathsf{Sets}$$\nwhere $\\mathsf {Aut}(x)\\text{-}\\mathsf{Sets}$ denotes the category of $\\mathsf {Aut}(x)$-Sets.\n\\end{Pro}\n\nIf we changed $\\mathsf{Sets}$ with $\\mathop{\\sf FSets}\\nolimits$, the category of finite sets, the above would still hold. Hence we can now generalise group actions to groupoid actions and for a groupoid $\\mathfrak{G}$, we write $\\mathfrak{G}$-$\\mathop{\\sf FSets}\\nolimits$ or $\\mathsf {Hom}_\\mathfrak{Cat}(\\mathfrak{G},\\mathop{\\sf FSets}\\nolimits)$. It should be emphasised that we use two different notations for the exact same category. The second notation will  mainly be used in calculations. If, however, $\\mathfrak{G}$ were a profinite groupoid, then we would only consider the continuous actions. For simplicity though, by abuse of notations, we will still refer to it as $\\mathsf {Hom}_\\mathfrak{Cat}(\\mathfrak{G},\\mathop{\\sf FSet}\\nolimits)$ or $\\mathfrak{G}$-$\\mathop{\\sf FSets}\\nolimits$. \n\n\\begin{Co}\\label{G.essentialsurj} Let $\\{\\mathcal{F}_i:\\mathcal{C}\\rightarrow \\mathop{\\sf FSets}\\nolimits\\}_{i\\in I}$ be a finitely connected Galois Category. Then it is equivalent to $\\mathfrak{G}$-$\\mathop{\\sf FSets}\\nolimits$, where $\\mathfrak{G}$ is a finitely connected profinite groupoid.\n\\end{Co}\n\n\\begin{proof} As proven in the above lemma (\\ref{G.MainLemma}), \n$$\\{\\mathcal{F}_i:\\mathcal{C}\\rightarrow \\mathop{\\sf FSets}\\nolimits\\}_{i\\in I}\\cong\\{\\mathcal{F}_j:\\prod_{j'\\in J} \\mathcal{C}_{j'}\\rightarrow \\mathop{\\sf FSets}\\nolimits\\}_{j\\in J}$$ \nsuch that $J$ is a finite set and $\\mathcal{F}_j(C_k)=\\emptyset$ for $k\\neq j$. Hence our Galois category is equivalent to $\\mathcal{F}_j:\\prod_{j\\in J}\\mathcal{C}_j\\rightarrow \\mathop{\\sf FSets}\\nolimits$. Again by the above lemma, we know that for each $j\\in J$, the functor $\\mathcal{F}_j:\\mathcal{C}_j\\rightarrow \\mathop{\\sf FSets}\\nolimits$ is a connected Galois category and hence, using Proposition \\ref{homrep}, we know that it is equivalent to $\\mathfrak{G}$-$\\mathop{\\sf FSets}\\nolimits$ where $\\mathfrak{G}$ is a connected, profinite groupoid. The result now follows from the fact that \n$$\\prod_{j\\in J}\\mathsf {Hom}_{\\mathfrak{Cat}}(\\mathfrak{G}_j,\\mathop{\\sf FSets}\\nolimits)\\cong \\mathsf {Hom}_{\\mathfrak{Cat}}(\\coprod_{j\\in J}\\mathfrak{G}_j,\\mathop{\\sf FSets}\\nolimits).$$\n\\end{proof}\n\n\\subsection{The 2-category of Galois Categories}\n\n\\begin{De} Let $\\{\\mathcal{F}_i:\\mathcal{C}\\rightarrow \\mathop{\\sf FSets}\\nolimits\\}_{i\\in I}$ and $\\{\\mathcal{G}_j:\\mathcal{D}\\rightarrow \\mathop{\\sf FSets}\\nolimits\\}_{j\\in J}$ be two Galois categories. A morphism of Galois categories consists of a map $f:J\\rightarrow I$, a functor $\\varphi:\\mathcal{C}\\rightarrow \\mathcal{D}$ preserving finite limits and finite colimits and a collection of isomorphisms $\\lambda_{j,\\varphi},j\\in J$, as given in the following diagram \n$$\\xymatrix{ \\mathcal{D}\\ar[rd]_{\\mathcal{G}_j} & \\dtwocell\\omit{\\lambda_{j,\\varphi}}& \\mathcal{C}\\ \\ar[ll]_{\\varphi}\\ar[dl]^{\\mathcal{F}_f(j)}  \\\\\n& {\\mathop{\\sf FSets}\\nolimits} & . }$$ \nWe will refer to it as $\\{(f,\\varphi,\\lambda_{j,\\varphi}):\\mathcal{F}_{f(j)}\\rightarrow \\mathcal{G}_j\\}$. \n\\end{De} \n\nTo define composition, we need to define the composition of the $\\lambda_{j,\\varphi}$'s. So say we now have \n$$\\xymatrix{ \\mathcal{E}\\ar[rrrrdd]_{\\mathcal{H}_k}& & \\drtwocell\\omit{\\ \\ \\ \\ \\lambda_{k,\\phi}} & & \\mathcal{D}\\ar[dd]^{\\mathcal{G}_{f(k)}}\\ar[llll]^{\\phi} & & \\dltwocell\\omit{\\lambda_{k,\\varphi}} & & \\mathcal{C}\\ \\ar[llll]^{\\varphi}\\ar[ddllll]^{\\ \\ \\ \\ \\mathcal{F}_g(i)} & &  \\\\ \n& & & & & & & & \\\\ \n& & & & {\\mathop{\\sf FSets}\\nolimits} & & & & .}$$ \nDefine $\\lambda_{k,\\phi}\\circ\\lambda_{k,\\varphi}(x)=\\lambda_{k,\\phi}(\\varphi(x))\\circ\\lambda_{k,\\varphi}(x).$ In more detail we have \n$$\\lambda_{k,\\phi}\\circ\\lambda_{k,\\varphi}(x):\\mathcal{F}(x)\\xrightarrow{\\lambda_{k,\\varphi}(x)}\\mathcal{G}(\\varphi(x))\\xrightarrow{\\lambda_{k,\\phi}(\\varphi(x))}\\mathcal{E}(\\phi\\circ\\varphi(x)).$$\n\nIt is easily verified that the above construction is strictly associative.\n\n\\begin{De} A 2-morphism between $\\{(f,\\varphi,\\lambda_{j,\\varphi}):\\mathcal{F}_{f(j)}\\rightarrow \\mathcal{G}_j\\}$ and \\newline $\\{(f,\\phi,\\lambda_{j,\\phi}):\\mathcal{F}_{f(j)}\\rightarrow \\mathcal{G}_j\\}$ is a natural transformation \n$$\\xymatrix{ \\mathcal{C}\\ar@<-1.5ex>[r]_{\\phi} \\ar@<1.5ex>[r]^{\\varphi}\\rtwocell\\omit_{\\ \\ \\ \\ \\ \\zeta} & \\mathcal{D} }$$\nsuch that additionally the following diagram \n$$\\xymatrix{\\mathcal{F}(x)\\ar[r]^{\\lambda_{j,\\varphi}(x)}\\ar[dr]_{\\lambda_{j,\\phi}(x)} & \\mathcal{G}(\\varphi(x))\\ar[d]^{\\zeta(x)} \\\\\n\t           & \\mathcal{G}(\\phi(x)) }$$\ncommutes for all $j$ and all $x$.\n\\end{De}\n\nThis shows that we can talk about the (strict) 2-category of Galois categories. We will denote it by $\\mathsf{GCat}$. \n\n\\begin{Le}\\label{hshjsjk} Let $\\{\\mathcal{F}_i:\\mathcal{C}_1\\times\\cdots\\times \\mathcal{C}_n\\rightarrow \\mathop{\\sf FSets}\\nolimits\\}_{i\\in I}$ be a finitely connected Galois category and $\\mathcal{G}:\\mathcal{D}\\rightarrow \\mathop{\\sf FSets}\\nolimits$ a connected Galois category. \n\nLet $\\mathcal{A}:\\mathcal{C_1}\\times\\cdots\\times \\mathcal{C}_n\\rightarrow \\mathcal{D}$ be a functor between the Galois categories preserving finite limits and finite colimits. Then there exists an $i\\in I$ and $\\mathcal{A}_i:\\hat{\\mathcal{C}_i}\\rightarrow \\hat{\\mathcal{D}}$, such that $\\mathcal{A}=\\mathcal{A_i}\\circ p_i$, where $p_i:\\mathcal{C}_1\\times\\cdots\\times \\mathcal{C}_n\\rightarrow \\mathcal{C}_i$ is the $i$-th projection.\n\\end{Le}\n\n\\begin{proof} Take the terminal object $t\\in \\mathcal{C}$. As in Lemma \\ref{G.MainLemma}, $t=\\coprod_{i\\in I}e_i$, where every $e_i$ is connected. Since $\\mathcal{D}$ is connected, its terminal object is connected. Since $\\mathcal{A}$ respects finite limits and colimits there exists $i\\in I$ such that $\\mathcal{A}(e_i)=\\star$ and $\\mathcal{A}(e_j)=\\emptyset$ for all $j\\neq i$. Recall now the equivalence $\\mathcal{C}\\cong \\mathcal{C}_1\\times\\cdots\\times \\mathcal{C}_n$ as constructed in Lemma \\ref{G.MainLemma}. For every $X\\in \\mathcal{C}$ we have $X\\cong \\coprod_{i}^n X_i$ where $X_i$ is the pullback of the diagram:\n$$\\xymatrix{ & X\\ar[d] \\\\ \n\t\te_i\\ar[r] & t=\\coprod_{i\\in I}e_i.}$$ \nSince $\\mathcal{A}$ respects pullbacks, for all $j\\neq i$ we have that $\\mathcal{A}(X_j)$ is the pullback of \n$$\\xymatrix{ & \\mathcal{A}(X)\\ar[d] \\\\ \n\t\t\\mathcal{A}(e_j)\\ar[r] & \\mathcal{A}(t)}$$ \nand since $Y\\rightarrow \\emptyset$ implies that $Y$ is the empty set, we get that $\\mathcal{A}(X_j)=\\emptyset$. So $\\mathcal{A}(X)=\\coprod\\limits_j \\mathcal{A}(X_j)=\\mathcal{A}(X_i)$. Hence $\\mathcal{A}$ factors through the projection $\\mathcal{C}_1\\times\\cdots\\times \\mathcal{C}_n\\rightarrow \\mathcal{C}_i$.\n\\end{proof}\n\n\\begin{Co}\\label{G.faithfullyflat} Let $\\mathfrak{G}_i$-s be finitely connected profinite groupoids indexed by a finite set $I$ and $\\mathfrak{H}$ be a connected profinite groupoid. Then we have an equivalence of categories: \n$$\\mathsf {Hom}_\\mathsf{GCat}(\\mathsf {Hom}_\\mathfrak{Cat}(\\coprod_{i\\in I}\\mathfrak{G}_i,\\mathop{\\sf FSets}\\nolimits),\\mathsf {Hom}_\\mathfrak{Cat}(\\mathfrak{H},\\mathop{\\sf FSets}\\nolimits))\\cong $$\n$$\\cong\\coprod_{i\\in I}\\mathsf {Hom}_{\\mathsf{GCat}}(\\mathsf {Hom}_\\mathfrak{Cat}(\\mathfrak{G}_i,\\mathop{\\sf FSets}\\nolimits),\\mathsf {Hom}_\\mathfrak{Cat}(\\mathfrak{H},\\mathop{\\sf FSets}\\nolimits)).$$\n\\end{Co}\n\n\\begin{proof} This follows from the Lemmas \\ref{hshjsjk} and \\ref{G.essentialsurj} and the fact that by definition functors in $\\mathsf{GCat}$ respect finite limits and finite colimits.\n\\end{proof}\n\n\\begin{Th}\\label{galois equivalence} The 2-category of finitely connected Galois categories is contravariantly 2-equivalent to the 2-category of profinite finitely connected groupoids.\n\\end{Th}\n\n\\begin{proof} This equivalence is given by associating to a profinite groupoid $\\mathfrak{G}$, the Galois category $\\mathsf {Hom}_{\\mathfrak{Cat}}(\\mathfrak{G},\\mathsf{Sets})$. On functors and natural transformations, the 2-functor is defined in the obvious way by composition. To show that this is a 2-equivalence, we only need to show that it's essentially surjective and full and faithful. Both of course in the 2-mathematical sense. Essential surjectivity is proven in Corollary \\ref{G.essentialsurj}. \n\\newline\n\\underline{Full and Faithful}: Let $\\mathfrak{G}$ and $\\mathfrak{H}$ be profinite and finitely connected groupoids. We have $\\mathfrak{G}\\cong\\coprod_{i\\in I}\\mathfrak{G}_i$ and $\\mathfrak{H}\\cong \\coprod_{j\\in J}\\mathfrak{H}_j$, where the $\\mathfrak{G}_i$-s and $\\mathfrak{H}_j$-s are profinite groupoids with one object. Hence \n\\begin{eqnarray*} \\mathsf {Hom}_\\mathfrak{Cat}(\\mathfrak{G},\\mathfrak{H})&\\cong&\\mathsf {Hom}_{\\mathfrak{Cat}}(\\coprod_{j\\in J}\\mathfrak{G}_i,\\coprod_{j\\in J}\\mathfrak{H}_j) \\\\\n\t\t\t\t\t     &\\cong&\\prod_{i\\in I}\\mathsf {Hom}_\\mathfrak{Cat}(\\mathfrak{G}_i,\\coprod_{j\\in J}\\mathfrak{H}_j) \\\\\n\t\t\t\t\t    &\\cong& \\prod_{i\\in I}\\coprod_{j\\in J}\\mathsf {Hom}_\\mathfrak{Cat}(\\mathfrak{G},\\mathfrak{H}_j).\n\\end{eqnarray*}\nThe last equivalence comes from the fact that the $\\mathfrak{G}_i$-s and $\\mathfrak{H}_j$-s have a single object. Hence any functor $\\mathfrak{G}_i$ can only go to a single $\\mathfrak{H}_j$. From \\cite[Corolarry 6.2. p.111]{SGAI} we get  that $\\mathsf {Hom}_\\mathfrak{Cat}(\\mathfrak{G}_i,\\mathfrak{H}_j)\\cong\\mathsf {Hom}_\\mathsf{GCat}(\\mathsf {Hom}_\\mathfrak{Cat}(\\mathfrak{H}_j,\\mathop{\\sf FSets}\\nolimits),\\mathsf {Hom}_\\mathfrak{Cat}(\\mathfrak{G}_i,\\mathop{\\sf FSets}\\nolimits)).$ On the other hand z\n\\begin{eqnarray*} & & \\mathsf {Hom}_\\mathsf{GCat}(\\mathsf {Hom}_\\mathfrak{Cat}(\\mathfrak{H},\\mathop{\\sf FSets}\\nolimits),\\mathsf {Hom}_\\mathfrak{Cat}(\\mathfrak{G},\\mathop{\\sf FSets}\\nolimits))\\\\ \n\t\t          &\\cong&\\mathsf {Hom}_\\mathsf{GCat}(\\mathsf {Hom}_\\mathfrak{Cat}(\\coprod_{j\\in J}\\mathfrak{H}_j,\\mathop{\\sf FSets}\\nolimits),\\mathsf {Hom}_\\mathfrak{Cat}(\\coprod_{i\\in I}\\mathfrak{G}_i,\\mathop{\\sf FSets}\\nolimits)) \\\\ \n\t                    &\\cong&\\mathsf {Hom}_\\mathsf{GCat}(\\mathsf {Hom}_\\mathfrak{Cat}(\\coprod_{j\\in J}\\mathfrak{H}_j,\\mathop{\\sf FSets}\\nolimits),\\prod_{i\\in I}\\mathsf {Hom}_\\mathfrak{Cat}(\\mathfrak{G},\\mathop{\\sf FSets}\\nolimits)) \\\\\n\t\t         &\\cong&\\prod_{i\\in I}\\mathsf {Hom}_\\mathsf{GCat}(\\mathsf {Hom}_\\mathfrak{Cat}(\\coprod_{j\\in J}\\mathfrak{H}_j,\\mathop{\\sf FSets}\\nolimits),\\mathsf {Hom}_\\mathfrak{Cat}(\\mathfrak{G},\\mathop{\\sf FSets}\\nolimits)).\n\\end{eqnarray*}\nUsing  Corollary \\ref{G.faithfullyflat} we get the desired result.\n\\end{proof}\n\n\\begin{Co}\\label{eq} Let $\\mathfrak{G}\\rightarrow \\mathfrak{H}$ be a functor between finitely connected, profinite groupoids. Assume that the induced functor $\\mathsf {Hom}_{\\mathfrak{Cat}}(\\mathfrak{H},\\mathop{\\sf FSets}\\nolimits)\\rightarrow \\mathsf {Hom}_{\\mathfrak{Cat}}(\\mathfrak{G},\\mathop{\\sf FSets}\\nolimits)$ is an equivalence of categories. Then $\\mathfrak{G}\\rightarrow \\mathfrak{H}$ was already an equivalence of categories.\n\\end{Co}\n\n\\begin{proof} From Theorem \\ref{galois equivalence} we get that \n$$\\mathsf{Eq}(\\mathfrak{G},\\mathfrak{H})\\cong \\mathsf{Eq}(\\mathfrak{G}\\text{-}\\mathop{\\sf FSets}\\nolimits,\\mathfrak{H}\\text{-}\\mathop{\\sf FSets}\\nolimits)$$\nwhere $\\mathsf{Eq}$ denotes the category of equivalences. This immediately implies the result.\n\\end{proof}\n\nUnfortunately, we can not take general 2-limits with values in $\\mathsf{GCat}$. So we can not talk about stacks with values in the 2-category of Galois categories for a general site. However, it follows from Proposition \\ref{galois2limit} below, that if we only consider sites where every covering can be replaced by a finite one, we can avoid that problem. \n\n\\begin{De} We call a site \\emph{finitely coverable}\\index{Site! Finitely coverable} if for every covering $\\{U_i\\rightarrow U\\}_{i\\in I}$ there exists a refinement $\\{V_j\\rightarrow U\\}_{j\\in J}$ such that $J$ is a finite set.\n\\end{De}\n\nA stack with values in the 2-category of Galois categories will be referred to as a \\emph{Galois stack}\\index{Galois Category! Stack}. Similarly for a prestack. If we have two 2-functors $\\mathfrak{F},\\mathfrak{G}:X\\rightarrow \\mathsf{GCat}$ from a finitely coverable site $X$ with values in Galois categories, then we will call a morphism between them that respects the structures a \\emph{Galois transformation}\\index{Galois Category! Transformation}, even though it would technically be a Galois functor. \n\n\\begin{Rem} It should be noted that for Galois 2-functors $\\mathfrak{F}$ and $\\mathfrak{G}$, when we write $\\mathsf {Hom}(\\mathfrak{F},\\mathfrak{G})$, we automatically assume that these are Galois transformations (i.e. respect finite limits and finite colimits), not just morphisms between 2-functors. \n\\end{Rem}\n\nLet $X$ be a site and $\\mathfrak{F}:X\\rightarrow \\sf{Groupoids}$ a covariant 2-functor. Then we denote by $\\mathfrak{F}_S$ the contravariant 2-functor given by $U\\mapsto Hom_{\\mathfrak{Cat}}(\\mathfrak{F}(U),\\mathsf{Sets})$. Now take two covariant 2-functors $\\mathfrak{E},\\mathfrak{F}:X\\rightarrow \\sf{Groupoids}$ and $F:\\mathfrak{E}\\rightarrow \\mathfrak{F}$ a natural transformation. Then it is clear that $F_S:\\mathfrak{F}_S\\rightarrow \\mathfrak{E}_S$ is a Galois transformation. But indeed Theorem \\nolinebreak \\ref{galois equivalence} shows that the reverse is also true. Hence we have the following as well:\n\n\\begin{Co}\\label{Galoisstackequivalence} Let $X$ be a site. Then the 2-category of fibred functors over $X$ with values in Galois categories, and morphisms and 2-morphisms preserving the Galois structure, is contravariantly equivalent to the 2-category of cofibered functors over $X$ with values in profinite groupoids.\n\\end{Co}\n\n\\subsection{Galois categories under stackification}\n\nFrom Proposition \\ref{direct} it follows that if a property is preserved by both 2-limits and filtered 2-colimits, it is preserved by stackification. Hence we have these following facts, which can be checked by routine methods for 2-limits and 2-colimits separately:\n\n\\begin{Pro}\\label{galois2limit} Let $I$ be a finite category and $\\mathfrak{F}:I\\rightarrow \\mathsf{GCat}$ be a 2-functor from $I$ to the 2-category of Galois categories given by $i\\mapsto \\{\\mathcal{F}_{ij}:\\mathcal{C}_i\\rightarrow\\mathop{\\sf FSets}\\nolimits\\}_{j\\in J_i}$. Then the 2-limit of $\\mathfrak{F}$ is again a Galois category and all the natural projections $2$-$\\mathop{\\sf lim}\\nolimits\\limits_i\\mathfrak{F}_i\\rightarrow \\mathfrak{F}_i$ preserve the Galois structure, i.e. are exact.\n\\end{Pro}\n\n\\begin{Le}\\label{Galois} Let $X$ be a site and $\\mathfrak{F}:X\\rightarrow \\mathfrak{Cat}$ be a 2-functor, $\\mathfrak{G}:X\\rightarrow \\mathfrak{Cat}$ a stack in $\\mathfrak{Cat}$ and $\\psi:\\mathfrak{F}\\rightarrow \\mathfrak{G}$ a natural transformation. Assume that $\\mathfrak{F}$ has an associated stack $\\hat{F}$ and let $\\hat{\\psi}:\\hat{\\mathfrak{F}}\\rightarrow \\mathfrak{G}$ be the associated natural transformation. Then the following hold:\n\\begin{itemize} \n\\item[i)] If $\\psi$ respects (finite) limits, so does $\\hat{\\psi}$;\n\\item[ii)] If $\\psi$ respects (finite) colimits, so does $\\hat{\\psi}$.\n\\end{itemize}\n\\end{Le}\n\nUsing  Proposition \\ref{galois2limit} we can talk about stacks with values in Galois categories (called Galois stacks). As such, using the universal definition, we still have the notion of an associated Galois stack, even if we don't prove that it always exists. We see then that the above can be reformulated in the following way: \n\n\\begin{Co}\\label{galois-fun-stackification} Let $\\mathfrak{F}:X\\rightarrow \\mathsf{GCat}$ be a 2-functor, $\\mathfrak{G}:X\\rightarrow \\mathsf{GCat}$ a Galois stack and $\\psi:\\mathfrak{F}\\rightarrow \\mathfrak{G}$ a Galois transformation. If $\\mathfrak{F}$ has an associated Galois stack $\\hat{\\mathfrak{F}}$, then $\\hat{\\psi}:\\hat{\\mathfrak{F}}\\rightarrow \\mathfrak{G}$ is a Galois transformation as well.\n\\end{Co}\n\n\n\\section{The \\'Etale Fundamental Groupoid}\\label{etale}\n\nIn this section we will state and prove the main result of this paper. We will prove that for any \\'etale  map $Y\\mapsto X$ of a noetherian scheme $X$, the association $\\Pi_1:Y\\mapsto\\Pi_1(Y)$ is the 2-terminal costack.\n\nThat is to say, the \\'etale fundamental groupoid is defined by the Seifert-van Kampen theorem, in complete analogue to the topological case, as proven in \\cite{top.groupoid}. The proof however is different.\n\nTo prove that the 2-functor $\\Pi_1$ is the 2-terminal costack over any noetherian scheme $X$ is equivalent to saying that it is the associated costack of the constant, covariant 2-functor taking the trivial groupoid as its value. Unfortunately we don't know much about costackification, so instead we will compose it with the functor $\\mathsf {Hom}(-,\\mathop{\\sf FSets}\\nolimits)$, and hence get a Galois category. \n\nSince by the definition of a costack, after composing with $\\mathsf {Hom}(-,\\mathop{\\sf FSets}\\nolimits)$ we will get a stack, we can use stackification to prove this theorem. This is also our main reason for studying properties preserved under 2-limits and filtered 2-colimits. \n\\newline\n\nLet $X$ be a noetherian scheme. We denote by $\\mathop{\\sf FEC}\\nolimits(X)$ the site of finite \\'etale coverings of $X$. It is a well known result that the category of finite \\'etale coverings is a finitely connected Galois category and that the following holds:\n\n\\begin{Th}\\label{Thm.8.0.2} The 2-functor $\\mathcal{_FE_C}:\\mathop{\\sf FEC}\\nolimits(X)\\rightarrow \\mathsf{GCat}$, where $\\mathop{\\sf FEC}\\nolimits$ denotes the category of finite \\'etale coverings, given by $Y\\mapsto\\mathop{\\sf FEC}\\nolimits(Y)$, forms a stack.\n\\end{Th}\n\n\\begin{De} Let $\\{\\mathcal{F}_j:\\mathcal{C}\\rightarrow\\mathop{\\sf FSets}\\nolimits\\}_{j\\in J}$ be a finitely connected Galois category. Then we can define the \\emph{fundamental groupoid}\\index{Fundamental groupoid! Galois category} $\\Pi_1(\\mathcal{C})$ of our Galois category as follows: \n\\begin{itemize} \n\\item Objects of $\\Pi_1(\\mathcal{C})$ are the functors $\\{\\mathcal{F}_j:\\mathcal{C}\\rightarrow \\mathop{\\sf FSets}\\nolimits\\}_{j\\in J}$;\n\\item For $\\mathcal{F}_i$ and $\\mathcal{F}_j$ we define $\\mathsf {Hom}_{\\Pi_1(\\mathcal{C})}(\\mathcal{F}_i,\\mathcal{F}_j):=\\mathsf{Iso}_{\\mathsf{Fnct}}(\\mathcal{F}_i,\\mathcal{F}_j)$ where $\\mathsf{Iso}_{\\mathsf{Fnct}}$ denotes the set of natural isomorphisms.\n\\end{itemize}\n\\end{De}\n\n\\begin{De} Let $X$ be a site. We define the \\emph{\\'etale fundamental groupoid}\\index{Fundamental groupoid! \\'Etale} $\\Pi_1(X)$ of $X$ to be the fundamental groupoid of the Galois category $\\mathcal{_FE_C}(X)$ of finite \\'etale coverings of $X$.\n\\end{De}\n\nEquivalently, Theorem \\ref{Thm.8.0.2} can be stated as follows:\n\n\\begin{Th}\\label{hom-stack} The 2-functor $\\mathcal{_FE_C}:\\mathop{\\sf FEC}\\nolimits(X)\\rightarrow \\mathsf{GCat}$ given by $Y\\mapsto\\Pi_1(Y)$-$\\mathop{\\sf FSets}\\nolimits$, where $\\Pi_1(Y)$ denotes the \\'etale fundamental groupoid of $Y$, forms a stack.\n\\end{Th}\n\n\\begin{Th}[Seifert-van Kampen Theorem] Let $X$ be a noetherian scheme. Then the assignment $Y\\mapsto \\Pi_1(Y)$ defines a costack on the site of finite \\'etale coverings of $X$.\n\\end{Th}\n\n\\begin{proof} For any covering $Z\\in \\mathop{\\sf Cov}\\nolimits(Y)$ we have the functor $2$-$\\mathop{\\sf colim}\\nolimits(Z,\\Pi_1)\\rightarrow \\Pi_1(Z)$, where $2$-$\\mathop{\\sf colim}\\nolimits(Z,\\Pi_1)$ denotes the 2-colimit of \n$$\\xymatrix{\\Pi_1(Z\\times_{X} Y)& \\Pi_1(Z\\times_{X} Y\\times_{X} Y)\\ar@<-.5ex>[l] \\ar@<.5ex>[l]& \\Pi_1(Z\\times_{X} Y\\times_{X}Y\\times_{X} Y).\\ar@<-.7ex>[l]\\ar[l]\\ar@<.7ex>[l]}$$ \nHence we get the associated functor $\\Pi_1(Z)$-$\\mathop{\\sf FSets}\\nolimits\\rightarrow2$-$\\mathop{\\sf colim}\\nolimits(Z,\\Pi_1)$-$\\mathop{\\sf FSets}\\nolimits$ where we denoted by $\\mathfrak{G}$-$\\mathop{\\sf FSets}\\nolimits$ the functor category $\\mathsf {Hom}_{\\mathfrak{Cat}}(\\mathfrak{G},\\mathop{\\sf FSets}\\nolimits)$. Since $\\mathsf {Hom}(-,\\mathop{\\sf FSets}\\nolimits)$ is left exact, we have \n$$2\\text{-}\\mathop{\\sf colim}\\nolimits(Z,\\Pi_1)\\text{-}\\mathop{\\sf FSets}\\nolimits\\cong 2\\text{-}\\mathop{\\sf lim}\\nolimits((Z,\\Pi_1)\\text{-}\\mathop{\\sf FSets}\\nolimits),$$ \nwhere $2$-$\\mathop{\\sf lim}\\nolimits((Z,\\Pi_1)$-$\\mathop{\\sf FSets}\\nolimits)$ denotes the 2-limit of \n$$\\xymatrix@=1.5em{\\Pi_1(Z\\times_X Y)\\text{-}\\mathop{\\sf FSets}\\nolimits\\ar@<-.5ex>[r] \\ar@<.5ex>[r] & \\Pi_1(Z\\times_X Y\\times_X Y)\\text{-}\\mathop{\\sf FSets}\\nolimits\\ar@<-.7ex>[r]\\ar[r]\\ar@<.7ex>[r] & \\Pi_1(Z\\times_X Y\\times_X Y\\times_X Y)\\text{-}\\mathop{\\sf FSets}\\nolimits.}$$ \nBy Theorem \\ref{hom-stack} the functor $\\Pi_1(Z)\\text{-}\\mathop{\\sf FSets}\\nolimits\\rightarrow 2$-$\\mathop{\\sf lim}\\nolimits((Z,\\Pi_1)$-$\\mathop{\\sf FSets}\\nolimits)$ is an equivalence of categories. Hence from Corollary \\ref{eq} $2$-$\\mathop{\\sf colim}\\nolimits(Z,\\Pi_1)\\rightarrow \\Pi_1(Z)$ was an equivalence of categories as well. This proves the assertion.\n\\end{proof}\n\n\\begin{De} Let $\\mathfrak{C}$ be a 2-category. We say that $\\mathfrak{T}$ is the \\emph{2-terminal object}\\index{Object! 2-Terminal} of $\\mathfrak{C}$, if for any other object $C\\in \\mathfrak{C}$, $Hom_{\\mathfrak{Cat}}(C,\\mathfrak{T})$ is equivalent to the 1-point category.\n\\end{De}\n\n\\begin{Th}\\label{main} Let $X$ be a noetherian scheme. Then the assignment $U\\mapsto\\Pi_1(U)$, $U\\in X$ is the 2-terminal costack over the site of \\'etale coverings of $X$.\n\\end{Th}\n\nTo prove this theorem, recall the following lemma:\n\n\\begin{Le}\\label{LCS} Consider the constant 2-functor $\\mathfrak{s}:U\\mapsto\\mathop{\\sf FSets}\\nolimits$, with morphisms in the contravariant way. Then the associated stack $\\hat{\\mathfrak{s}}$ is given by $\\hat{\\mathfrak{s}}(U)=LCS(U)$, where $LCS(U)$ denotes the category of locally constant sheaves on $U$.\n\\end{Le}\n\nLet $A$ be a covariant 2-functor. Recall that we denoted by $A_S$ the contravariant 2-functor given by $U\\mapsto Hom(A(U),\\mathop{\\sf FSets}\\nolimits)$.\n\n\\begin{proof}[Proof of Thm. \\ref{main}] We have already shown that the assignment $U\\rightarrow \\Pi_1(U)$ forms a costack. Hence to prove this theorem, we essentially have to show that for every costack $C$ we have an essentially unique map $C\\rightarrow \\Pi_1$.\nDenote by $P$ the constant, covariant assignment $U\\mapsto \\mathsf{pt}$. It is clear that we have a map $C\\rightarrow P$ and hence a map $P_S\\rightarrow C_S$ between Galois 2-functors, which is a Galois transformation as it's induced by a functor between groupoids. As shown in Lemma \\ref{LCS}, the stackification of $P_S$ exists and is $U\\mapsto LCS(U)$, where $LCS(U)$ denotes the category of locally constant sheaves on $U$. In the case of noetherian schemes, $LCS(U)$ is equivalent to $\\Pi_1(U)$-$\\mathop{\\sf FSets}\\nolimits$, which is a Galois category. Since $C$ was a costack, $C_S$ is a Galois stack and hence the map $P_S\\rightarrow C_S$ factors through $\\Pi_{1S}$. Hence, using Corollary \\ref{galois-fun-stackification}, we know that the map $\\Pi_{1S}\\rightarrow C_S$ is a Galois transformation. \n\nHence by the uniqueness of the associated stack (see Definition \\ref{uniquestack}) and Corollary \\ref{Galoisstackequivalence} respectively, we have the following equivalences of categories: \n$$\\mathsf {Hom}_{\\mathsf{Gal}}(P_S,C_S)\\cong\\mathsf {Hom}_{\\mathsf{Gal}}(\\Pi_{1S},C_S)\\cong \\mathsf {Hom}_\\mathfrak{Cat}(C,\\Pi_1).$$\nHere $\\mathsf {Hom}_\\mathsf{Gal}$ denotes the category of Galois transformations (i.e. of natural transformations respecting finite limits and colimits). Uniqueness and existence now comes from the fact that we have precisely one exact functor $P_S\\rightarrow C_S$. The last claim is true because $P_S$ is equivalent to $\\mathop{\\sf FSets}\\nolimits$ and hence a functor respecting finite colimits is defined by its value on the singleton $\\star$, which has to map to the terminal object of $C_S$. \n\\end{proof}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe purpose of this work is to study a parapolar space which is locally of type $A_{7,4}$ and which is subject to an extra assumption called the Weak Hexagon Axiom. The present results complete a more general analysis of those parapolar spaces which are locally of type $A_{n-1, 4}({\\mathbb K})$, with $n$ a suitably chosen integer and ${\\mathbb K}$ a field, performed by the author and described below. To simplify the notation we will omit the field ${\\mathbb K}$.\n\n\\medskip\nIn a previous paper \\cite{on1}, we studied parapolar spaces ${\\widetilde \\Gamma}$ which were locally of type $A_{n-1,4}$ (with $n > 6$ and $ n \\not =  8$) and which satisfied:\n\n\\medskip\n{\\it The Weak Hexagon Axiom (WHA): Let $H = (p_1, \\ldots p_6)$ be a\n$6$-circuit, isometrically embedded in ${\\widetilde \\Gamma}$, this means that $p_i \\in p^{\\perp}_{i+1}$, indices taken mod $6$, and all the other pairs are not collinear. Also assume that at least one of the pairs of points at distance two, say $\\lbrace p_1, p_3 \\rbrace$, is polar. Then there exists a point $w \\in p_1^{\\perp} \\cap p_3^{\\perp} \\cap p_5^{\\perp}$.}\n\n\\medskip\nFirst it was proved that ${\\widetilde \\Gamma}$ can be enriched to a locally truncated geometry of rank $5$. Using the fact that ${\\widetilde \\Gamma}$ inherits from the local structure two classes of maximal singular subspaces, a new family of objects ${\\widetilde {\\mathcal D}}$ was constructed. The elements of ${\\widetilde {\\mathcal D}}$ are $2$-convex subspaces of type $D_{6,6}$; see \\cite[Theorem 2]{on1}. Second, using a sheaf theoretic argument it was shown that ${\\widetilde \\Gamma}$ was a homomorphic image of a truncated building; see \\cite[Theorem 3.b]{on1}.\n\n\\medskip\nThe case with $n=7$ which was not considered in \\cite{on1} is the subject of this work. The difficulty of this case relies on the fact that all the maximal singular subspaces have the same singular rank, hence a partition of the maximal singular subspaces according to dimension is no longer attainable and therefore many of the arguments based on the local properties cannot be used. In order to overcome this difficulty, we shall construct a locally truncated diagram geometry {$\\Gamma = ({\\mathcal P}, {\\mathcal L})$} which is a covering of ${\\widetilde \\Gamma}$ and which is locally isomorphic to ${\\widetilde \\Gamma}$, thus of local type $A_{7,4}$, satisfies {\\it (WHA)} and in addition, has the property that the maximal singular subspaces can be separated in two families which we shall denote $\\mathcal{A}$ and $\\mathcal{B}$.\n\n\\medskip\nFrom this point our approach is similar to the one used in \\cite{on1}. We start by constructing two new collections of $2$-convex subspaces ${\\mathcal D}_{\\mathcal A}$ and  ${\\mathcal D}_{\\mathcal B}$ whose elements are of type $D_{6,6}$. We prove that every symplecton $S \\in \\mathcal{S}$ of $\\Gamma$ is contained in exactly one element of ${\\mathcal D}_{\\mathcal A}$ and one element of ${\\mathcal D}_{\\mathcal B}$. Therefore the space $\\Gamma$ can be enriched to a rank six geometry $(\\mathcal{P, L, A, B, D_A, D_B})$. Then we construct a sheaf over this locally truncated diagram geometry and we prove that $\\Gamma$ is a homomorphic image of a truncated building of affine type $E_7$.\n\n\\medskip\nIn Section $2$ we provide the reader with a list of definitions and basic results which will be used in the latter sections. In Section $3$, we use a theorem by Kasikova and Shult to construct the covering $\\Gamma$. The main result of Section $4$ is contained in Theorem $4.1$ which attests the existence of two families of subspaces ${\\mathcal D}_{\\mathcal A}$ and ${\\mathcal D}_{\\mathcal B}$ and that every symplecton of $\\Gamma$ lies in exactly two subspaces, one from each family. In Section $5$, we prove the main result of the paper:\n\n\\begin{t5}Let ${\\widetilde \\Gamma} = ({\\widetilde {\\mathcal P}}, {\\widetilde {\\mathcal L}})$ be a parapolar space which is locally of type $A_{7, 4}$. Let $I = \\lbrace 1, \\ldots 8 \\rbrace, \\; J = \\lbrace 1, 8 \\rbrace$ and $K = I \\setminus J$. Assume that ${\\widetilde \\Gamma}$ satisfies the Weak Hexagon Axiom. Then there is a residually connected $J$-locally truncated diagram geometry $\\Gamma$ belonging to the diagram:\\\\\n\\begin{picture}(1000, 50)(0,0)\n\\put(0,25){$\\mathcal{Y}$}\n\\put(60,25){$\\qed$}\n\\put(75,15){\\scriptsize 1}\n\\put(80,29){\\line(1,0){32}}\n\\put(112,26){$\\circ$}\n\\put(113,15){\\scriptsize 2}\n\\put(113,36){\\scriptsize ${\\mathcal D}_{\\mathcal B}$}\n\\put(118,29){\\line(1,0){30}}\n\\put(148,26){$\\circ$}\n\\put(150,36){\\scriptsize ${\\mathcal B}$}\n\\put(150,15){\\scriptsize 3}\n\\put(153,29){\\line(1,0){30}}\n\\put(183,26){$\\circ$}\n\\put(190,36){\\scriptsize ${\\mathcal L}$}\n\\put(190,15){\\scriptsize 5}\n\\put(186,-3){\\line(0,1){30}}\n\\put(183,-9){$\\circ$}\n\\put(171,-10){\\scriptsize ${\\mathcal P}$}\n\\put(190,-10){\\scriptsize 4}\n\\put(188,29){\\line(1,0){30}}\n\\put(218,26){$\\circ$}\n\\put(220,36){\\scriptsize ${\\mathcal A}$}\n\\put(220,15){\\scriptsize 6}\n\\put(223,29){\\line(1,0){30}}\n\\put(253,26){$\\circ$}\n\\put(253,36){\\scriptsize ${\\mathcal D}_{\\mathcal A}$}\n\\put(255,15){\\scriptsize 7}\n\\put(258,29){\\line(1,0){30}}\n\\put(276,25){$\\qed$}\n\\put(290,15){\\scriptsize 8}\n\\end{picture}\n\n\\vspace{.4cm}\nwhose universal covering is the truncation of a building. Therefore, ${\\widetilde \\Gamma}$ is also the homomorphic image of a truncated building.\n\\end{t5}\n\n\\section{Preliminaries and definitions}\n\\subsection{Geometries} In this section the basic definitions related to geometries are given; for an expository treatment see \\cite[Chapter 3]{hnbk}.\n\n\\medskip\nA {\\it geometry over} $I$ is a system $\\Gamma = (V, *, t)$ consisting of set $V$, a binary, symmetric, reflexive relation on $V$ and a mapping $t: V \\rightarrow I$. The elements of $V$ are called {\\it objects}, $*$ is called the {\\it incidence relation} and $t$ is the {\\it type function} of $\\Gamma$.\n\n\\medskip\nA {\\it flag} $F$ of $\\Gamma$ is a (possibly empty) subset of pairwise incident objects of $\\Gamma$. The set $t(F)$ is the {\\it type} of $F$ and the set $I \\setminus t(F)$ is its {\\it cotype}. The cardinalities of these sets are the {\\it rank} and the {\\it corank} of $F$. The {\\it residue} of $F$ in $\\Gamma$ is the geometry $\\res{\\Gamma}{F} = (V_F, *_{|V_F}, t _{|V_F})$ over $I \\setminus t(F)$, where $V_F$ is the set of all members of $V \\setminus F$ incident with each element of $F$. The corank of a flag $F$ is the rank of $\\res {\\Gamma}{F}$. A geometry $\\Gamma$ is {\\it residually connected} if and only if for every flag $F$ of corank at least one, $\\res{\\Gamma}{F}$ is not empty and if, for each flag of corank at least two, $\\res{\\Gamma}{F}$ is connected.\n\n\\medskip\nLet $\\Gamma _k = (V_k, *_k, t_k)$ with $k \\in K$ some index set and where each $\\Gamma_k$ is a geometry over $I_k$. Assume that $\\lbrace I_k \\rbrace _{k \\in K}$ is a family of pairwise disjoint sets. The {\\it direct sum of geometries} is the geometry denoted by $\\Gamma = \\bigoplus _{k \\in K} \\Gamma _k = (V, *, t)$, where $V = \\bigcup _{k \\in K} V_k$. The incidence is defined as follows: $*_{|V_k}\\;  :=  *_k$ and $x * y$ for any two objects $x \\in V_{k_1}$ and $y \\in V_{k_2}$ with $k_1 \\not = k_2$. Finally $t _{|V_k}\\;  := t_k$.\n\n\\medskip\nLet ${\\mathcal Geom}_I$ denote the category whose objects are\ngeometries with typeset $I$ and whose morphisms are the type preserving graph morphisms. A {\\it fibering morphism of geometries} $\\phi: \\Gamma_1 \\rightarrow \\Gamma_2$ is an object surjective morphism which is object bijective when restricted to the residue of each object in $\\Gamma_1$.\n\n\\subsection{Locally truncated geometries}\nWe gather in this subsection the necessary notions on locally truncated geometries needed in this paper; for a more detailed account of the concepts from this subsection the reader is referred to Brouwer and Cohen \\cite{loc} and Ronan \\cite{ron}.\n\n\\medskip\nLet $I$ be an index set and let $J \\subset I$. The {\\it truncation of type $J$} of $\\Gamma$, denoted by $^J \\Gamma$ is the geometry obtained by restricting the typeset of\n$\\Gamma$ to $J$. The truncation is a functor $^J {\\bf Tr}: {\\mathcal Geom}_I \\rightarrow {\\mathcal Geom}_J$ from the category of geometries over $I$\nto the category of geometries over $J$.\n\n\\medskip\nThe {\\it $J$-truncation of $\\Gamma$}, denoted by $_J \\Gamma$, has as objects the objects of $\\Gamma$ whose types\nare in $I \\setminus J$, incidence and type function are those from\n$\\Gamma$ but restricted to $I \\setminus J$. Differently said, the\n$J$-truncation of $\\Gamma$ is the truncation of type $I \\setminus J$ of $\\Gamma$, that is $_J \\Gamma = \\; ^{I \\setminus J}\\Gamma$.\n\n\\medskip\nA {\\it diagram} $D$ over $I$ is a mapping which assigns to each $2$-subset $\\lbrace i, j \\rbrace$ of $I$, a class $D(i,j)$ of rank $2$ geometries.  A geometry $\\Delta$ over $I$ {\\it belongs to the diagram} $D$ if and only if every\nresidue of type $\\lbrace i, j \\rbrace$ of $\\Delta$ is a geometry from $D(i,j)$.\n\n\\medskip\nA geometry $\\Gamma$ over $I \\setminus J$ is said to be $J$-{\\it locally truncated of type $D$} (or {\\it with diagram $D$}) over\n$I$ if and only if for every nonempty flag $F$ of $\\Gamma$, the residue $\\res{\\Gamma}{F}$ is isomorphic to the truncation of type $I \\setminus ( J \\cup t(F))$ of a geometry belonging to the diagram $D_{I \\setminus t(F)}$, the restriction of $D$\nto the typeset $I \\setminus t(F)$. If $\\Gamma$ is the truncation of type $I \\setminus J$ of a geometry\n$\\Delta$ of type $D$ over $I$ then $\\Gamma$ it is a geometry of $J$-locally\ntruncated type $D$. The converse is in general not true; see Brouwer and Cohen\n\\cite{loc} and Ronan \\cite{ron}.\n\n\\medskip\nLet $M=(m_{ij})$ be a Coxeter matrix with rows and columns indexed by $I$. The {\\it\ndiagram} of $M$, denoted $D(M)$, assigns to each $2$-subset $\\lbrace i, j \\rbrace$ of $I$, the class $D(i,j)$ of generalized $m_{i j }$-gons. If $\\Gamma$ is a residually connected geometry with diagram $D(M)$, that is every residue of type $\\lbrace i, j \\rbrace$ of $\\Gamma$ is a generalized $m_{i j}$-gon, then $\\Gamma$ is called a {\\it geometry of type} $M$.\n\n\\subsection{Chamber systems}\nThe notion of chamber system was introduced by Tits \\cite{tits}. We give below basic definitions and results which will be used in Section $5$; for a detailed account on chamber systems see \\cite{cp}.\n\n\\medskip\nA {\\it chamber system} ${\\mathcal C} = (C, E, \\lambda, I)$ over $I$ is a simple graph $(C, E)$ together with an edge-labeling $\\lambda: E \\rightarrow 2^{I} \\setminus \\lbrace \\emptyset \\rbrace$ by nonempty subsets of $I$ such that, if $a, b, c \\in C$ are three pairwise adjacent vertices, then $\\lambda (a,b) \\cap \\lambda (b,\nc) \\subseteq \\lambda (a,c)$. The elements of $C$ are called\n{\\it chambers}. Two distinct chambers $a$ and $b$ are $i$-{\\it adjacent} iff $(a,b) \\in E$ is an edge and $i \\in \\lambda (a,b)$, for any $i \\in I$. The rank of ${\\mathcal C}$ is the cardinality of the index set $I$. The chamber systems over $I$ together with the appropriate morphisms form a category denoted ${\\mathcal Chamb}_I$.\n\n\\medskip\nFor $J$ a subset of $I$, the {\\it residue of ${\\mathcal C}$ of type $J$} or the $J$-{\\it residue}, is a connected component of the graph $(C, E_J, \\lambda _J, J)$, with\n$\\lambda_J$ the restriction of $\\lambda$ to $\\lambda^{-1}(2^J) \\subseteq E$, where $2^J$ is the codomain of $\\lambda _J$, and $E_J =\\lbrace e \\in E \\mid \\lambda _J (e) \\not=\n\\emptyset \\rbrace $. A $J$-residue $R$ is a chamber system over the typeset $J$. The set $I \\setminus J$ is called the {\\it cotype} of $R$. The {\\it rank of the residue} $R$ is $| J|$ and its {\\it corank} is $| I \\setminus J |$.\n\n\\medskip\nA chamber system over $I$ is {\\it residually connected} if and only if for every subset $J \\subseteq I$ and for every family $\\lbrace R_j: j\\in J \\rbrace$ of residues of cotype $j$, with the property that any two have nonempty intersection, it follows that $\\cap _{j \\in J}R_j$ is a nonempty residue of type $I \\setminus J$.\n\n\\medskip\nLet $\\Gamma$ be a geometry over $I$. Denote by ${\\bf C}(\\Gamma)$ the set of its chamber flags, that is, the flags of type $I$. Two chamber flags $c$ and $d$ are said to be $i$-adjacent whenever they have the same element of type $j$ for all $j \\not = i$, with $i, j \\in I$. Then ${\\bf C}(\\Gamma)$, with the above adjacency relation, is a chamber system of type $I$. Starting with a chamber system ${\\mathcal C}$ over $I$, we define a geometry ${\\bf G}({\\mathcal C}) = (C_i, i\\in I, *, t)$. The objects of this geometry are the elements $C_i, i \\in I$, the collection of all corank one residues of type $I \\setminus \\lbrace i \\rbrace$ of ${\\mathcal C}$. Two objects are incident if they have nonempty intersection. The above construction gives rise to a pair of functors ${\\mathbf G}: {\\mathcal Chamb}_I \\rightarrow\n{\\mathcal Geom}_I$ and\n${\\mathbf C}: {\\mathcal Geom}_I \\rightarrow {\\mathcal Chamb}_I$ such that ${\\mathbf G}( {\\bf C}(\\Gamma)) = \\Gamma$, if $\\Gamma$ is a residually connected geometry and $I$ is finite, and ${\\mathbf C}({\\mathbf G}({\\mathcal C}))= {\\mathcal C}$, if ${\\mathcal C}$ is a residually connected chamber system. For more details see \\cite{asp}.\n\n\\medskip\nFor a connected chamber system ${\\mathcal C}$ over $I$, a {\\it $2$-cover} of ${\\mathcal C}$ is a\nconnected chamber system $\\widetilde {\\mathcal C}$ together with a chamber system\nmorphism $h : \\widetilde {\\mathcal C} \\rightarrow {\\mathcal C}$ which is surjective on chambers and is an isomorphism when restricted to any residues of rank at most $2$ of $\\widetilde {\\mathcal C}$. A $2$-cover $h: \\widetilde {\\mathcal C} \\rightarrow {\\mathcal C}$ is\nsaid to be {\\it universal} if for any other $2$-cover $\\varphi : {\\mathcal C}'\n\\rightarrow {\\mathcal C}$ there is a $2$-cover $\\psi : \\widetilde {\\mathcal C} \\rightarrow\n{\\mathcal C}'$ such that $h \\circ \\psi = \\varphi$. It can be proved that\nchamber systems always have universal coverings.\n\n\\medskip\nA {\\it chamber system ${\\mathcal C}$ over $I$ belongs to the\ndiagram $D(M)$}, with $M$ a Coxeter matrix, if and only if every residue of ${\\mathcal C}$ of type $\\lbrace i, j \\rbrace \\subseteq I$ is the chamber system of a generalized\n$m_{ij}$-gon; one also says that ${\\mathcal C}$ is a chamber system of type $M$. {\\it Buildings} are chamber systems of type $M$ which satisfy extra axioms. For a complete definition of the buildings see \\cite{tits}. The following result, known as Tits' Local Approach Theorem, can be found in \\cite[Corollary 3]{tits}:\n\n\\begin{thm}[Tits]Suppose ${\\mathcal C}$ is a chamber system of type $M$ with $M$ a Coxeter\nmatrix, and suppose that for every rank $3$ residue, the universal $2$-cover is a building. Then the universal $2$-cover of ${\\mathcal C}$ is a building ${\\mathcal B}$ of type $M$.\n\\end{thm}\n\nIn particular, the chamber system ${\\mathcal C}$ is obtained from ${\\mathcal B}$ by factoring out a group\nof automorphisms in which no non-trivial element fixes any rank $2$\nresidue of ${\\mathcal B}$.\n\n\\subsection{Sheaves}\nLet $I$ be an index set, $J \\subset I$ and set $K = I \\setminus J$. Let $\\Gamma$ be a geometry over $K$ which is locally truncated of type $D$ over $I$ and let ${\\mathcal F}$ be a family of nonempty flags of $\\Gamma$. A {\\it sheaf over the geometry} $\\Gamma$ is a class of geometries $\\lbrace \\Sigma(F) \\; \\text{for} \\; F \\in {\\mathcal F} \\rbrace$ together with isomorphisms\n\\begin{center}\n$\\varphi_F: \\res{\\Gamma}{F} \\rightarrow \\; _J \\Sigma (F)$\n\\end{center}\nof geometries over $I \\setminus t(F)$. Given a pair of incident flags $F_1 \\subseteq F_2$ in ${\\mathcal F}$ the connecting homomorphisms of the sheaf are the maps $\\varphi _{F_1, F_2} : \\Sigma (F_2) \\rightarrow \\Sigma (F_1)$ with the property that\n\\begin{center} $\\varphi_{F_1, F_2}(\\Sigma(F_2)) \\simeq \\res{\\Sigma(F_1)}{F_2 \\setminus F_1}$.\n\\end{center}\nFurthermore, they are subject to the following conditions:\n\\begin{center}\n$\\varphi _{F_1, F_2} \\circ \\varphi _{F_2, F_3} = \\varphi _{F_1, F_3}$,\n\\end{center}\nfor $F_1, F_2, F_3 \\in {\\mathcal F}$ with $F_1 \\subseteq F_2 \\subseteq F_3$. To simplify the notation, we will omit the connecting isomorphisms $\\varphi _F$ and write $\\res{\\Gamma}{F} = \\; _J \\Sigma(F)$ instead.\n\n\\medskip\nA sheaf $\\Sigma$ is {\\it residually connected} if and only if for each\nobject $x$ of the geometry $\\Gamma$ the sheaf geometry $\\Sigma (x)$ is\nresidually connected. Due to the functorial relation between the category of geometries and the\ncategory of chamber systems, whenever a sheaf $\\Sigma$ exists, there\nis a chamber system associated to it \\cite[Lemma 1]{loc}.\n\n\\subsection{Spaces}\nThis section is devoted to the basic concepts used in Section $3$; for a good review on spaces and related topics the reader is referred to \\cite[Chapter 12]{hnbk}.\n\n\\medskip\nA {\\it space} $\\Gamma$ is a pair $({\\mathcal P}, {\\mathcal L})$ consisting of a nonempty set ${\\mathcal P}$, whose members are called {\\it points}, and a collection ${\\mathcal L}$ of subsets of ${\\mathcal P}$ of cardinality at least two, whose members are called {\\it lines}. Let $p, q \\in {\\mathcal P}$ be two distinct points. We say that $p$ {\\it is collinear to} $q$ if both $p$ and $q$ lie in some line $L \\in {\\mathcal L}$. The set of all points of ${\\mathcal P}$ collinear with $p$, including $p$ itself, will be denoted $p^{\\perp}$; it is called {\\it the perp of} $p$. A space is a {\\it partial linear space} if two distinct points lie in at most one line. A space is a {\\it gamma space} if it is a partial linear space and for any $p \\in {\\mathcal P}$ and $L \\in {\\mathcal L}$ the set $p^{\\perp} \\cap L$ is empty, a point or $L$.\n\n\\medskip\nThe {\\it collinearity graph} of a space is the graph whose vertex set is ${\\mathcal P}$ and in which two points are adjacent if they are distinct and collinear. Given two points $p, q \\in {\\mathcal P}$, the distance between $p$ and $q$ in the collinearity graph will be denoted $d(p,q)$.\n\n\\medskip\nA subset $X$ of the point set ${\\mathcal P}$ is a {\\it subspace} of $\\Gamma$ if every line $L \\in {\\mathcal L}$ meeting $X$ in at least two points entirely belongs to $X$. A subspace is {\\it singular} if every two points of $X$ are collinear. A subspace is $2$-{\\it convex} if for all pairs of points $p, q \\in X$ with $d(p,q)=2$, each point which is collinear with both $p$ and $q$ is also contained in $X$.\n\n\\medskip\nThe {\\it singular rank} of a space $\\Gamma$ is the length of the longest chain of distinct nonempty singular subspaces $X_0 \\subset X_1 \\subset \\ldots \\subset X_n$. In what follows the term {\\it rank} will denote the projective rank of a singular subspace of $\\Gamma$, that is one less than the vector space dimension. A polar space is said to be of rank $n$ if its singular rank is $n-1$.\n\n\\medskip\nA {\\it parapolar space} is a connected partial linear gamma space possessing a collection of $2$-convex subspaces $\\mathcal S$, called {\\it symplecta}, isomorphic to nondegenerate polar spaces of rank at least $2$, with the properties that each line is\ncontained in a symplecton and each quadrangle is contained in a unique symplecton. A parapolar space in which every pair of points at distance $2$ belongs to a symplecton, is also called a {\\it strong parapolar space}. Let $p, q \\in {\\mathcal P}$ be two points at distance two in a parapolar space. If $|p ^{\\perp} \\cap q^{\\perp}|=1$ then $\\lbrace p, q \\rbrace$ is called a {\\it special pair}; if $|p^{\\perp} \\cap q^{\\perp}|>1$ then $\\lbrace p, q \\rbrace$ is a {\\it polar pair}. If $\\lbrace p, q \\rbrace$ is a polar pair, then the convex closure of $p$ and $q$ is the unique symplecton containing the two points which will be denoted $\\ll p, q \\gg$. In a parapolar space all the singular subspaces are projective spaces. In a parapolar space {$\\Gamma = ({\\mathcal P}, {\\mathcal L})$}, the {\\it residue of a point} $p \\in {\\mathcal P}$ is the space ${\\rm Res}_{\\Gamma}p = ({\\mathcal L}_p, \\pi_p)$ induced on the lines and the planes which contain $p$.\n\n\\medskip\nLet ${\\mathfrak D}_n$ be a Coxeter diagram with $n$ nodes and select a node $i$ in the diagram. A space is said to be of type ${\\mathfrak D}_{n,i}$ if it is the shadow space over $i$ of a building of type ${\\mathfrak D}_n$; see \\cite[Chapter 12, Section 4.7]{hnbk}.\n\n\\subsubsection{The space of type $A_{7,4}$}\nLet $V$ be an $8$-dimensional vector space over some division ring $\\mathbb{K}$. Define the space {$\\Gamma = ({\\mathcal P}, {\\mathcal L})$}  whose points $\\mathcal{P}$ are the $4$-subspaces of $V$ and whose lines $\\mathcal{L}$ are the $(3,5)$-flags of $V$. Then $\\Gamma$ is a strong parapolar space whose symplecta are polar spaces of type $D_{3,1}$. Let $ \\mathcal S$ denote the family of symplecta. The maximal singular subspaces ${\\mathcal M}$ have rank $4$ and can be partitioned into two classes ${\\mathcal A}$ and ${\\mathcal B}$, according to the property {\\it(G1)} below. We list some of the properties of the space of type $A_{7,4}$; these properties can be derived by linear algebra arguments. A characterization of the spaces of type $A_{n,j}$ can be found in \\cite{coh}.\n\n\\begin{itemize}\n\\item[{\\it(G1)}] If $M_1,M_2 \\in {\\mathcal M}$ are two distinct maximal singular\nsubspaces belonging to the same class then $  M_1 \\cap  M_2$ is either empty or a point. If they belong to different classes then $  M_1 \\cap   M_2$ is either empty or a line.\n\\item[{\\it(G2)}] If $p \\in {\\mathcal P}$, $S \\in {\\mathcal S}$, $p \\not\\in S$ then $ p^{\\perp} \\cap S$ is empty, a point or a plane.\n\\item[{\\it(G3)}] If $S \\in {\\mathcal S}$ and $M \\in {\\mathcal M}$ then $S \\cap M$ is empty, a point or a plane.\n\\item[{\\it(G4)}] If $( p,M) \\in {\\mathcal P} \\times {\\mathcal M}$ and $ p \\not\\in  M$ then $p^{\\perp} \\cap  M$ is either empty or a line.\n\\end{itemize}\n\n\\subsubsection{The space of type $D_{6,6}$}\nThis is the space {$\\Gamma = ({\\mathcal P}, {\\mathcal L})$} whose points ${\\mathcal P}$ are one class of maximal singular subspaces of the polar space of type $D_{6,1}$. The lines ${\\mathcal L}$ are the rank $3$ singular subspaces of the same polar space. Then $\\Gamma$ is a strong parapolar space, with a family ${\\mathcal S}$ of symplecta which are polar spaces of type $D_{4,1}$. Let ${\\mathcal S}$ denote the collection of symplecta. There are two classes of maximal singular subspaces ${\\mathcal A}$, whose elements\nhave rank $5$ and ${\\mathcal B}$, a class of rank $3$ singular subspaces.\nThe diagram of a space of type $D_{6,6}$ is given below:\\\\\n\n\\begin{picture}(1000, 50)(0,0)\n\\put(0,25){$D_{6,6}$}\n\\put(74,26){$\\circ$}\n\\put(76,8){${\\mathcal C}$}\n\\put(80,29){\\line(1,0){40}}\n\\put(120,26){$\\circ$}\n\\put(122,8){${\\mathcal S}$}\n\\put(126,29){\\line(1,0){40}}\n\\put(166,26){$\\circ$}\n\\put(168,8){${\\mathcal B}$}\n\\put(172,29){\\line(1,0){40}}\n\\put(212,26){$\\circ$}\n\\put(220,8){${\\mathcal L}$}\n\\put(215,-10){\\line(0,1){37}}\n\\put(212,-16){$\\circ$}\n\\put(220,-24){${\\mathcal P}$}\n\\put(218,29){\\line(1,0){40}}\n\\put(258,26){$\\circ$}\n\\put(260,8){${\\mathcal A}$}\n\\end{picture}\n\n\\vspace*{1cm}\nThe space of type $D_{6,6}$ was initially characterized by Cohen and Cooperstein \\cite[Theorem 4]{cc}:\n\n\\begin{thm}[Cohen and Cooperstein]Let {$\\Gamma = ({\\mathcal P}, {\\mathcal L})$} be a strong parapolar space of singular rank $5$, which is not a polar space and whose symplecta have rank $4$. Assume that, given a point-symplecton pair $(p, S) \\in {\\mathcal P} \\times {\\mathcal S}$ with $p \\not \\in S$, the set $p^{\\perp} \\cap S$ is either a point or a maximal singular subspace of $S$. Then $\\Gamma$ is a space of type $D_{6,6}$.\\\\\n\\end{thm}\n\n\\section{The maximal singular subspaces}\n\nSuppose that ${\\widetilde \\Gamma} = ({\\widetilde{\\mathcal P}}, {\\widetilde{\\mathcal L}})$ is parapolar space which is locally of type $A_{2n-1,n}$ with $n$ a positive integer.\n\n\\smallskip\nThe Grassmann space $A_{2n-1, n}(\\mathbb{K})$ and its quotient $A_{2n-1, n}(\\mathbb{K})\/\\langle \\sigma \\rangle$, where $\\mathbb{K}$ is an infinite division ring and where $\\sigma$ is an involutory automorphism of $A_{2n-1,n}(\\mathbb{K})$ induced by a polarity of the underlying projective space of Witt index at most $n-4$, have the same diagram. In this case ${\\widetilde \\Gamma}$ can have point residuals of both types $A_{2n-1, n}(\\mathbb{K})$ and $A_{2n-1, n}(\\mathbb{K})\/ \\langle \\sigma \\rangle$. In the second geometry the maximal singular subspaces are fused in one single family; see \\cite[Section 6]{coh} for example. Since the case of interest here is $n=4$, the Grassmann space of type $A_{7,4}$ is too ``small\" and it does not have nontrivial quotients which are parapolar spaces, thus a local partition of the maximal singular subspaces into two classes can be attained (see Section 2.5.1 for details).\n\n\\medskip\nHowever, all the maximal singular subspaces of ${\\widetilde \\Gamma}$ have the same rank and there is no global partition in two classes according to the rank. In order to overcome this difficulty, we shall use a result of Kasikova and Shult \\cite[Section 3.4]{ks} and we shall construct a covering of ${\\widetilde \\Gamma}$ in which the partition of the maximal singular subspaces in two global classes can be realized. We adapt Theorem $12$ from \\cite{ks} to the case of interest here and for completeness we also provide a proof following \\cite{ks}.\n\n\\begin{thm}[Kasikova and Shult] Let ${\\widetilde \\Gamma} = ({\\widetilde {\\mathcal P}}, {\\widetilde {\\mathcal L}})$ be a parapolar space which is locally of type $A_{7,4}$. Let $I = \\lbrace 1, \\ldots 8 \\rbrace$ and $J = \\lbrace 1,2,7,8 \\rbrace$.  Then there is a $J$-locally truncated connected geometry $\\Gamma$ with diagram:\\\\\n\\begin{picture}(1000, 50)(0,0)\n\\put(0,25){$ \\mathcal{Y'}$}\n\\put(45,25){$\\qed$}\n\\put(60,15){\\scriptsize $1$}\n\\put(65,29){\\line(1,0){40}}\n\\put(95,25){$\\qed$}\n\\put(110,15){\\scriptsize ${2}$}\n\\put(115,29){\\line(1,0){40}}\n\\put(155,26){$\\circ$}\n\\put(157,15){\\scriptsize ${3}$}\n\\put(160,29){\\line(1,0){40}}\n\\put(200,26){$\\circ$}\n\\put(210,15){\\scriptsize $5$}\n\\put(203,-3){\\line(0,1){30}}\n\\put(200,-9){$\\circ$}\n\\put(210,-10){\\scriptsize $4$}\n\\put(206,29){\\line(1,0){40}}\n\\put(246,26){$\\circ$}\n\\put(248,15){\\scriptsize $6$}\n\\put(252,29){\\line(1,0){40}}\n\\put(280,25){$\\qed$}\n\\put(295,15){\\scriptsize $7$}\n\\put(300,29){\\line(1,0){40}}\n\\put(330,25){$\\qed$}\n\\put(345,15){\\scriptsize $8$}\n\\end{picture}\n\n\\vspace{.5cm}\nand a fibering morphism of geometries $\\phi_{\\Gamma}: \\Gamma \\rightarrow {\\widetilde \\Gamma}$ which induces a one- or two-fold $\\mathcal {T}$-covering $\\varphi _{\\Gamma}: \\Delta _{\\Gamma} \\rightarrow \\Delta_{\\widetilde \\Gamma}$ of the point-collinearity graph of ${\\widetilde \\Gamma}$.\n\\end{thm}\n\n\\begin{proof} Let ${\\widetilde \\Gamma} = ({\\widetilde{\\mathcal P}}, {\\widetilde{\\mathcal L}})$ be a parapolar space which is locally of type $A_{7,4}$. Then for each point ${\\widetilde p} \\in {\\widetilde{\\mathcal P}}$ the residue ${\\rm Res}_{\\widetilde \\Gamma}{\\widetilde p}$ contains two classes of maximal singular subspaces ${\\widetilde{\\mathcal A}}_{\\widetilde p}$ and ${\\widetilde {\\mathcal B}}_{\\widetilde p}$ whose elements are projective spaces of rank $5$. Let ${\\widetilde {\\mathcal M}}_{\\widetilde p} = {\\widetilde {\\mathcal A}}_{\\widetilde p} \\cup {\\widetilde {\\mathcal B}}_{\\widetilde p}$ and set ${\\widetilde {\\mathcal M}} = \\lbrace {\\widetilde {\\mathcal M}} _{\\widetilde p} \\mid {\\widetilde p} \\in {\\widetilde {\\mathcal P}} \\rbrace$. Each plane ${\\widetilde \\pi}$ on ${\\widetilde p}$ lies in exactly one element from each class. Let ${\\mathbb P}$ denote the set of pairs $({\\widetilde p}, {\\widetilde{\\mathcal X}}_{\\widetilde p})$ where ${\\widetilde p} \\in {\\widetilde{\\mathcal P}}$ and ${\\widetilde{\\mathcal X}}_{\\widetilde p}$ is one of the classes ${\\widetilde{\\mathcal A}}_{\\widetilde p}$ or ${\\widetilde{\\mathcal B}}_{\\widetilde p}$. Let ${\\mathbb L}$ denote the collection of pairs $({\\widetilde L}, {\\widetilde{\\mathcal X}}_{\\widetilde L})$ with ${\\widetilde L} \\in {\\widetilde {\\mathcal L}}$ and ${\\widetilde{\\mathcal X}}_{\\widetilde L}$ a class of maximal singular subspaces containing ${\\widetilde L}$. We say that $({\\widetilde p}, {\\widetilde{\\mathcal X}}_{\\widetilde p})$ is incident with $({\\widetilde L}, {\\widetilde{\\mathcal X}}_{\\widetilde L})$ if and only if ${\\widetilde p}$ is incident with ${\\widetilde L}$ in ${\\widetilde \\Gamma}$ and ${\\widetilde{\\mathcal X}}_{\\widetilde L} \\subseteq {\\widetilde{\\mathcal X}}_{\\widetilde p}$. Thus $\\Upsilon = ({\\mathbb P},{\\mathbb L})$ is a point-line geometry.\n\n\\medskip\nThe geometry morphism $\\phi: \\Upsilon \\rightarrow {\\widetilde \\Gamma}$ induced by the projection onto the first coordinate is vertex surjective. Let now $({\\widetilde p}, {\\widetilde{\\mathcal X}}_{\\widetilde p}) \\in {\\mathbb P}$ and $({\\widetilde L}_i, {\\widetilde{\\mathcal X}}_{{\\widetilde L}_i}) \\in {\\mathbb L}$ with $i=1,2$, be two lines incident with $({\\widetilde p}, {\\widetilde{\\mathcal X}}_{\\widetilde p})$. Observe that $\\phi(({\\widetilde L}_1, {\\widetilde{\\mathcal X}}_{{\\widetilde L}_1})) = \\phi(({\\widetilde L}_2, {\\widetilde{\\mathcal X}}_{{\\widetilde L}_2}))$ implies that ${\\widetilde L}_1 = {\\widetilde L}_2$. Since $({\\widetilde p}, {\\widetilde{\\mathcal X}}_{\\widetilde p})$ is incident with both lines we must have  ${\\widetilde {\\mathcal X}}_{{\\widetilde L}_1} = {\\widetilde{\\mathcal X}}_{{\\widetilde L}_2}$ and therefore $({\\widetilde L}_1, {\\widetilde{\\mathcal X}}_{{\\widetilde L}_1}) = ({\\widetilde L}_2, {\\widetilde{\\mathcal L}}_{{\\widetilde L}_2})$. A similar argument can be applied if we start with a line in ${\\mathbb L}$ and two points in ${\\mathbb P}$ incident with it. Thus $\\phi$ is one-to-one when restricted to neighbor graphs, the set of all the lines (or points) which are incident with a fixed point (or line respectively).\n\n\\medskip\nThe map $\\phi$ also induces a vertex surjective morphism on the point-collinearity graphs: $\\varphi: \\Delta_{\\Upsilon} \\rightarrow \\Delta_{\\widetilde \\Gamma}$ which restricts to an isomorphism $({\\widetilde p}, {\\widetilde{\\mathcal X}}_{\\widetilde p})^{\\perp_{\\Upsilon}} \\rightarrow {\\widetilde p}^{\\perp_{\\widetilde{\\Gamma}}}$. Furthermore, all $3$-circuits in $\\Delta_{\\widetilde \\Gamma}$ lift to $3$-circuits in $\\Delta_{\\Upsilon}$. In the language of \\cite[Section 2]{ks}, the restriction $\\varphi _{\\Gamma}$ of $\\varphi$ to a connected component of $\\Delta_{\\Upsilon}$ is a $\\mathcal{T}$-covering.\n\n\\medskip\nIn $\\Upsilon$ there are two global classes of maximal singular subspaces. They can be defined as follows: for any ${\\widetilde M} \\in {\\widetilde {\\mathcal M}}$ let $\\mathcal{A}_{\\widetilde M} = \\lbrace ({\\widetilde p}, {\\widetilde{\\mathcal X}}_{\\widetilde p}({\\widetilde M}))| {\\widetilde p} \\in {\\widetilde M} \\rbrace$ where ${\\widetilde {\\mathcal X}}_{\\widetilde p}({\\widetilde M})$ denotes the class ${\\widetilde{\\mathcal A}}_{\\widetilde p}$ or ${\\widetilde{\\mathcal B}}_{\\widetilde p}$ which contains ${\\widetilde M}$ and $\\mathcal{B}_{\\widetilde M} = \\lbrace ({\\widetilde p}, {\\widetilde{\\mathcal X}'}_{\\widetilde p}({\\widetilde M}))| {\\widetilde p} \\in {\\widetilde M} \\rbrace$ where ${\\widetilde {\\mathcal X}'}_{\\widetilde p}({\\widetilde M})$ denotes the class which does not contain ${\\widetilde M}$. Then the two families of maximal singular subspaces are ${\\mathcal A} = \\lbrace {\\mathcal A}_{\\widetilde M} | {\\widetilde M} \\in {\\widetilde {\\mathcal M}} \\rbrace$ and $\\mathcal{B} = \\lbrace {\\widetilde {\\mathcal B}}_{\\widetilde M} | {\\widetilde M} \\in {\\widetilde {\\mathcal M}} \\rbrace$.\n\n\\medskip\nIt is quite clear that $\\Upsilon$ can be either connected (when the local classes of maximal singular subspaces of ${\\widetilde \\Gamma}$ are fused) or disconnected (when a global separation is possible) and has two connected components. Let $\\Gamma$ be a connected component of $\\Upsilon$. This is mapped by $\\phi$ as either a one-to-one mapping or a two-to-one mapping depending on whether $\\Gamma$ is a proper subgeometry or not. So if we denote by $\\phi_{\\Gamma}$ the restriction of $\\phi$ to $\\Gamma$ we obtain the morphism from the conclusion of the theorem.\n\\end{proof}\n\n\\begin{rem} The point-line geometry $\\Gamma = ({\\mathcal P}, {\\mathcal L})$ is connected and locally isomorphic to ${\\widetilde \\Gamma}$, which means that $\\Gamma$ is locally of type $A_{7,4}$, hence it belongs to diagram $\\mathcal{Y}'$. The maximal singular subspaces of $\\Gamma$ partition in two classes which we shall denote ${\\mathcal A}$ and ${\\mathcal B}$. Then $\\Gamma$ is a parapolar space with a class of symplecta ${\\mathcal S}$ of type $D_{4,1}$.\n\\end{rem}\n\n\\begin{rem} Assume ${\\widetilde \\Gamma}$ satisfies the Weak Hexagon Axiom {\\it (WHA)}. It is a consequence of the definition of the fibering morphism of geometries that ({\\it WHA}) is also valid in $\\Gamma$. Therefore, in our analysis we can replace the space ${\\widetilde \\Gamma}$ with its covering $\\Gamma$, gaining in this way the advantage of being able to partition the maximal singular subspaces into two classes.\n\\end{rem}\n\n\\section{The family ${\\mathcal D}$ of subspaces}\n\nThroughout this section {$\\Gamma = ({\\mathcal P}, {\\mathcal L})$} will denote a parapolar space which is locally of type $A_{7,4}$ and it is constructed as in the proof of Theorem $3.1$ of the previous section. The local assumption means that if $p \\in {\\mathcal P}$ then ${\\rm Res}_{\\Gamma} p$ is a space of type $A_{7,4}$. Recall the basic notions on spaces and related structures from Section 2.5. The result of the previous section allows to obtain a global partition of the maximal singular subspaces into two classes; see {\\it (P1)} below for notations and details. Next, $\\Gamma$ contains a set of symplecta ${\\mathcal S}$ whose elements are (non-degenerate) polar spaces of type $D_{4,1}$.\n\n\\medskip\nLet $S$ be a symplecton in $\\Gamma$ and let ${\\mathcal X} \\in \\lbrace {\\mathcal A}, {\\mathcal B} \\rbrace$. We define the following two sets associated to $S$:\n\n$$N_{\\mathcal X}(S)= \\lbrace p \\in {\\mathcal P} \\setminus S \\mid\np^{\\perp}\n\\cap S \\in M_{\\mathcal X}(S) \\rbrace$$\nand\n\\begin{equation*}\n\\begin{split}\nG_{\\mathcal X}(S) & = \\lbrace p \\in {\\mathcal P} \\setminus S \\mid\np^{\\perp} \\cap S =\n\\lbrace q \\rbrace, \\text{ a point};\\; \\text{for any}\\; r \\in q^{\\perp}\\cap S\\\\\n&\\quad \\text{the pair} \\; \\lbrace p,r \\rbrace \\; \\text{is polar}; \\; \\text{for some} \\; A \\in {\\mathcal A}_q \\cap {\\mathcal A}(S), \\; p^{\\perp} \\cap A \\; \\text{is a plane} \\rbrace.\n\\end{split}\n\\end{equation*}\n\n\\smallskip\nThen set\n$$D_{\\mathcal X}(S)= S \\cup N_{\\mathcal X}(S) \\cup G_{\\mathcal X}(S)$$\nand let ${\\mathcal D}_{\\mathcal X}= \\lbrace D_{\\mathcal X}(S)|\\; S \\in {\\mathcal S} \\rbrace$. Finally denote ${\\mathcal D} = {\\mathcal D}_{\\mathcal A} \\cup {\\mathcal D}_{\\mathcal B}$.\n\n\\medskip\nThe main result of this section is:\n\n\\begin{thm} Let {$\\Gamma = ({\\mathcal P}, {\\mathcal L})$} be a parapolar space  which is locally of type $A_{7,4}$ and which satisfies:\n\n\\smallskip\nThe Weak Hexagon Axiom {\\it (WHA)}: Let $H = (p_1, \\ldots p_6)$ be a\n$6$-circuit, isometrically embedded in $\\Gamma$, this means that $p_i \\in p^{\\perp}_{i+1}$, indices taken mod $6$, and all the other pairs are not collinear. Also assume that at least one of the pairs of points at distance two, say $\\lbrace p_1, p_3 \\rbrace$, is polar. Then there exists a point $w \\in p_1^{\\perp} \\cap p_3^{\\perp} \\cap p_5^{\\perp}$.\n\n\\smallskip\nThen there exist two collections of $2$-convex subspaces ${\\mathcal D}_{\\mathcal A}$ and ${\\mathcal D}_{\\mathcal B}$, whose elements are spaces of type $D_{6,6}$. Every symplecton $S \\in {\\mathcal S}$ lies in exactly one element from each class.\n\\end{thm}\n\nIn order to prove the theorem we first show that each $D_{\\mathcal X}$\nis a $2$-convex subspace which is a strong parapolar space in its own. Then we use the Cohen-Cooperstein characterization theorem, see Section 2.5.2, to conclude that $D_{\\mathcal X}$ is of type $D_{6,6}$.\n\n\\medskip\nThe following properties of $\\Gamma$ are consequences of the theory of parapolar spaces and of the local assumption; see also the properties {\\it (G1-G4)} given in Section 2.5.1.\n\n\\medskip\n{\\it (P1)} There are two classes of maximal singular subspaces\n${\\mathcal A}$ and ${\\mathcal B}$. The set ${\\mathcal M} = {\\mathcal\nA} \\cup\n{\\mathcal B}$ is a collection of singular subspaces of rank $5$. Two distinct maximal singular\nsubspaces which belong to the same class can meet at a line, a\npoint or the empty set. Two maximal singular subspaces from different\nclasses can meet at a plane, a point or they are disjoint.\\\\\n{\\it (P2)} If $(S, M) \\in {\\mathcal S} \\times {\\mathcal M}$ then the set $S\n\\cap M$ can be empty, a point, a line  or a\nmaximal singular subspace of $S$.\\\\\n{\\it (P3)} For $S \\in {\\mathcal S}$ let ${\\mathcal  M}(S)$ be the family of maximal singular subspaces with largest intersection with $S$; observe that ${\\mathcal M}(S) = {\\mathcal A}(S) \\cup {\\mathcal B}(S)$. For $M_i \\in {\\mathcal M}(S),\\; i=1,2$, denote ${\\overline M}_i = M_i \\cap S$. Then ${\\overline M}_1 \\cap {\\overline M}_2$ can be a\npoint or a plane if ${\\overline M}_1, {\\overline M}_2$ belong to different classes; ${\\overline M}_1 \\cap {\\overline M}_2$ can be empty or a line when ${\\overline M}_1$ and ${\\overline M}_2$ are disjoint and belong to the same class.\\\\\n{\\it (P4)} If $M \\in {\\mathcal M}$ and $p \\in\n{\\mathcal P} \\setminus  M$ then\n$p^{\\perp} \\cap M$ can be empty, a point or a plane.\\\\\n{\\it (P5)} If $S \\in {\\mathcal S}$ and $p \\in {\\mathcal P} \\setminus S$\nthen $p^{\\perp} \\cap S$ can be empty, a point, a line or a maximal singular subspace of\n$S$.\n\n\\begin{ntn} For $S \\in {\\mathcal S}$ and ${\\mathcal X} \\in \\lbrace {\\mathcal A}, {\\mathcal B}\n\\rbrace$ let $M_{\\mathcal X}(S)= \\lbrace {\\overline X} \\mid {\\overline X} = X \\cap  S\n\\; \\; \\text{for some} \\; X \\in {\\mathcal X}(S) \\rbrace$ be the two classes of maximal singular subspaces of $S$.\n\\end{ntn}\n\n\\begin{ntn} In what follows if $p \\in {\\mathcal\nP}$ and $S \\in {\\mathcal S}$ are such that  $p ^{\\perp} \\cap S \\in M_{\\mathcal X}(S)$,\nthen we denote ${\\overline X}_p:= p^{\\perp} \\cap S$ and the maximal singular\nsubspace containing it $X_p$. If $p, q \\in {\\mathcal P}$ form a polar pair in $\\Gamma$, the unique symplecton containing them will be denoted $\\ll p, q \\gg$.\n\\end{ntn}\n\n\\begin{lem}\nLet $S$ be a symplecton in $\\Gamma$.\\\\\na). If $p \\in N_{\\mathcal X}(S)$ and $q \\in p^{\\perp}$ is such that\n$q^{\\perp} \\cap S \\setminus {\\overline X}_p \\not= \\emptyset$, then $q \\in S \\cup N_{\\mathcal X}(S)$.\\\\\nb). The set $S \\cup N_{\\mathcal X}(S)$ is a subspace of $\\Gamma$.\n\\end{lem}\n\n\\begin{proof} a). As the statement is obviously true when $q \\in S$, one may assume that $q \\not\\in S$. Let $r \\in q^{\\perp} \\cap S \\setminus {\\overline X}_p$. Set $R= \\ll r,p \\gg$ and note that $q \\in R$. Hence $L:=q^{\\perp} \\cap r^{\\perp} \\cap \\overline X_p$ is a line and $q^{\\perp} \\cap S$ contains the plane $\\langle r, L \\rangle $. According to $\\it{(P5)}$, $q^{\\perp} \\cap S = {\\overline M}$ is a maximal singular subspace of $S$. Now $\\langle qr, {\\overline M}\\cap {\\overline X}_p\\rangle$ has rank $3$ and lies in $R$, thus it is a maximal singular subspace of $R$. It follows that ${\\overline M}\\cap {\\overline X}_p=L$ and according to $\\it{(P3)}$, ${\\overline M}$ and ${\\overline X}_p$ belong to the same class. Hence $q \\in N_{\\mathcal X}(S)$.\n\n\\medskip\nb). One has to show that if $p$ and $q$ are two collinear points in $S \\cup N_{\\mathcal X}(S)$, then the line $pq$ lies entirely in $S \\cup N_{\\mathcal X}(S)$. If at least one of the points $p$ or $q$ is in $S$ the conclusion follows at\nonce. Assume $p, q \\not\\in S$. Set ${\\overline X}_p = p^{\\perp} \\cap S$ and ${\\overline X}_q = q^{\\perp} \\cap S$, two\nmaximal singular subspaces of $S$ belonging to the same class. According to ${\\it (P3)}$\nthere are three cases to consider:\n\n\\smallskip\n$(i)$. ${\\overline X}_p = {\\overline X}_q$ in which case $X_p = X_q$ and\nthus $pq \\subset N_{\\mathcal X}(S)$.\n\n\\smallskip\n$(ii)$. ${\\overline X}_p \\cap {\\overline X}_q = L$ is a line in $S$.\nLet $r \\in pq \\setminus \\lbrace p, q \\rbrace$. Clearly $L \\subset r^{\\perp}$. Let $w \\in {\\overline X}_q \\setminus r^{\\perp}$. Hence ${\\overline N}= \\langle w, w^{\\perp} \\cap {\\overline X}_p \\rangle$ is a maximal singular subspace of $S$, not in the\nsame class with ${\\overline X}_p$ and ${\\overline X}_q$ since ${\\overline N} \\cap {\\overline X}_p$ and ${\\overline N} \\cap {\\overline\nX}_q$ are both planes in $S$. Set $R =  \\ll w,p \\gg$ and observe that $\\overline N \\subseteq R \\cap S$. In $R$, the set $r^{\\perp} \\cap {\\overline N}$ is a plane and by ${\\it(P5)}$, $r ^{\\perp}\\cap S = {\\overline X}_r$ is a maximal singular subspace of $S$. It remains to show that ${\\overline X}_r \\in M_{\\mathcal X}(S)$. It suffices to prove: ${\\overline X}_p \\cap {\\overline X}_r =L$. Now ${\\overline X}_r$ meets ${\\overline N}$ at the plane $r^{\\perp} \\cap {\\overline N}$. So, the family of ${\\overline X}_r$ is not the same as that of ${\\overline N}$, which in its turn is not the same as that of ${\\overline X}_p$ and ${\\overline X}_q$. Hence ${\\overline X}_r$ and ${\\overline X}_p$, being different, meet at a line, which must be $L$.\n\n\\smallskip\n$(iii)$. The singular subspaces ${\\overline X}_p$ and ${\\overline X}_q$ are disjoint. Let $t \\in {\\overline X}_p$ and set $T =  \\ll t, q \\gg$. Then the plane $t^{\\perp} \\cap {\\overline X}_q$ lies in $T$ and therefore $p^{\\perp} \\cap t^{\\perp} \\cap {\\overline X}_q$ is a line. But this contradicts the assumption that ${\\overline X}_p \\cap {\\overline X}_q = \\emptyset$. Thus\neither ${\\overline X}_p \\cap {\\overline X}_q \\not= \\emptyset$ or $p$ is not collinear with $q$.\n\\end{proof}\n\n\\begin{lem} Let $p \\in G_{\\mathcal X}(S)$ with $\\lbrace q \\rbrace = p^{\\perp} \\cap S$.\\\\\na). For any $X \\in {\\mathcal X}_q \\cap {\\mathcal X}(S)$ the set $p^{\\perp} \\cap X$ is a plane.\\\\\nb). Let $r \\in N_{\\mathcal X}(S)$ be such that $q \\in r^{\\perp} \\cap S$. Then either $r \\in p^{\\perp}$ or $\\lbrace p, r \\rbrace$ is a polar pair.\n\\end{lem}\n\n\\begin{proof}\na). The result is a consequence of the local structure. For a proof see \\cite[Lemma 5.1]{on1}.\\\\\nb). Follows directly from part a) of the lemma.\n\\end{proof}\n\nIn what follows, for ease of exposition, we shall let ${\\mathcal X}={\\mathcal A}$ and show that $D_{\\mathcal A}(S)$ is a space of type $D_{6,6}$. Afterwards, we can replace ${\\mathcal A}$ with ${\\mathcal B}$ and repeat the arguments to obtain that $D_{\\mathcal B}(S)$ is also a space of type $D_{6,6}$.\n\n\\begin{lem} Let $R, S \\in {\\mathcal S}$. If $R \\cap S= \\overline B$ is a maximal\nsingular subspace in $M_{\\mathcal B}(S) \\cap M_{\\mathcal B}(R)$ then $D_{\\mathcal A}(S) = D_{\\mathcal A}(R)$.\n\\end{lem}\n\n\\begin{proof} It suffices to prove $D_{\\mathcal A}(S) \\subseteq D_{\\mathcal A}(R)$.\n\n\\medskip\n$(i)$. Let $p \\in S \\setminus {\\overline  B}$. Then $p ^{\\perp} \\cap \\overline B$ is a plane and, by {\\it (P5)}, it follows that $p^{\\perp} \\cap R = \\overline A_p$ is a maximal singular subspace of $R$ in $M_{\\mathcal A}(R)$ (since it has a plane in common with $\\overline B$). Thus $p \\in N_{\\mathcal A}(R) \\subset D_{\\mathcal A}(R)$ and $S \\subset D_{\\mathcal A}(R)$.\n\n\\medskip\n$(ii)$. Let now $p \\in N_{\\mathcal A}(S)$. If $p \\in R$ we are done so we may assume that $p \\not \\in R$. Then $\\overline A_p = p^{\\perp} \\cap S$ can have either a plane or a point in common with $\\overline B$. If $\\overline A_p \\cap \\overline B$ is a plane, then $p \\in N_{\\mathcal A}(R)$ by a similar argument to that used in $(i)$. Next let $\\overline A_p \\cap \\overline B = \\lbrace q \\rbrace$ be a single point.\nIf there exists a point $r \\in p^{\\perp} \\cap R \\setminus \\lbrace q \\rbrace$ then $r \\not \\in A_p$. Assume to the contrary that $r \\in A_p$. Then $r ^{\\perp} \\cap S \\supseteq \\langle {\\overline A}_p, r^{\\perp} \\cap {\\overline B} \\rangle \\supset {\\overline A}_p$ and since ${\\overline A}_p$ is a maximal singular subspace of $S$ we obtain a contradiction with {\\it (P5)}. Thus $ r \\not \\in A_p$. Let $t \\in {\\overline A}_p \\setminus r^{\\perp}$, which, according to Step $(i)$, is in $N_{\\mathcal A}(R)$. An application of Lemma $4.4(a)$, to the pair $\\lbrace p, t \\rbrace$ and symplecton $R$, gives that $p \\in N_{\\mathcal A}(S)$. Next assume that $p^{\\perp} \\cap R = \\lbrace q \\rbrace$, a single point. {\\it Claim}: $p \\in G_{\\mathcal A}(R)$. Let $s \\in q^{\\perp} \\cap R \\setminus S$ so $s \\in R \\subset N_{\\mathcal A}(S)$ and the pair $\\lbrace p, s \\rbrace$ is polar. Let ${\\overline A}_s = s^{\\perp} \\cap S$ with $A_s$ be the maximal singular subspace containing ${\\overline A}_s$. Since $p^{\\perp} \\cap A_s$ contains the line ${\\overline A}_p \\cap {\\overline A}_s$, it follows, by {\\it (P4)}, that $p^{\\perp} \\cap A_s$ is a plane. This concludes the proof of the claim.\n\n\\medskip\n$(iii)$. Let $p \\in G_{\\mathcal A}(S)$ be such that $p^{\\perp} \\cap {\\overline B} = \\lbrace q \\rbrace$ is a point. Assume first that $p^{\\perp} \\cap R = \\lbrace q \\rbrace$. Let $r \\in q^{\\perp} \\cap R$, so $r \\in N_{\\mathcal A}(S)$. By Lemma $4.5(b)$ the pair $\\lbrace p, r  \\rbrace$ is polar. Further, if $A \\in {\\mathcal A}(S) \\cap {\\mathcal A}(R) \\cap {\\mathcal A}_q$ then $p^{\\perp} \\cap A$ is a plane. Thus $p \\in G_{\\mathcal A}(R)$. Assume now $p^{\\perp} \\cap R \\supseteq qs$, a line. Consider ${\\rm Res}_{\\Gamma}(q)$. For a subspace  $F$ of $\\Gamma$ which contains $q$, ${\\widetilde F}$ will denote the corresponding subspace in $\\res{\\Gamma}{q}$. In $\\res{\\Gamma}{q}$, ${\\widetilde R}$ and ${\\widetilde S}$ are two ``symplecta\" which meet at a plane of $\\widetilde {\\mathcal B}$-type. Also $\\tilde p$ is a ``point\" at distance two from every single ``point\" in $\\widetilde S$ and $\\tilde s \\in \\tilde p ^{\\tilde \\perp} \\cap \\widetilde R$. According to the result of \\cite[Lemma 2.3]{on1}, it follows that $\\tilde p^{\\tilde \\perp} \\cap \\widetilde R$ is a ``plane\". Then, back in $\\Gamma$, $p^{\\perp} \\cap R$ is a maximal singular subspace of $R$. Now $(p^{\\perp} \\cap R) \\cap \\overline B = \\lbrace q \\rbrace$ which implies $p^{\\perp} \\cap R \\in M_{\\mathcal A}(R)$. Thus $p \\in N_{\\mathcal A}(R)$.\n\n\\medskip\n$(iv)$. Let now $p \\in G_{\\mathcal A}(S)$ be such that $p^{\\perp} \\cap {\\overline B} = \\emptyset$ and let $\\lbrace q \\rbrace = p^{\\perp} \\cap S$. Recall from Step $(i)$ that $q\\in N_{\\mathcal A}(R)$. Let $A_q \\in {\\mathcal A}(R) \\cap {\\mathcal A}_q$ be the maximal singular subspace which contains $\\langle q, q^{\\perp} \\cap R \\rangle$. Note that $A_q \\in {\\mathcal A}(S)$ also, since $A_q \\cap S = \\langle q, q^{\\perp} \\cap {\\overline B}\\rangle $. By Lemma $4.5(a), \\; p^{\\perp} \\cap A_q$ is a plane which intersects $R$ nontrivially. Note that $p^{\\perp} \\cap A_q \\cap R$ cannot contain a line, since this would imply $p^{\\perp} \\cap {\\overline B} \\not= \\emptyset$, contrary to our choice of $p$. Let $\\lbrace r \\rbrace = p^{\\perp} \\cap A_q \\cap R$. {\\it Claim}: $p \\in G_{\\mathcal A}(R)$. We first prove that $p^{\\perp} \\cap R = \\lbrace r \\rbrace$ is a single point. Assume by contradiction that there is a point $s \\in p^{\\perp} \\cap R \\setminus \\lbrace r \\rbrace$. According to the above argument, $s \\not \\in A_q$. Then, by Lemma $4.4(a)$, applied to the pair $\\lbrace p, q \\rbrace$ and symplecton $R$ we get $p \\in N_{\\mathcal A}(R)$. But this implies $p^{\\perp} \\cap R \\cap {\\overline B} \\not = \\emptyset$, which contradicts the fact that $p^{\\perp} \\cap {\\overline B} = \\emptyset$. Therefore $p^{\\perp} \\cap R = \\lbrace r \\rbrace$. Since $p^{\\perp} \\cap A_q$ is a plane, it remains to prove that given any point $t \\in r^{\\perp} \\cap R$, the pair $\\lbrace p, t \\rbrace$ is polar. This is clearly true for any $t \\in r^{\\perp} \\cap {\\overline B} = q^{\\perp} \\cap {\\overline B}$, by the definition of $G_{\\mathcal A}(S)$. So let us assume that $t \\not \\in {\\overline B}$. Now $t \\in N_{\\mathcal A}(S)$ and since $q^{\\perp} \\cap t^{\\perp} \\cap S$ contains a plane, the pair $\\lbrace q, t \\rbrace$ is polar. Set $T = \\ll q, t \\gg$ and note that $T \\cap S = \\langle q, q^{\\perp} \\cap t^{\\perp} \\cap S\\rangle  \\in M_{\\mathcal B}(S)$. Now $p \\in G_{\\mathcal A}(S)$, $p^{\\perp} \\cap S \\cap T \\not = \\emptyset$ and $p^{\\perp} \\cap T \\supset qr$ and according to Step $(iii)$, $p \\in N_{\\mathcal A}(T)$. Therefore $p^{\\perp} \\cap T \\cap t^{\\perp}$ contains a plane and this proves that $\\lbrace p,t \\rbrace$ is a polar pair.\n\\end{proof}\n\n\\begin{lem} a). Let $S, R \\in {\\mathcal S}$ be such that $S \\cap R = L$ is a line. Assume that $R \\cap {\\mathcal A}(S) \\not= \\emptyset$. Then $D_{\\mathcal A}(S) = D_{\\mathcal A}(R)$.\\\\\nb). Let $p \\in G_{\\mathcal A}(S)$ with $\\lbrace q \\rbrace = p^{\\perp} \\cap S$. For $r \\in q^{\\perp} \\cap S$ set $R = \\ll p, r \\gg$. Then $D_{\\mathcal A}(S) = D_{\\mathcal A}(R)$.\n\\end{lem}\n\n\\begin{proof}a). Let $S, R \\in {\\mathcal S}$ be such that $S \\cap R = L$ is a line. Let\n$p \\in R \\cap G_{\\mathcal A}(S)$. Denote $p ^{\\perp} \\cap S = \\lbrace q\n\\rbrace$ and observe that $ q \\in L$. Let $A \\in {\\mathcal A}(S)$ be such that $L \\subset A$. Then, according to {\\it (P2)}, $R \\cap A$ can be a line or a maximal singular subspace of $R$. Since $p \\in G_{\\mathcal A}(S),\\; p^{\\perp} \\cap A $ is a plane. Let $r \\in p^{\\perp} \\cap A \\setminus \\lbrace q \\rbrace$ and let $t \\in L \\setminus \\lbrace q \\rbrace$. Thus $R = \\ll p, t \\gg$ and $r \\in R$. Then $A \\cap R \\supseteq \\langle r, L \\rangle $ and by {\\it (P2)} it follows that $A \\in {\\mathcal A}(R)$. We proved that if $A \\in {\\mathcal A}(S)$ is such that $L \\subset A$ then $A \\in {\\mathcal A}(R)$ as well.\\\\\nLet now $A_1, A_2 \\in {\\mathcal A}(S) \\cap {\\mathcal A}(R)$ be two distinct maximal singular subspaces such that $L \\subseteq A_1 \\cap A_2$. The line $L$ is contained in the symplecton $S$ and so is the intersection of two maximal singular subspaces of $S$ of the same class. Let $p_1 \\in A_1 \\cap S \\setminus L$ and $p_2 \\in A_2 \\cap R \\setminus L$. Set $T = \\ll p_1, p_2 \\gg$. Note that $T \\cap S = \\langle p_1, p_1^{\\perp} \\cap A_2 \\cap S \\rangle $ and $T \\cap R = \\langle p_2, p_2^{\\perp} \\cap A_1 \\cap R \\rangle $ both singular subspaces of ${\\mathcal B}$-type. Then, by Lemma $4.6$: $D_{\\mathcal A}(S) = D_{\\mathcal A}(T) = D_{\\mathcal A}(R)$.\n\n\\medskip\nb). Let $S$ and $R$ be two symplecta as in the hypothesis. We claim that $R \\cap S = pq$, a line. If $R \\cap S$ contains a plane then $p^{\\perp} \\cap S$ contains a line, contradicting the properties of $p \\in G_{\\mathcal A}(S)$. Since $p \\in G_{\\mathcal A}(S) \\cap R$, part a) applies and the result follows.\n\\end{proof}\n\n\\begin{prop} Let $S \\in {\\mathcal S}$ then $D_{\\mathcal A}(S)$ is a subspace of $\\Gamma$.\n\\end{prop}\n\n\\begin{proof}Let $p, q \\in D_{\\mathcal A}(S)$ be two collinear points. We have to prove that $pq \\subset D_{\\mathcal A}(S)$.\n\n\\medskip\n$(i)$. If $p,q \\in S \\cup N_{\\mathcal A}(S)$ the result follows from Lemma $4.4(b)$.\n\n\\medskip\n$(ii)$. If $p \\in G_{\\mathcal A}(S)$ and $ q \\in S$ then $\\lbrace q \\rbrace = p^{\\perp} \\cap S$. In this case $pq \\subset S \\cup G_{\\mathcal A}(S)$ follows from the gamma space property of $\\Gamma$; see Section 2.5.\n\n\\medskip\n$(iii)$. Assume now that $p \\in N_{\\mathcal A}(S)$ and $q \\in G_{\\mathcal A}(S)$.\nLet ${\\overline A}_p = p^{\\perp} \\cap S$ and $\\lbrace r \\rbrace = q^{\\perp} \\cap S$. It follows, from Lemma $4.4(a)$, that $r \\in {\\overline A}_p$. Let $t \\in r^{\\perp} \\cap S \\setminus {\\overline A}_p$. Set $R = \\ll p, t \\gg$. Then, by Lemma $4.6$, $D_{\\mathcal A}(S) = D_{\\mathcal A}(R)$. Now $q \\in G_{\\mathcal A}(S) \\subset D_{\\mathcal A}(R)$ and $q^{\\perp} \\cap R$ contains the line $pr$, hence $q \\in N_{\\mathcal A}(R)$. Thus $p \\in R,\\; q \\in N_{\\mathcal A}(R)$ and, according to Lemma $4.4(b), \\; pq \\subset D_{\\mathcal A}(R) = D_{\\mathcal A}(S)$.\n\n\\medskip\n$(iv)$. Let $p, q \\in G_{\\mathcal A}(S)$. First assume $p^{\\perp} \\cap\nq^{\\perp} \\cap S = \\lbrace r \\rbrace$. Let $t \\in r^{\\perp} \\cap S$ and set $R = \\ll p, t\\gg$. Then, according to Lemma $4.7(b)$, $D_{\\mathcal A}(S) = D_{\\mathcal A}(R)$. Thus $p \\in R$ and $q \\in R \\cup N_{\\mathcal A}(R)$ implies $pq \\subset D_{\\mathcal A}(R)= D_{\\mathcal A}(S)$. Let now $p^{\\perp} \\cap S = \\lbrace r \\rbrace$ and $q^{\\perp} \\cap R = \\lbrace t \\rbrace$ be such that $r \\not = t$. If $r \\in t^{\\perp}$ then set $R = \\ll p, t \\gg$ and, by Lemma $4.7(b)$, $D_{\\mathcal A}(R) = D_{\\mathcal A}(S)$. Therefore $pq \\subset R \\subset D_{\\mathcal A}(S)$. If $r \\not \\in t^{\\perp}$ then take $w \\in r^{\\perp} \\cap t^{\\perp}$ and set $T = \\ll p, w \\gg$. Again, by Lemma $4.7(b)$ $D_{\\mathcal A}(T) = D_{\\mathcal A}(S)$. So $p \\in T,\\; q \\in D_{\\mathcal A}(T)$ and by previous results of this Proposition, $pq \\subset D_{\\mathcal A}(T) = D_{\\mathcal A}(S)$.\n\\end{proof}\n\nThe assumption {\\it (WHA)} from the statement of Theorem $4.1$ is used in the proof of the following proposition.\n\n\\begin{prop} Let $S \\in {\\mathcal S}$, then $D_{\\mathcal A}(S)$ is a $2$-convex\nsubspace of $\\Gamma$.\n\\end{prop}\n\n\\begin{proof}Let $p, q \\in D_{\\mathcal A}(S)$ be two points at distance $2$. We have to prove that $p^{\\perp} \\cap q^{\\perp} \\subset D_{\\mathcal A}(S)$.\n\n\\medskip\n$(i)$. If $p,q \\in S$ then obviously $p^{\\perp} \\cap q^{\\perp} \\subset\nS$ since $S$ is a $2$-convex subspace of $\\Gamma$.\n\n\\medskip\n$(ii)$. Let now $p \\in N_{\\mathcal A}(S)$ and $q \\in S$. The pair $\\lbrace p,q \\rbrace$ is polar. Set $R = \\ll p,q \\gg$. Since $R \\cap S \\in \\mss{\\mathcal B}{S}$ it follows, by Lemma $4.6$ that $D_{\\mathcal A}(S) = D_{\\mathcal A}(R)$. Hence $p^{\\perp} \\cap q^{\\perp} \\subset R \\subset D_{\\mathcal A}(S)$.\n\n\\medskip\n$(iii)$. Consider now the case when $p,q \\in N_{\\mathcal A}(S)$. Then ${\\overline A}_p = p^{\\perp} \\cap S$ and ${\\overline A}_q =\nq^{\\perp} \\cap S$ are two maximal singular subspaces of $S$ from the same\nclass. There are two cases to consider:\n\n\\smallskip\n$(iii.a)$. Assume ${\\overline A}_p \\cap {\\overline A}_q =L$ is a line. Let $r \\in {\\overline A}_q \\setminus L$ and set $R = \\ll r,p \\gg$. Then $R \\cap S \\in M_{\\mathcal B}(S)$ and, by Lemma $4.6,\\; D_{\\mathcal A}(S) = D_{\\mathcal A}(R)$. So $q \\in D_{\\mathcal A}(R)$ and because $q^{\\perp} \\cap R \\supseteq \\langle r, L \\rangle $ it follows $q \\in N_{\\mathcal A}(R)$. Note that $q \\not \\in R$ since $A_q \\cap R = \\langle r, p^{\\perp} \\cap A_q \\rangle$ has rank $3$ and thus is already maximal in $R$. Now $p \\in R,\\; q \\in N_{\\mathcal A}(R)$ and according to Step $(ii)$ of this Proposition, $p^{\\perp} \\cap q^{\\perp} \\subset D_{\\mathcal A}(R) =D_{\\mathcal A}(S)$.\n\n\\smallskip\n$(iii.b)$. Assume now that ${\\overline A}_p$ and ${\\overline A}_q$ are disjoint maximal singular subspaces of $S$. Let $r \\in p^{\\perp} \\cap q^{\\perp}$. Take $p_1 \\in {\\overline A}_p \\setminus r^{\\perp}$ and $q_1 \\in {\\overline A}_q \\setminus ( r^{\\perp} \\cup p_1^{\\perp})$. Also let $r_1 \\in p_1^{\\perp} \\cap q_1^{\\perp}$ be such that $r_1 \\not \\in p^{\\perp} \\cup q^{\\perp}$. Now $H = (p_1, r_1, q_1, q,r,p)$ is a minimal $6$-circuit in $\\Gamma$ which contains at least one polar pair $\\lbrace p_1, q_1 \\rbrace$. Then {\\it (WHA)} applies and there exists a point $w \\in r^{\\perp} \\cap p_1^{\\perp} \\cap q_1^{\\perp} \\subset S$. Thus, according to Lemma $4.4(a)$, $r \\in N_{\\mathcal A}(S) \\subset D_{\\mathcal A}(S)$.\n\n\\medskip\n$(iv)$. Let $p \\in G_{\\mathcal A}(S)$ and $q \\in S$. Let $\\lbrace t \\rbrace = p^{\\perp} \\cap S$. If $q \\in t^{\\perp}$ then $R = \\ll p,q \\gg$ is contained in $D_{\\mathcal A}(S)$; see Lemma $4.7(b)$. So let us assume $q \\not \\in t^{\\perp}$. Take $A \\in {\\mathcal A}(S) \\cap {\\mathcal A}_t$. Then according to Lemma $4.5$, $p^{\\perp} \\cap A$ is a plane. Let $p_1 \\in p^{\\perp} \\cap A \\setminus \\lbrace t \\rbrace$ and let $r \\in t^{\\perp} \\cap S \\setminus A$. Set $R = \\ll p_1, r \\gg$. Since $R \\cap S = \\langle r, r^{\\perp} \\cap A \\rangle  \\in \\mss{\\mathcal B}{R} \\cap \\mss{\\mathcal B}{S}$ it follows, by Lemma $4.6$ that $D_{\\mathcal A}(S) = D_{\\mathcal A}(R)$. Now $p^{\\perp} \\cap R \\supset p_1 t$ so $p \\in N_{\\mathcal A}(R)$. Also $q \\in N_{\\mathcal A}(R)$. According to Step $(iii)$ of this Proposition $p^{\\perp} \\cap q^{\\perp} \\subset D_{\\mathcal A}(R) = D_{\\mathcal A}(S)$.\n\n\\medskip\n$(v)$. Let $p \\in N_{\\mathcal A}(S), \\; q \\in G_{\\mathcal A}(S)$. Let\n$\\lbrace r \\rbrace = q^{\\perp} \\cap S$ and ${\\overline A}_p = p^{\\perp} \\cap S$. Assume first that $r \\not\\in {\\overline A}_p$. Set $R = \\ll p, r \\gg$. Then $R \\cap S\n\\in M_{\\mathcal B}(S)$ and according to Lemma $4.6, \\; D_{\\mathcal A}(R) = D_{\\mathcal A}(S)$. Now $p \\in R,\\; q \\in D_{\\mathcal A}(R)$ and using the above results of this Proposition, it follows $p^{\\perp} \\cap q^{\\perp} \\subset D_{\\mathcal A}(R) = D_{\\mathcal A}(S)$. Next consider the case $r \\in {\\overline A}_p$. Let $t \\in {\\overline A}_p$ be a point and set $T = \\ll q,t \\gg$. By Lemma $4.7(b),\\; D_{\\mathcal A}(S) = D_{\\mathcal A}(T)$. So we may apply the results of the Steps $(i)-(iii)$ of this Proposition to the pair of points $ p \\in D_{\\mathcal A}(T),\\; q \\in T$ and conclude that $p^{\\perp} \\cap q^{\\perp} \\subset D_{\\mathcal A}(T) = D_{\\mathcal A}(S)$.\n\n\\medskip\n$(vi)$. Let now $p,q \\in G_{\\mathcal A}(S)$ be such that $p^{\\perp} \\cap S =\n\\lbrace p_1 \\rbrace$ and $q^{\\perp} \\cap S = \\lbrace q_1 \\rbrace$. If $p_1 \\not \\in q_1^{\\perp}$ then let $r \\in p_1^{\\perp} \\cap q_1^{\\perp}$; if $p_1 \\in q_1^{\\perp} \\setminus \\lbrace q_1 \\rbrace$ then we let $r = q_1$ and if $p_1 = q_1$ then take $r \\in p_1^{\\perp} \\cap S$. Set $R =  \\ll p, r \\gg$. By Lemma $4.7(b),\\; D_{\\mathcal A}(S) = D_{\\mathcal A}(R)$. Now $q \\in G_{\\mathcal A}(S) \\subset D_{\\mathcal A}(R)$ and $p \\in R$ hence using previous results of this Proposition, $p^{\\perp} \\cap q^{\\perp} \\subset D_{\\mathcal A}(R) = D_{\\mathcal A}(S)$.\n\\end{proof}\n\n\\begin{lem} Let $p \\in G_{\\mathcal A}(S)$ with $\\lbrace p_1 \\rbrace = p^{\\perp} \\cap S$ and let $w \\in S$. Then $d(p,w) = 1+d(p_1,w)$.\n\\end{lem}\n\n\\begin{proof} We start by proving the following {\\it claim}: if $p,q \\in G_{\\mathcal A}(S)$ are two collinear points with $\\lbrace p_1 \\rbrace = p^{\\perp} \\cap S$ and $\\lbrace q_1 \\rbrace = q^{\\perp} \\cap S$ then $q_1 \\in p_1^{\\perp}$. Assume by contradiction that $q_1 \\not \\in p_1^{\\perp}$. Let $A \\in {\\mathcal A}_{p_1} \\cap{\\mathcal A}(S)$. Then, according to Lemma $4.5,\\; p^{\\perp} \\cap A$ is a plane. Let $x \\in p^{\\perp} \\cap A \\setminus \\lbrace p_1 \\rbrace$. Also let $r \\in p_1^{\\perp} \\cap q_1^{\\perp} \\setminus A$. Set $R = \\ll r, x \\gg$ and observe that $R \\cap S = \\langle r, r^{\\perp} \\cap A \\cap S\\rangle  \\in M_{\\mathcal B}(S)$ and, by Lemma $4.6, \\; D_{\\mathcal A}(R) = D_{\\mathcal A}(S)$. Note that $p^{\\perp} \\cap R \\supset p_1x$ and consequently $p \\in N_{\\mathcal A}(R)$. Set $\\overline A_p = p^{\\perp} \\cap R$. Also $q \\in G_{\\mathcal A}(S) \\subset D_{\\mathcal A}(R)$ implies that there is a point $y \\in q^{\\perp} \\cap R$. If $y \\not \\in {\\overline A}_p$, since $p$ and $q$ are collinear, it follows, by Lemma $4.4(a)$, that $q \\in N_{\\mathcal A}(R)$. But then $q^{\\perp} \\cap R$ and $R \\cap S$ are maximal singular subspaces in $R$ from different families and therefore they have a point in common. Since $y \\not \\in R \\cap S$ it follows that $q^{\\perp} \\cap R$ contains more than a point, which is a contradiction with the fact that $q \\in G_{\\mathcal A}(S)$. Therefore $y \\in {\\overline A}_p$. Next $q_1 \\in S$ and, by Lemma $4.6$, $q_1 \\in N_{\\mathcal A}(R)$. Let $\\overline A _{q_1} = q_1^{\\perp} \\cap R$ and observe that $y \\in {\\overline A}_{q_1}$ because otherwise another application of Lemma $4.4(a)$ to the pair $\\lbrace q, q_1 \\rbrace$ and symplecton $R$ would give $q \\in N_{\\mathcal A}(R)$, a contradiction. Therefore $y \\in {\\overline A}_p \\cap {\\overline A}_{q_1}$ which, using {\\it (P3)} implies ${\\overline A}_p \\cap {\\overline A}_{q_1} = L$ is a line. But $r \\not \\in x^{\\perp}$ so $x \\not \\in L$. Since ${\\overline A}_{q_1}$ has rank $3$ and since $S \\cap R$ meets ${\\overline A}_{q_1}$ in a plane it follows that $L \\cap (R \\cap S) \\not = \\emptyset$. But this implies that $p^{\\perp} \\cap S$ contains more than a point. We reach a contradiction with the fact that $p \\in G_{\\mathcal A}(S)$. Therefore the assumption made was false and $p_1 \\in q_1^{\\perp}$; the claim is proved.\n\n\\medskip\nIn order to prove the Lemma it suffices to show that, if $p \\in G_{\\mathcal A}(S)$ and $w \\in S \\setminus p_1^{\\perp}$ then $d(p, w) = 3$. Assume by contradiction that $d(p, w) = 2$. Then there exists a point $t \\in p^{\\perp} \\cap w^{\\perp}$. Note that $t$ cannot be in $S$. According to Proposition $4.10,\\; t \\in D_{\\mathcal A}(S)$. Moreover $t \\not \\in N_{\\mathcal A}(S)$, because since $w \\not \\in p_1^{\\perp}$, Lemma $4.4(a)$ would imply $p \\in N_{\\mathcal A}(S)$, a contradiction. So we must have $t \\in G_{\\mathcal A}(S)$. But now, according to the previous paragraph $w \\in p_1^{\\perp} \\cap S$, which contradicts the hypothesis on $w$. Therefore the assumption made was false and in this case $d(p,w)=3$.\n\\end{proof}\n\n\\begin{prop} For any $S \\in {\\mathcal S},\\ D_{\\mathcal A}(S)$ is a strong parapolar\nsubspace of $\\Gamma$.\n\\end{prop}\n\n\\begin{proof} We have to prove that if $p,q \\in D_{\\mathcal A}(S)$ are two points at distance two, the pair $\\lbrace p,q \\rbrace$ is polar, that is $p^{\\perp} \\cap q^{\\perp}$ contains at least two points.\n\n\\medskip\n$(i)$. Let $p \\in D_{\\mathcal A}(S)$ and $q \\in S$. If both $p$ and $q$ are in $S$ then $S = \\ll p, q \\gg$ and we are done. Assume next $p \\in N_{\\mathcal A}(S)$. Then $q^{\\perp} \\cap p^{\\perp} \\cap S$ contains a plane and therefore $\\lbrace p,q \\rbrace$ is a polar pair. Let now $p \\in G_{\\mathcal A}(S)$ with $\\lbrace p_1 \\rbrace = p^{\\perp} \\cap S$. In this case, since $d(p,q) = 2$, by Lemma $4.10$ it follows that $q \\in p_1^{\\perp}$. Then the fact that $\\lbrace p, q \\rbrace$ is polar pair follows from the definition of $G_{\\mathcal A}(S)$.\n\n\\medskip\n$(ii)$. Assume now $p,q \\in N_{\\mathcal A}(S)$. Let ${\\overline A}_p = p^{\\perp} \\cap S$ and ${\\overline A}_q = q^{\\perp} \\cap S$. If ${\\overline A}_p \\cap {\\overline A}_q \\not = \\emptyset$ the result is immediate. So we may assume that ${\\overline A}_p \\cap {\\overline A}_q = \\emptyset$. Let $r \\in {\\overline A}_p$. Then set $R = \\ll q, r \\gg$ which, by Lemma $4.6$, is such that $D_{\\mathcal A}(R) = D_{\\mathcal A}(S)$. Thus $p \\in G_{\\mathcal A}(S) \\subset D_{\\mathcal A}(R),\\; q \\in R$ and by Step $(i)$ of this Proposition it follows that $\\lbrace p, q \\rbrace$ is a polar pair.\n\n\\medskip\n$(iii)$. Let $p \\in N_{\\mathcal A}(S)$ and $q \\in G_{\\mathcal A}(S)$. Let ${\\overline A}_p = p^{\\perp} \\cap S$ and $\\lbrace t \\rbrace = q^{\\perp} \\cap S$. If $t \\in {\\overline A}_p$ then, according to Lemma $4.5(b),\\; \\lbrace p, q \\rbrace$ is a polar pair. Assume next that $t \\not \\in {\\overline A}_p$. Take a point $r \\in t^{\\perp} \\cap {\\overline A}_p$ and set $R = \\ll q, r \\gg$. By Lemma $4.7(b), \\; D_{\\mathcal A}(R) = D_{\\mathcal A}(S)$. So $q \\in R,\\; p \\in N_{\\mathcal A}(S) \\subset D_{\\mathcal A}(R)$ and $\\lbrace p,q \\rbrace$ is a polar pair by Step $(i)$ of this Proposition.\n\n\\medskip\n$(iv)$. Let now $p,q \\in G_{\\mathcal A}(S)$ with $\\lbrace r \\rbrace = p^{\\perp} \\cap S$ and $\\lbrace t \\rbrace = q^{\\perp} \\cap S$. If $r \\in t^{\\perp} \\setminus \\lbrace t \\rbrace$ set $R = \\ll r, q \\gg$. If $r = t$ then take $w \\in r^{\\perp}$ and if $r \\not \\in t^{\\perp}$ take $w \\in r^{\\perp} \\cap t^{\\perp}$ and set $R =\\ll w, q \\gg$. Then, by Lemma $4.7(b),\\; D_{\\mathcal A}(R) = D_{\\mathcal A}(S)$ and since $p \\in G_{\\mathcal A}(S) \\subset D_{\\mathcal A}(R), q \\in R$, the pair $\\lbrace p,q \\rbrace$ is polar by Step $(i)$ of this Proposition.\n\\end{proof}\n\n\\begin{prop}Given $R \\in {\\mathcal S}$ there exists a unique element\n$D_{\\mathcal A}(S) \\in {\\mathcal D}_{\\mathcal A}$, for some $S \\in {\\mathcal S}$ containing $R$.\n\\end{prop}\n\n\\begin{proof}\nLet $R$ be a symplecton and assume that $R \\subset D_{\\mathcal A}(S)$ for some $S \\in {\\mathcal S}$. Write $R=\\ll p,q \\gg$ with $p$ and $q$ two points of $R$, at distance two in the collinearity graph. We analyze the possible relations between $D_{\\mathcal A}(S)$ and $D_{\\mathcal A}(R)$.\n\n\\medskip\n$(i)$. Assume $p \\in D_{\\mathcal A}(S)$ and $q \\in S$. If $p \\in S$ then $S= \\ll p,q \\gg = R$. Next let $p \\in N_{\\mathcal A}(S)$ and set ${\\overline A}_p = p^{\\perp} \\cap S$. Then $R \\cap S = \\langle q, q^{\\perp} \\cap {\\overline A}_p \\rangle  \\in M_{\\mathcal B}(S)$. Hence $D_{\\mathcal A}(S) = D_{\\mathcal A}(R)$, by Lemma $4.6$. Let now $p \\in G_{\\mathcal A}(S)$ with $\\lbrace r \\rbrace = p^{\\perp} \\cap S$. By Lemma $4.11$, $q \\in r^{\\perp} \\cap S$ and by Lemma $4.7(b)$ it follows that $D_{\\mathcal A}(R) = D_{\\mathcal A}(S)$.\n\n\\medskip\n$(ii)$. Let $p, q \\in N_{\\mathcal A}(S)$ with ${\\overline A}_p = p^{\\perp} \\cap S$ and ${\\overline A}_q = q^{\\perp} \\cap S$.\n\n\\smallskip\n$(ii.a)$. Assume ${\\overline A}_p \\cap {\\overline A}_q = L$ is a line. {\\it Claim}: $R \\cap S = L$. Suppose by contradiction that $R \\cap S$ contains a plane $\\langle r,L \\rangle $ where $r$ is a point not on $L$. Since $r \\not \\in A_p \\cap A_q$ we can assume, without loss of generality, that $r \\not \\in A_q$. If $r \\in {\\overline A}_p$ then $\\langle r, q^{\\perp} \\cap A_p, p \\rangle$ is a singular subspace of rank $4$ in $R$, a contradiction. So let us assume that $r \\in S \\setminus ({\\overline A}_p \\cup {\\overline A}_q)$. Then $\\langle r, r^{\\perp} \\cap {\\overline A}_p, r^{\\perp} \\cap {\\overline A}_q \\rangle  \\subset R \\cap S$ which is a contradiction with the fact that $R \\cap S$ can be at most a common maximal singular subspace. Therefore the claim is proved. Now pick a point $z \\in p^{\\perp} \\cap q^{\\perp} \\setminus L$ such that $z^{\\perp} \\cap L = \\lbrace t \\rbrace$, a single point. We intend to prove that $z \\in G_{\\mathcal A}(S)$. According to Proposition $4.11, \\; z\\in D_{\\mathcal A}(S)$, so it suffices to prove that $z^{\\perp} \\cap S = \\lbrace t \\rbrace$. Assume by contradiction that $z \\in N_{\\mathcal A}(S)$ with ${\\overline A}_z = z^{\\perp} \\cap S$. Then, according to {\\it (P3)}, ${\\overline A}_z \\cap {\\overline A}_p$ is a line $L'$ distinct from $L$. So if $w \\in L \\setminus \\lbrace t \\rbrace$ then $\\langle L, L' \\rangle \\subset z^{\\perp} \\cap w^{\\perp} \\subset R \\cap S$ and we reach a contradiction with the previous result that $R \\cap S = L$. Therefore $z^{\\perp} \\cap S = \\lbrace p \\rbrace$ and $z \\in G_{\\mathcal A}(S)$. Now $R$ and $S$ are as in Lemma $4.7(a)$ and consequently $D_{\\mathcal A}(R) = D_{\\mathcal A}(S)$.\n\n\\smallskip\n$(ii.b)$. Let ${\\overline A}_p \\cap {\\overline A}_q = \\emptyset$. Let $t \\in {\\overline A}_p$ and set $T = \\ll t,q \\gg$. Thus $T \\cap S = \\langle  t, t^{\\perp} \\cap {\\overline A}_q \\rangle  \\in M_{\\mathcal B}(S)$ and by Lemma $4.6,\\; D_{\\mathcal A}(T) = D_{\\mathcal A}(S)$. Now $p \\in D_{\\mathcal A}(T), q \\in T$ so by Step $1$, $R = \\ll p, q \\gg$ is such that $D_{\\mathcal A}(R) = D_{\\mathcal A}(T) = D_{\\mathcal A}(S)$.\n\n\\medskip\n$(iii)$. Let $p \\in N_{\\mathcal A}(S), \\; q \\in G_{\\mathcal A}(S)$ with $q^{\\perp} \\cap S = \\lbrace r \\rbrace$. If $r \\in {\\overline A}_p$ then let another point $t \\in {\\overline A}_p$ and set $T = \\ll q, t \\gg$. By Lemma $4.7b$, $D_{\\mathcal A}(T) = D_{\\mathcal A}(S)$. Apply Step $(i)$ to $p \\in D_{\\mathcal A}(T),\\; q \\in T$ to get $D_{\\mathcal A}(R) = D_{\\mathcal A}(T) = D_{\\mathcal A}(S)$. If $r \\not \\in {\\overline A}_p$ set $T = \\ll p, r \\gg$ and since $T \\cap S \\in M_{\\mathcal B}(S)$, Lemma $4.6$ gives $D_{\\mathcal A}(T)=D_{\\mathcal A}(S)$. Now $p \\in T$ and $q \\in D_{\\mathcal A}(T)$ and using Step $(i)$ again we get $D_{\\mathcal A}(R) = D_{\\mathcal A}(T) = D_{\\mathcal A}(S)$.\n\n\\medskip\n$(iv)$. Let $p,q \\in G_{\\mathcal A}(S)$ with $\\lbrace p_1 \\rbrace =\np^{\\perp} \\cap S$ and $\\lbrace q_1 \\rbrace = q^{\\perp} \\cap S$. If $p_1 = q_1$ take $r \\in p_1^{\\perp} \\cap S$, set $T = \\ll r, p \\gg$. If $p_1 \\in q_1^{\\perp} \\setminus \\lbrace q_1 \\rbrace$ take $T = \\ll p, q_1 \\gg$. If $p_1 \\not \\in q_1^{\\perp}$ then take $r \\in p_1^{\\perp} \\cap q_1^{\\perp}$ and set $T =  \\ll p,r \\gg$. Then apply Step $(i)$ to get $D_{\\mathcal A}(R) = D_{\\mathcal A}(T) = D_{\\mathcal A}(S)$.\n\\end{proof}\n\n\\medskip\n\\begin{proof}[{\\it Proof Theorem 4.1}] Let {$\\Gamma = ({\\mathcal P}, {\\mathcal L})$} be a parapolar space\nwhich is locally $A_{7,4}$. Assume that the maximal singular subspaces of $\\Gamma$ partition in two classes. In addition, assume that $\\Gamma$ satisfies the Weak Hexagon Axiom {\\it (WHA)}. For any symplecton $S \\in {\\mathcal S}$ we defined two sets $D_{\\mathcal X}(S)$ with ${\\mathcal X} \\in \\lbrace {\\mathcal A}, {\\mathcal B} \\rbrace$; see the beginning of Section $4$. Then $D_{\\mathcal A}(S)$ is a $2$-convex subspace of\n$\\Gamma$, see Propositions $4.8$ and $4.9$, which is strong parapolar, by\nProposition $4.11$. Further, by Proposition $4.12$, every symplecton lies in a unique element of ${\\mathcal D}_{\\mathcal A}$.\n\n\\medskip\nConsider a point-symplecton pair $(p, R)$ in $D_{\\mathcal A}(S)$. Since $R \\subset D_{\\mathcal A}(S)$ it follows, by Proposition $4.12$ that $D_{\\mathcal A}(R)=D_{\\mathcal A}(S)$. Therefore, either $p \\in R$ or $p^{\\perp} \\cap R$ is a single point or a maximal singular subspace of $R$. By the characterization theorem of Cohen and\nCooperstein \\cite{cc}, see section $2.4$ also, $D_{\\mathcal A}(S)$ is a space of type $D_{6,6}$.\n\n\\medskip\nNext, replace ${\\mathcal A}$ with ${\\mathcal B}$. All of the above arguments can be repeated and we can conclude that $D_{\\mathcal B}(S)$ is a subspace of $\\Gamma$ which is of type $D_{6,6}$. Also, every symplecton $S \\in {\\mathcal S}$ is contained in exactly one element from the family ${\\mathcal D}_{\\mathcal B}$.\n\\end{proof}\n\n\\section{The sheaf theoretic characterization}\n\nIn this Section, we combine Ronan-Brouwer-Cohen sheaf theory with Tits' Local Approach Theorem and prove the following:\n\n\\begin{thm}Let ${\\widetilde \\Gamma}= ( {\\widetilde {\\mathcal P}}, {\\widetilde {\\mathcal L}})$ be a parapolar space which is locally of type $A_{7, 4}$. Let $I = \\lbrace 1, \\ldots 8 \\rbrace, \\; J = \\lbrace 1, 8 \\rbrace$ and $K = I \\setminus J$. Assume that ${\\widetilde \\Gamma}$ satisfies the Weak Hexagon Axiom. Then there is a residually connected $J$-locally truncated diagram geometry $\\Gamma$ which belongs to the diagram:\\\\\n\\begin{picture}(1000, 50)(0,0)\n\\put(0,25){$\\mathcal{Y}$}\n\\put(60,25){$\\qed$}\n\\put(75,15){\\scriptsize 1}\n\\put(80,29){\\line(1,0){32}}\n\\put(112,26){$\\circ$}\n\\put(113,15){\\scriptsize 2}\n\\put(113,36){\\scriptsize ${\\mathcal D}_{\\mathcal B}$}\n\\put(118,29){\\line(1,0){30}}\n\\put(148,26){$\\circ$}\n\\put(150,36){\\scriptsize ${\\mathcal B}$}\n\\put(150,15){\\scriptsize 3}\n\\put(153,29){\\line(1,0){30}}\n\\put(183,26){$\\circ$}\n\\put(190,36){\\scriptsize ${\\mathcal L}$}\n\\put(190,15){\\scriptsize 5}\n\\put(186,-3){\\line(0,1){30}}\n\\put(183,-9){$\\circ$}\n\\put(171,-10){\\scriptsize ${\\mathcal P}$}\n\\put(190,-10){\\scriptsize 4}\n\\put(188,29){\\line(1,0){30}}\n\\put(218,26){$\\circ$}\n\\put(220,36){\\scriptsize ${\\mathcal A}$}\n\\put(220,15){\\scriptsize 6}\n\\put(223,29){\\line(1,0){30}}\n\\put(253,26){$\\circ$}\n\\put(253,36){\\scriptsize ${\\mathcal D}_{\\mathcal A}$}\n\\put(255,15){\\scriptsize 7}\n\\put(258,29){\\line(1,0){30}}\n\\put(276,25){$\\qed$}\n\\put(290,15){\\scriptsize 8}\n\\end{picture}\n\n\\vspace{.4cm}\nwhose universal $2$-cover is the truncation of a building. Therefore, ${\\widetilde \\Gamma}$ is the homomorphic image of a truncated building, also.\n\\end{thm}\n\n\\begin{proof}Let us assume that ${\\widetilde \\Gamma}= ( {\\widetilde {\\mathcal P}}, {\\widetilde {\\mathcal L}})$ is a parapolar space, locally of type $A_{7,4}$ and which satisfies {\\it(WHA)}. Construct the space {$\\Gamma = ({\\mathcal P}, {\\mathcal L})$} according to the method described in Section $3$. Then $\\Gamma$ is also locally of type $A_{7,4}$ and satisfies {\\it(WHA)}. Furthermore, $\\Gamma$ has two classes of maximal singular subspaces ${\\mathcal A}$ and ${\\mathcal B}$, whose elements are projective spaces of rank $5$. It was proved in Section $4$ that in $\\Gamma$ there are two more families of subspaces, denoted ${\\mathcal D}_{\\mathcal A}$ and ${\\mathcal D}_{\\mathcal A}$, whose elements are of type $D_{6,6}$. Define incidence as follows:\n\\begin{itemize}\n\\item[(i).]An object in $\\mathcal{A}$ is incident with an object in $\\mathcal{B}$ if they intersect at a plane.\n\\item[(ii).] An object in $\\mathcal{B}$ (or $\\mathcal{A}$) is incident with an object from $\\mathcal{D}_{\\mathcal A}$ ($\\mathcal{D}_{\\mathcal B}$ respectively) if they intersect at a singular subspace of rank $3$.\n\\item[(iii).] An object from $\\mathcal{D}_{\\mathcal A}$ is incident with an object from $\\mathcal{D}_{\\mathcal B}$ if they have a symplecton in common.\n\\item[(iv).] Inclusion for all the remaining cases.\n\\end{itemize}\n\n\\medskip\nHence, the space $\\Gamma$ can be enriched to a rank $6$ geometry $\\Gamma = (\\mathcal{P, L, A, B}, {\\mathcal D}_{\\mathcal A}, {\\mathcal D}_{\\mathcal B})$. It follows from Theorem $3.1$ that $\\Gamma$ is a covering of ${\\widetilde \\Gamma}$. Furthermore, $\\Gamma$ is a geometry over $K$ which is $J$-locally truncated with respect to the diagram $\\mathcal{Y}$.\n\n\\medskip\nIn the sequel we will use the terminology given in Sections $2.1$-$2.3$; the definition of a sheaf over a locally truncated geometry was given in Section $2.4$.\n\n\\medskip\nNext we construct a sheaf $\\Sigma$ over the collection $\\mathcal{F}$ of flags of rank $1$ and $2$ of $\\Gamma$. It is a consequence of a result of Ellard and Shult \\cite{els}, see also \\cite[Section 6]{on1}, that it suffices to work with the collection $\\mathcal{F}$ of flags, instead of the full family of nonempty flags of $\\Gamma$.\n\n\\medskip\n${\\mathfrak 1})$. For $a \\in {}^{2}\\Gamma$, an object of type $2$ in $\\Gamma$, define $\\Sigma ^R(a)$ to be the geometry of $\\lbrace 8 \\rbrace$-locally truncated type belonging to the diagram $\\mathcal{Y}_{I \\setminus \\lbrace 1,2 \\rbrace}$ and which satisfies the property that $\\res {\\Gamma}{a} =\\;  _{\\lbrace 8 \\rbrace}\\Sigma ^R(a)$. This is well defined since $\\Sigma ^R (a)$ is of $\\lbrace 8 \\rbrace$-locally truncated type $D_{6,6}$, which, up to the relabeling of the nodes is unique. In $\\Sigma\n^R (a)$ the objects of types in $K$ are the same as in $\\Gamma$ with the corresponding incidence. The objects of type $\\lbrace 8 \\rbrace$ are collections of objects in $\\Gamma$ with their flags, which are incident with a given object of type in $K$; the incidence between objects of type $\\lbrace 8 \\rbrace$ is given by symmetrized containment.\n\n\\medskip\n${\\mathfrak 2})$. For $b \\in \\; ^{4}\\Gamma$ define $\\Sigma(b)$ to be such that $\\res{\\Gamma}{ b} =\\;_J \\Sigma(b)$  Observe that $\\res{\\Gamma}{b}$ is\nthe $J$-truncation of a geometry belonging to the diagram $D_{I\n\\setminus \\lbrace 4 \\rbrace}$ of type $A_{7,4}$. But this is\nuniquely determined by its truncation, \\cite[Theorem 1]{loc}, and thus $\\Sigma(b)$ is unambiguously defined. There are two types of objects in $\\Sigma\n(b)$: those inherited from $\\Gamma$, with their incidence and the objects with types\nin $J$, which are defined as in ${\\mathfrak 1})$.\n\n\\medskip\n${\\mathfrak 3})$. Let now $l \\in \\lbrace 3, \\dots, 7 \\rbrace$ and let $x_l \\in \\;\n^{l}\\Gamma$. Denote by $F = \\lbrace a,x \\rbrace $ a $\\lbrace 2, l-1 \\rbrace $-flag in $\\res{\\Gamma}{x_l}$. For $l=3$, $F = \\lbrace a \\rbrace$ is just an object of type $2$. Define recursively: $\\Sigma ^R (x_l) = \\Sigma ^R (a, x_l) := \\res{\\Sigma ^R (a,x)}{x_l}$. Let $F'$ be another flag of type\n$\\lbrace 2, l-1 \\rbrace$ in $\\res{\\Gamma }{x_l}$ which is $i$-adjacent to $F$ in $C$, the chamber system associated to $^{\\lbrace 2, l-1 \\rbrace}\\res {\\Gamma} {x_l}$ with  $i \\in \\lbrace 2, l-1 \\rbrace$. Then: $\\res{\\Sigma ^R (F)}{x_l} = \\res{\\Sigma ^R (F \\cap F')} {(F \\setminus F') \\cup \\lbrace x_l \\rbrace } = \\res{\\Sigma ^R (F') }{x_l}$ which proves the well-definedness in this case. If $F$ and $F'$ are not\n$i$-adjacent, since the chamber system $C$ is connected, there is a\ngallery from $F$ to $F'$, and by repeated applications of the above\nargument we get the result.\n\n\\medskip\n${\\mathfrak 4})$. Let $f \\in \\; ^{7} \\Gamma$. Define $\\Sigma ^L (f)$ to be the\ngeometry of $\\lbrace 1 \\rbrace$-locally truncated type, with diagram $\\mathcal{Y}_{I \\setminus \\lbrace 7,8 \\rbrace}$, and such that $\\res{\\Gamma}{ f} = \\;\n_{\\{1\\}} \\Sigma ^L (f)$; this is uniquely defined since it has diagram of type $D_{6,6}$. The objects in $\\Sigma ^L(f)$ and their incidence are defined as in the first step.\n\n\\medskip\n${\\mathfrak 5})$. Let $l \\in \\lbrace 6, \\ldots 2 \\}$, taken in this order. Let $x_l \\in\n\\; ^{l}\\Gamma$ and $F = \\lbrace x, f \\rbrace$ a $\\lbrace l+1, 7 \\rbrace$-flag in $\\res{\\Gamma}{x_l}$. Recursively define: $\\Sigma ^L (x_l) = \\Sigma ^L (x_l, f) := \\res{\\Sigma ^L(x, f)} {x_l}$. The proof of well-definedness is similar to the one given in the third step. The objects in $\\Sigma ^L(x_l)$ and their incidence are defined using the method from the first step.\n\n\\medskip\nThe sheaf values over the rank $1$ flags of $\\Gamma$ can be written as\nfollows:\n\\begin{itemize}\n\\item[ (i).] \\quad $\\Sigma (x) := \\Sigma ^L (x) \\oplus \\Sigma ^R (x),\\;   \\text{for\nany object}\\ x\\  \\text{of type in}\\  K \\setminus \\lbrace 4 \\rbrace$;\n\\item[(ii).] \\quad $\\Sigma (b) \\; \\text{for any object}\\ b \\in \\; ^{4}\\Gamma$.\n\\end{itemize}\n\nNext, we construct the sheaf the sheaf over the rank $2$ flags of $\\Gamma$ and we check the compatibility condition: $\\res{\\Sigma(x)}{y} = \\Sigma(x,y) = \\res{\\Sigma(y)}{x}$, where $\\lbrace x, y \\rbrace$ is a nonempty rank $2$ flag in $\\Gamma$.\n\n\\medskip\n${\\mathfrak 6})$. For $\\lbrace x_i, x_j \\rbrace$, a rank $2$ flag of type $\\lbrace i, j\\rbrace$ in $\\Gamma$ with $i, j \\in K \\setminus \\lbrace 4 \\rbrace$\nand $i<j$, set\n$$\\Sigma (x_i, x_j) := \\Sigma ^{L} (x_i) \\oplus \\;\n^{(ij)} \\Gamma \\oplus \\Sigma ^{R}(x_j)$$\nwhere $^{(ij)}\\Gamma$ denotes the\ntruncation of $\\Gamma$ to those objects of type $l \\in K$ with $i<l<j$. Since\n\\begin{equation*}\n\\begin{split}\n&\\res{\\Sigma(x_i)}{x_j} = \\res {\\Sigma ^L (x_i)}{x_j} \\oplus \\res {\\Sigma ^R(x_i)}{x_j} = \\Sigma ^L (x_i) \\oplus \\; ^{(ij)}\\Gamma \\oplus \\Sigma ^R (x_j)\\\\\n&= \\res {\\Sigma ^L (x_j)}{x_i} \\oplus \\Sigma ^R (x_j) = \\res{\\Sigma (x_j)}{x_i}\n\\end{split}\n\\end{equation*}\nit follows that the compatibility condition is satisfied in this case.\n\n\\medskip\n${\\mathfrak 7})$. There are two remaining cases, as follows:\n\\begin{itemize}\n\\item[a).] For $\\lbrace a,b \\rbrace$, a $\\{ 3,4\\}$-flag, define $\\Sigma (a,b) := \\Sigma ^L(a) \\oplus \\Sigma ^R (b)$.\n\\item[b).] For a flag $ \\lbrace b, x_j \\rbrace$ of type $\\{4, j \\}$, with $j \\in \\{5,6,7\\}$ set $\\Sigma(b, x_j) := \\res{\\Sigma ^L(x_j)}{b} \\oplus \\Sigma ^R(x_j)$.\n\\end{itemize}\nThe compatibility condition can be checked now, the proof is similar to the one given in the previous step.\n\n\\medskip\nTherefore $\\Sigma$ is defined for all rank $1$ and rank $2$ flags of\n$\\Gamma$. Now using the aforementioned Lemma, see \\cite[Section 6]{on1}, of Ellard and Shult, we can extend the sheaf $\\Sigma$ to a sheaf (denoted the same) over all nonempty flags of $\\Gamma$.\n\n\\medskip\nThe sheaf $\\Sigma$ is residually connected; this follows from the fact that all sheaf values at rank $1$ and $2$ flags of $\\Gamma$ are $J$-truncated buildings or products of $J$-truncated buildings.\n\n\\medskip\nAccording to a result of Brouwer and Cohen \\cite{loc}, there exists a canonically defined chamber system $\\mathcal{C}(\\Sigma)$ over $I$, which is residually connected and belongs to diagram $\\mathcal{Y}$. It satisfies the property that $_J {\\mathcal C}(\\Sigma) \\; \\simeq \\; {\\mathcal C}(\\Gamma)$, an isomorphism of chamber systems over $K$, with the term on the right being the chamber system corresponding to the geometry $\\Gamma$. Moreover, all rank $3$ residues are covered by truncations of buildings. Therefore by Tits' Local Approach Theorem \\cite{tits}, see also Theorem $2.1$,  the universal $2$-cover of ${\\mathcal C}(\\Sigma)$ is the chamber system ${\\mathcal B}$ of a truncated building belonging to diagram $\\mathcal{Y}$.\n\n\\medskip\nThere is a $J$-locally truncated building geometry $\\widetilde \\Delta $ which belongs to the diagram $\\mathcal{Y}$ and which is the image under the functor ${\\mathbf G}: {\\mathcal\nChamb}_I \\rightarrow {\\mathcal Geom}_I$ of $\\mathcal{B}$; see Section 2.3. The chamber system ${\\mathcal C}(\\Sigma)$ also has a corresponding geometry which we shall denote $\\Delta$. Since the chamber system ${\\mathcal C}(\\Sigma)$ is\nresidually connected, it follows that $\\Delta$ is also residually connected. The $2$-cover map of chamber systems ${\\mathcal B} \\rightarrow \\mathcal{C}(\\Sigma)$ functorially induces a covering of geometries $\\widetilde{\\Delta} \\rightarrow \\Delta$. The fact that $\\widetilde{\\Delta}$ is the universal cover of $\\Delta$ follows from the fact that $\\mathcal{B}$ is the universal cover of $\\mathcal{C}(\\Sigma)$. Moreover the geometry $\\Delta$ has the property that $ _J \\Delta \\simeq \\Gamma$; for details see \\cite{on1}. Combining the previous results with Theorem $3.1$, it follows that $\\widetilde{\\Gamma}$ is the homomorphic image of a building.\n\\end{proof}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{intro}\n\nCataclysmic Variables (CVs) are binary systems where a late-type main-sequence donor star transfers mass onto a primary white dwarf (WD). According to standard evolutionary theory, loss of angular momentum drives CVs to initially evolve from longer to shorter orbital periods. All systems are predicted to reach a minimum orbital period roughly when the donor stops its hydrogen burning and becomes degenerate. The observed minimum period is found at $\\approx$ 83 minutes~\\cite{2009MNRAS.397.2170G}. At about the point when the donor reaches this evolutionary state, the thermal time scale becomes longer than the mass-transfer timescale, and the donor cannot shrink fast enough in response to continued mass loss. As a result, the brown-dwarf donor will expand, and the system will move towards longer orbital periods. \n\nStandard evolutionary theory also predicts that about 70\\% of all CVs should have passed their minimum period and have sub-stellar donors (\\citealp{1993A&A...271..149K}; \\citealp{1997MNRAS.287..929H}). However, few such systems have been successfully confirmed, and until recently, almost no CVs containing donors with masses below the hydrogen-burning limit, had been found. Systems that have passed their minimum period (post-bounce systems) should be very faint and are therefore difficult to find, in general. Nevertheless, in the course of the Sloan Digital Sky Survey (SDSS), a few post-bounce CVs were discovered (\\citealp{2008MNRAS.388.1582L}), among them the eclipsing system SDSS J1507+52 (hereafter J1507). \n\nThis particular system was quickly recognised to be odd due to its short orbital period of about 67 minutes (\\citealp{2005AJ....129.2386S}), which is well below the minimum period characteristic of normal CVs. A few other systems are occasionally found to have periods below the usual minimum period, but, in general, these systems show evidence for abnormally hot and bright donor stars in their optical spectra, suggesting that their secondaries are nuclear-evolved objects. This is not the case in J1507, whose secondary is not visible at all in optical spectroscopy (\\citealp{2005AJ....129.2386S}; \\citealp{2007MNRAS.381..827L}). Together with its short orbital period, this indicates a system with a faint disc, a low accretion rate and a relatively unevolved, low-mass donor.\n\n\\cite{2007MNRAS.381..827L} performed eclipse analysis of J1507 and found a donor mass of 0.056 $\\pm$ 0.001 M$_{\\odot}$, clearly below the hydrogen-burning limit. However, because of its anomalously short orbital period, the system is not consistent with the standard mass-period relation for CVs (\\citealp{2006MNRAS.373..484K}). In line with this,~\\cite{2007MNRAS.381..827L} found the donor radius to be smaller than predicted by standard CV evolution sequences. They therefore suggested that the donor might be unusually young, i.e. that J1507 might be a CV that formed recently from a previously detached WD -- brown dwarf close binary system. A young brown dwarf has a higher density, and therefore a smaller radius, since the donor has not yet had a chance to expand in response to mass loss. This would explain the short orbital period found in J1507, since according to the period-density relation, a higher density and subsequently a smaller radius, implies a shorter orbital period. The list of photometrically inferred system parameters provided by~\\cite{2007MNRAS.381..827L} also includes the effective temperature of the white dwarf, which they estimate to be $T_{\\text{eff}}$ = 11000 $\\pm$ 500 K. \n\nConcurrently,~\\cite{2008PASP..120..510P} showed that the system has an unusually high space velocity, similar to the velocities of stars in the Galactic halo. A typical star of 1 M$_{\\odot}$ in the Galactic disc has a space velocity below 50 km\\,s$^{-1}$, while J1507 has a velocity of about 167 km\\,s$^{-1}$. If J1507 is a member of the Galactic halo, its donor will be a Population II object with low metallicity. Due to their lower atmospheric opacity, such objects have significantly smaller radii than their solar metallicity Population I counterparts. Membership of the Galactic halo would therefore also account for J1507's small donor radius and short orbital period. However, Population II stars only represent about 0.5 \\% of the stars in the Solar neighbourhood, which would make J1507 a rare system.~\\cite{2008PASP..120..510P} estimated a slightly higher effective temperature of the WD in J1507, $T_{\\text{eff}}$ = 11500 $\\pm$ 700 K. They also found the system to exhibit multi-periodic variability on time-scales of minutes, which they interpreted as non-radial pulsations originating from the primary white dwarf.\n\n\nBoth hypothesis imply that J1507 is an interesting and important system that can shed light on poorly understood aspects of CV evolution. However, since the models are very different, it is clearly important to determine which -- if either -- of them is correct. As noted above, eclipse-based estimates for the effective temperature of the WD suggested $T_{\\text{eff}} \\simeq$ 10000 K -- 12000 K. At these temperatures, the ultraviolet (UV) spectrum of a WD accreting metal-rich material would be expected to show strong absorption features due to Fe II and Fe III (c.f. \\citealp{1994ApJ...426..294H}). We have therefore obtained far- and near-UV spectroscopy of J1507 with COS and STIS onboard the Hubble Space Telescope, with the aim of measuring the metallicity of the system and establishing its status as either a Pop I (disk) or a Pop II (halo) object.\n\n\n\\begin{figure}\n\\includegraphics[width=9cm]{cos_light.eps}\n\\caption{The Gaussian smoothed and flux-calibrated light curve obtained by COS\/HST, constructed from the monochromatic flux averaged over the region 1425\\,\\AA\\, -- 1900\\,\\AA. Bottom panel shows the phase-folded light curve in the vicinity of the eclipse.} \n\\label{fig:light}\n\\end{figure}\n\n\n\n\n\\section{Observations and Reduction}\n\nFar- and near-UV observations of J1507 were performed in February 2010, using the HST. The observations were obtained in the so-called time-tag mode. In this mode, each photon event is tagged with a corresponding event time, along with their wavelength and spatial position on the detector. This allows us to rebin the data over any time interval and wavelength region. \n\nFar-UV observations were carried out with the Cosmic Origins Spectrograph (COS), for a total exposure time of about 3 hours, covering 3 eclipses in three successive HST orbits. The maximum time resolution of the COS TIME-TAG mode is 32 ms. The G140L grating was used, which gave a spectral range of 1230\\,\\AA\\ -- 2378\\,\\AA\\,and a spectral resolution of 0.5\\,\\AA.  \n\nNear-UV observations were obtained by the Space Telescope Imaging Spectrograph (STIS), for a total exposure time of about 1.4 hours, at a maximum time resolution 125 $\\mu$s. The G230L grating was used, resulting in a spectral range of 1650\\,\\AA\\ -- 3150\\,\\AA\\,and a spectral resolution of 3.16\\,\\AA. The data set consists of 3 sub-exposures (of length, 1600 s, 1918 s and 1708 s). No near-UV data was obtained during the eclipses. Also, gaps in the data are cased by interruptions coming from Earth occultation. \n\nAll data were reduced and calibrated using the \\textsc{Pyraf} package \\textsc{Stsdas}, provided by STScI (the Space Telescope Science Institute)\\footnote{\\textsc{Stsdas} and \\textsc{Pyraf} are products of the Space Telescope Science Institute, which is operated by AURA for NASA.}.\n\n\n\n\n\\begin{figure*}\n\\includegraphics[width=11.0cm]{spec.eps}\n\\caption{The two top panels show the mean out-of-eclipse spectrum for both the far (left) and near (right) UV regions. A model constructed from parameters obtained from the far UV-region is plotted in red (see Table~\\ref{pars}). A model corresponding to values from Littlefair et al. (2007), $T_{\\text{eff}}$=11000 K and $\\log g$=8.5, is plotted in blue. Underneath each wavelength region is a plot showing the residuals. The dips at (1460, 1530, 1600 and 1670)\\,\\AA\\, coincide with regions of bad pixels. The 1-$\\sigma$ flux errors are plotted in green.} \n\\label{fig:spec}\n\\end{figure*}\n\n\n\\section{Analysis and Results}\n\nFigure~\\ref{fig:light} shows the Gaussian smoothed and flux-calibrated light curve from the far-UV data. This light curve was constructed by dividing the larger exposures into one-second bins, and by average the fluxes over the wavelength range, 1425\\,\\AA\\, -- 1900\\,\\AA. The light curve shows sharp, square-shaped eclipses, close to zero flux at mid-eclipse, and a very flat out-of-eclipse continuum, strongly suggesting that the UV flux is dominated by emission from (or very close to) the accreting WD. The folded light curve constructed from the combined data for all three HST orbits is presented in the bottom panel. All further analysis was carried out on the unsmoothed data, excluding the eclipses. Ground-based optical observations were performed at the same time as the HST observations, yielding a magnitude for J1507 of $g = 18.44 \\pm 0.02$. This is consistent with SDSS, showing that the system was in quiescence during the HST observations. \n\nThe two top panels of Figure~\\ref{fig:spec} show the mean out-of eclipse spectrum (in black), for both the near- and far-UV. The reduction was performed separately for the COS and STIS data, and the mean spectrum for each region were joined at 1900 \\AA\\,without overlap for a total range of 1268\\,\\AA\\, -- 3150\\,\\AA. The COS data was rebinned slightly, to 0.2\\,\\AA\/pix, to increase the S\/N and provide sufficient sampling of the G140L grating's spectral resolution. Different dispersions were retained for the the far- and near-UV regions (0.2 \\,\\AA\/pix and 1.58 \\,\\AA\/pix, respectively). The overall UV spectrum for J1507 is relatively sparse in spectral lines. We identify CII at 1335, SiII at 1526.7 HeII at 1640.5, AlII at 1671 and MgII at 2800 \\,\\AA. The strong emission feature near 1300 \\AA\\, is almost certainly due to geocoronal emission in the O I 1304 \\AA\\, line that could not be cleanly removed by the pipeline background subtraction.\n\n\n\\subsection{Spectral Modelling}\n\nA model grid spanning the four key atmospheric parameters, effective temperature ($T_{\\text{eff}}$), surface gravity ($\\log g$), metallicity ($[Fe\/H]$) and rotational velocity ($v \\sin i$), was constructed in order to fit the far- and near-UV spectrum. Atmospheric structures were calculated with \\textsc{Tlusty}, with the spectral synthesis being done with \\textsc{Synspec} (\\citealp{1988CoPhC..52..103H}; \\citealp{1995ApJ...439..875H}). The grid consisted of; 12500 K $\\mathrel{ \\rlap{\\raise.5ex\\hbox{$<$} T_{\\text{eff}} \\mathrel{ \\rlap{\\raise.5ex\\hbox{$<$}$ 20000 K in steps of 500 K, 7.5  $\\mathrel{ \\rlap{\\raise.5ex\\hbox{$<$} \\log g \\mathrel{ \\rlap{\\raise.5ex\\hbox{$<$} 9.5$ in steps of 0.25, -2.0 $\\mathrel{ \\rlap{\\raise.5ex\\hbox{$<$} [Fe\/H]  \\mathrel{ \\rlap{\\raise.5ex\\hbox{$<$}$ 0.0 in steps of 0.25, and 0 km\\,s$^{-1} \\mathrel{ \\rlap{\\raise.5ex\\hbox{$<$} v \\sin i \\mathrel{ \\rlap{\\raise.5ex\\hbox{$<$}$ 1000 km\\,s$^{-1}$ in steps of 50 km\\,s$^{-1}$. Models at intermediate parameter values were constructed by linear interpolation on this grid. Pixels with non-zero data quality flags set were excluded from our fits, as well as the regions around the spectral lines, OI at 1304\\,\\AA\\, CII at 1335\\,\\AA\\,and the quasi-molecular H feature at $\\sim$ 1400\\,\\AA. \n\nBefore our models could be compared to the observed spectrum, they were convolved with a Gaussian filter to the spectral resolution appropriate to the COS (< 1900\\,\\AA) and STIS (> 1900\\,\\AA) data, and linearly interpolated onto the observational wavelength scale. In general, our model fits did represent the main features of the data quite well, but the formal $\\chi^{2}$ was somewhat high. In order to obtain more realistic formal parameter errors, we therefore added an intrinsic dispersion term to the flux errors in such a way that $\\chi^{2}_{\\nu}$ = 1. This dispersion corresponds to about 1\\,\\% of the flux at 1900\\,\\AA. In order to explore the systematic uncertainties affecting our fits, we tried fitting many different wavelength regions, such as the far-UV (COS), near-UV (STIS), regions with a high line concentration and also the whole wavelength range spanning both the far- and near-UV.   \n\n\n\\begin{figure*}\n\\includegraphics[width=11.0cm]{met_vel.eps}\n\\caption{The top panel shows the mean out-of-eclipse spectrum for the far UV region, where models at three different $[Fe\/H]$, 0 (green), -1.2 (red) and -2 (blue), are plotted on top. The bottom left panel shows a  zoom of the region around the Si II, where the coloured lines correspond to the same values of the metallicity as in the top panel. In the bottom right panel, the colours correspond to $v \\sin i$ at 0 km\\,$s^{-1}$ (green), 180 km\\,$s^{-1}$ (red), 500 km\\,$s^{-1}$ (blue).} \n\\label{fig:met}\n\\end{figure*}\n\n\n\\subsection{Temperature and Metallicity}\n\nA fit to the far UV-region 1268\\,\\AA\\, -- 1900\\,\\AA\\,\\,yields $T_{\\text{eff}} = 14200 \\pm 50$ K, $\\log g = 8.2 \\pm 0.04, [Fe\/H] = -1.2 \\pm 0.05 $ and $v \\sin i = 180 \\pm 20$ km\\,s$^{-1}$. These fit parameters provide a good approximation also for the near-UV spectrum out to 2800\\,\\AA\\, where the spectrum may begin to show disc features. We adopt these parameters as our best-fit parameters. The quoted errors are formal errors from the $\\chi^{2}$ fitting. Table~\\ref{pars} presents the best parameters along with their uncertainties. These uncertainties are larger than the formal errors, because they include our best estimates of the systematic uncertainties associated with different choices of fitting windows. However, there are several reasons for adopting the parameters inferred from the fit to the COS far-UV data set as our best-bet estimates. This region includes almost all line features, and is also most sensitive to changes in $T_{\\text{eff}}$ and $\\log g$. Also, the far UV-region is more likely to represent pure light from the WD, while towards redder wavelengths, the spectrum could be affected by the disc and the bright spot. Finally, fitting only the COS data removes any possibility that a mis-match in the flux calibration of COS and STIS could affect our results.\n\nThe two top panels in Figure~\\ref{fig:spec} show the best-fit model in red, along with the mean out-of-eclipse spectrum for J1507. The corresponding residuals between model and data are plotted underneath each of the COS and STIS spectral regions, together with the 1-$\\sigma$ error range marked in green. The dips in the residuals at (1460, 1530, 1600 and 1670)\\,\\AA\\,, are caused by bad pixels in the data and these regions were excluded from our fits. As a comparison, a model showing the parameters by~\\cite{2007MNRAS.381..827L}, $T_{\\text{eff}}$ = 11000 K and $\\log g$ = 8.5, is plotted in blue. Clearly, this temperature is far too low to match the data.\n \n The top panel in Figure~\\ref{fig:met}, shows the mean spectrum plotted together with models at three different values of $[Fe\/H]$, 0 (green), -1.2 (red) and -2 (blue). It is clear that a model of solar metallicity (green line) does not match the data and overestimates both the number and strength of visible absorption lines in the spectrum. The left panel shows a zoom of the region around the Si II line (the coloured lines correspond to the same values of the metallicity as given for the top panel). To the right is a plot showing the dependence of the model on $v \\sin i$, where the coloured lines in green, red and blue represent rotational velocities of 0 km\\,s$^{-1}$, 180 km\\,s$^{-1}$ and 500 km\\,s$^{-1}$, respectively. In all panels, the red lines show the best fitted parameters. The metallicity and rotational velocity affect both the line depths and widths. Therefore, as a check, we fixed $T_{\\text{eff}}$ and $\\log g$ to our best values, and made model fits to smaller wavelength regions specifically chosen because they contain strong and\/or many spectral lines, with $v \\sin i$ and $[Fe\/H]$ as free parameters. We find consistent values of $v \\sin i$ and $[Fe\/H]$, independent of the wavelength region we perform the fits on.   \n  \nWe compared the metallicity found for J1507 with the metallicity distribution for Galactic halo stars at the same space velocity. ~\\cite{2008PASP..120..510P} calculated a space velocity for J1507 of 167 km\\,s$^{-1}$, which can be broken into the Galactic velocity components U, V and W, where $\\sqrt{U^{2} + W^{2}}$ =139 km\\,s$^{-1}$ for J1507.~\\cite{1996AJ....112..668C} presented a study of the metallicity distribution for the Galactic halo stars. They find that for stars with velocities in the range 100 <  $\\sqrt{U^{2} + W^{2}}$ < 140, the metallicity distribution peaks around -0.7, but has a long tail towards metallicities as low as -1.5 (see Figure 6 in their paper). This is in good agreement with our metallicity of -1.2. Therefore, we conclude that J1507 is most likely a Galactic halo CV.\n\n\n\\subsection{Possible Extinction Effects}\n\nUp to now, we have assumed that the observed far-UV spectrum of SDSS J1507 is unaffected by interstellar extinction. Based on the absence of the well-known 2175\\,\\AA\\,absorption feature in the data, we can set E(B-V) $\\leq$ 0.05 as a fairly conservative limit on the amount of reddening that may be present. In order to test the effect of extinction at this level on our conclusions, we carried out additional model fits after dereddening the data by E(B-V) = 0.05. We find that none of our inferred parameters would change significantly if extinction at this level were present.\n\n\n\n\\subsection{Distance Estimates}\n\nThe theoretical relationship between observed flux ($F$) and the Eddington flux ($H$) provided by the \\textsc{Synspec} models can be used to estimate the distance ($d$) towards J1507. More specifically, $F =  4 \\pi R_{\\text{wd}}^{2}H\/\\,d^{2}$, where $R_{\\text{wd}}$ is the WD radius. This means that the normalisation factor needed to optimally match a model spectrum to the data is a direct measure of $R_{\\text{wd}}^{2}\/d^{2}$. WDs also follow the well-known mass-radius relation (with only a weak temperature sensitivity), so the surface gravity of the model, $g = G\\,M_{\\text{wd}}\/R_{\\text{wd}}^{2}$, uniquely fixes $R_{\\text{wd}}$. Thus for a WD model with given $T_{\\text{eff}}$ and $\\log g$, the normalisation factor required to fit the data is a unique measure of distance. \n\nIn practice, we estimated the distance by first fitting a linear function to the relationship between $M_{\\text{wd}}$ and $\\log g$ in Pierre Bergeron's WD cooling models\\footnote{Cooling models by Pierre Bergeron are found at: \\newline http:\/\/www.astro.umontreal.ca\/$\\sim$bergeron\/CoolingModels\/}, at $T_{\\text{eff}}$ = 14500 K. We then use this function to estimate $M_{\\text{wd}}$, and hence $R_{\\text{wd}}$, for given $\\log g$. Rough values of the WD mass and radius are found at $M_{\\text{wd}} = 0.75 \\pm 0.15$ M$_{\\odot}$ and $R_{\\text{wd}} = 0.011 \\pm 0.002$ R$_{\\odot}$. This is consistent with both ~\\cite{2007MNRAS.381..827L} and ~\\cite{2008PASP..120..510P}. The normalisation constant of the model combined with $R_{\\text{wd}}$, yield the corresponding distance estimate. We find a distance towards J1507 of $d = 250 \\pm 50$ pc (the effect of reddening is allowed for in the quoted error).\n\n\n\\begin{table}\n  \\centering\n  \\begin{minipage}{70mm}\n  \\caption{Best-fit parameters for J1507 obtained from model fitting to the far-UV region. The errors are defined as the whole range for which we find good fits, irrespective of wavelength region. Formal 1-$\\sigma$ errors are given in parenthesis.}\n \\begin{tabular}{ll}\n \\hline\n \\hline\n$T_{\\text{eff}}$: & 14200 $\\pm$ 500 (50) K  \\\\\n$\\log g$: & 8.2 $\\pm$ 0.3  (0.04)\\\\\n$[Fe\/H]$: & -1.2 $\\pm$ 0.2 (0.05)   \\\\\n$v \\sin i$: & 180 $\\pm$  20 (20) km\\,s$^{-1}$\\\\\n\\hline\n\\hline\n\\label{pars} \n\\end{tabular}\n\\end{minipage}\n\\end{table}\n\n\n\\section{Discussion and Summary}\n\nWe have obtained HST observations  in the UV spectral-range of the cataclysmic variable J1507, with the aim of measuring its metallicity to determine whether or not the system is a member of the Galactic halo. \n\nBy comparing the observed spectrum to synthetic spectra described by the four parameters, $T_{\\text{eff}}$, $\\log g$, $[Fe\/H]$ and $v \\sin i$, a best fit is found at $T_{\\text{eff}}$ = $14200 \\pm 500$ K. This value of $T_{\\text{eff}}$ is based on the assumption that the WD is the only component contributing to the UV flux in J1507. If a hot, optically thick boundary layer would be present, it would bias our estimate of $T_{\\text{eff}}$. However, the single-temperature WD models presented here seem to provide a good fit to the data, and we expect that the boundary layer in a low $\\dot{M}$-system (such as J1507) would be optically thin and thus not contribute significantly to the UV flux. \n\nOur best model fit give a $T_{\\text{eff}}$ that is much higher than previous estimates, which implies that the accretion rate is higher than previously suggested. More specifically, since $\\dot{M} \\propto T^{4}$, the 30\\% increase in the estimated  $T_{\\text{eff}}$ corresponds to an increase in $\\dot{M}$ by almost a factor of 3 (\\citealp{2003ApJ...596L.227T}). This may have significant implications for the evolution of this system, including the question of whether gravitational radiation alone is sufficient to drive this accretion rate (see \\citealp{2009ApJ...693.1007T}; \\citealp{2011arXiv1102.2440K}).\n\nAt the higher $T_{\\text{eff}}$ we infer, Fe II and III are no longer dominant contributors to the atmospheric UV opacity, making it more difficult than expected to measure the metallicity of the system. Nevertheless, model fits to the data clearly favour a significantly sub-solar metallicity, $[Fe\/H] = -1.2 \\pm 0.2$, comparable to the typical metallicity found for halo stars at the same high space velocity as J1507.\n\n\\cite{2008PASP..120..510P} found pulsations in J1507, which they identified as non-radial pulsations originating from the primary WD. Non-accreting CVs show pulsations within a well-defined temperature range, the so-called, instability strip at 10900 K -- 12200 K (\\citealp{2006AJ....132..831G}). We find that the effective temperature in J1507 is well above the instability strip for non-accreting pulsating WDs. However, it is already becoming clear that non-radially pulsating WDs in CVs are found across a wider temperature range, often towards higher temperatures (\\citealt{2010ApJ...710...64S}). The amplitudes of these pulsations are expected to vary with wavelength, and a search for the UV counterpart of the optical WD pulsations will be presented in Uthas et al. (in preparation). The fact that non-radial pulsations are present in J1507 is certainly interesting and potentially important since a higher metallicity in the outer envelope of the accretors might be able to push the instability strip towards higher temperatures (\\citealt{2006ApJ...643L.119A}). In this context, it is interesting to note that J1507 has a very low metallicity, but is nevertheless managing to pulsate at an effective temperature of above 14000 K.\n   \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Training Details of our Full Model and Ablations}\n\\label{appsec:models}\n\\subsection{Tracking Models}\nThe hidden size of the instruction encoder is 100, the embedding sizes of action functions and entities are 30. We use dropout with a rate of 0.3 before any non-recurrent fully connected layers \\cite{dropout}. We use the Adam optimizer \\citep{adam} with a learning rate of .001 and decay by a factor of 0.1 if we see no improvement on validation loss over three epochs. We stop training early if the development loss does not decrease for five epochs. The batch size is 64. We use two instruction encoders, one for the entity selector, and one for the action selector. Word embeddings and entity embeddings are initialized with skipgram embeddings \\citep{skipgram1,skipgram2} using a word2vec model trained on the training set. We use a vocabulary size of 7358 for words, and 2996 for entities. Gradients with respect to the coverage loss (Eq.~\\ref{eq:train:usage}) are only backpropagated in steps where no entity is annotated as being selected. To account for the false negatives in the training data due to the heuristic generation of the labels, gradients with respect to the entity selection loss are zeroed when no entity label is present.\n\n\\subsection{Generation Model}\n\\label{app:ssec:gen}\nThe hidden size of the context encoder is 200. The hidden size of the state vector encoder is 200. State vectors have dimensionality 30 (the same as in the neural process network). Dropout of 0.3 is used during training in the decoder. The context and state representations are projected jointly using an element-wise product followed by a linear projection \\cite{hadamard}. Both encoders and the decoder are single layer. The learning rate is 0.0003 initially and is halved every 5 epochs. The model is trained with the Adam optimizer.\n\n\\section{Training Details of Baselines}\n\\label{appsec:baseline}\n\\subsection{Tracking Baselines}\n\\label{app:ssec:tracking}\n\\paragraph{Joint Gated Recurrent Unit}\nThe hidden state of the GRU is 100. We use a dropout with a rate of 0.3 before any non-recurrent fully connected layers. We use the Adam optimizer with a learning rate of .001 and decay by a factor of 0.1 if we see no improvement on validation loss over a single epoch. We stop training early if the development loss does not decrease for five epochs. The batch size is 64. We use encoders, one for the entity selector, and one for the state change predictors. Word embeddings are initialized with skipgram embeddings using a word2vec model trained on the training set. We use a vocabulary size of 7358 for words.\n\n\\paragraph{Recurrent Entity Networks}\nMemory cells are tied to the entities in the document. For a recipe with 12 ingredients, 12 entity cells are initialized. All hyperparameters are the same as the in the bAbI task from \\cite{ren}. The learning rate start at 0.01 and is halved every 25 epochs. Entity cells and word embeddings are 100 dimensional. The encoder is a multiplicative mask initialized the same as in \\cite{ren}. Intermediate supervision from the weak labels is provided to help predict entities. A separate encoder is used for computing the attention over memory cells and the content to write to the memory. Dropout of 0.3 is used in the encoders. The batch size is 64. We use a vocabulary size of 7358 for words, and 2996 for entities.\n\n\\subsection{Generation Baselines}\n\\paragraph{Seq2seq}\n\nThe encoder and decoder are both single-layer GRUs with hidden size 200. We use dropout with probability 0.3 in the decoder. We train with the Adam optimizer starting with a learning rate 0.0003 that is halved every 5 epochs. The encoder is bidirectional. The model is trained to minimize the negative loglikelihood of predicting the next word.\n\n\\paragraph{Attentive Seq2seq}\n\nThe encoder is the same as in the seq2seq baseline. A multiplicative attention between the decoder hidden state and the context vectors is used to compute the attention over the context at every decoder time step. The model is trained with the same learning rate, learning schedule and loss function as the seq2seq baseline.\n\n\\paragraph{EntNet Generator}\n\nThe model is trained in the same way as the NPN generator model in Appendix~\\ref{app:ssec:gen} except that the state representations used as input are produced from by EntNet baseline described in Section~\\ref{ssec:exps:intrinsic} and Appendix~\\ref{app:ssec:tracking}.\n\n\\section{Annotations}\n\\label{appsec:ann}\n\n\\subsection{Annotating State Changes} \n\\label{appssec:annsc}\nWe provide workers with a verb, its definition, an illustrative image of the action, and a set of sentences where the verb is mentioned. Workers are provided a checklist of the six state change types and instructed to identify which of them the verb causes. They are free to identify multiple changes. Seven workers annotate each verb and we assign a state change based on majority vote. Of the set of 384 verbs extracted, only 342 have a state change type identified with them. Of those, 74 entail multiple state change types.\n\n\\subsection{Annotating End States} \n\\label{appssec:annes}\nWe give workers a verb, a state change type, and an example with the verb and ask them to provide an end state for the ingredient the verb is applied to in the example. We then use the answers to manually aggregate a set of end states for each state change type. These end states are used as labels when the model predicting state changes. For example, a \\textsc{location} change might lead to an end state of ``pan,'' ``pot'', or ``oven.'' End states for each state change type are provided in Table~\\ref{table:end_states}.\n\n\\begin{table}[H]\n    \\centering\n    \\begin{tabular}{l | l}\n        State Change Type & End States \\\\\n        \\hline\n        Temperature & hot; cold; room \\\\\n        Composition & composed; not composed\\\\\n        Cleanliness & clean; dirty; dry\\\\\n        Cookedness & cooked; raw \\\\\n        Shape & molded; hit; deformed; separated \\\\\n        Location & pan, pot, cupboard, screen, scale, \\\\\n        &garbage, 260 more \\\\\n    \\end{tabular}\n    \\caption{End states for each state change type}\n    \\label{table:end_states}\n\\end{table}\n\n\\subsection{Annotating Development and Test Sets} \n\\label{appssec:devtest}\nAnnotators are instructed to note any entities that undergo one of the six state changes in each step, as well as to identify new combinations of ingredients that are created. For example, the sentence ``Cut the tomatoes and add to the onions'' would involve a \\textsc{shape} change for the tomatoes and a combination created from the ``tomatoes'' and ``onions''. In a separate task, three workers are asked to identify the actions performed in every sentence of the development and test set recipes. If an action receives a majority vote that it is performed, it is included in the annotations.\n\n\\section{Additional Results}\n\n\\subsection{Removing Training data}\n\n\\begin{table}\n\\centering\n\\begin{tabular}{l || r | r | r || r | r  }\n\\multirow{2}{*}{Model} & \\multicolumn{3}{c}{Entity Selection} & \\multicolumn{2}{c}{State Change}  \\\\\n                      & {\\sc F1} & {\\sc UR} & {\\sc CR} & {\\sc F1} & {\\sc Acc}  \\\\\n\\hline\\hline\n25\\% training data kept & 54.34 & 75.71 & 21.17 & 2.52  & 50.23   \\\\\n50\\% training data kept & 55.12 & 76.04 & 19.05 & 36.34  & 54.66 \\\\\n75\\% training data kept & 56.64 & 76.03 & 21.00 & 48.86  & 57.62   \\\\\n\\hline\n100\\% training data & 56.84 & 74.98 & 21.14 & 50.56 & 57.87  \\\\\n\\end{tabular}\n\\caption{Results for entity selection and state change selection on the development set when randomly dropping a percentage of the training labels}\n\\label{table:tp_remove}\n\\end{table}\n\\section{Conclusion}\n\\vspace{-1mm}\n\\label{sec:conclusions}\nWe introduced the \\emph{Neural Process Network} for modeling a process of actions and their causal effects on entities by learning action transformations that change entity state representations. The model maintains a recurrent memory structure to track entity states and is trained to predict the state changes that entities undergo. Empirical results demonstrate that our model can learn the causal effects of action semantics in the cooking domain and track the dynamic state changes of entities, showing advantages over competitive baselines.\n\n\\section*{Acknowledgments}\nWe thank Yonatan Bisk, Peter Clark, Bhavana Dalvi, Niket Tandon, and Yuyin Sun for helpful discussions at various stages of this work. This research was supported in part by NSF (IIS-1524371, IIS-1714566, NRI-1525251), DARPA under the CwC program through the ARO (W911NF-15-1-0543), Samsung Research, and gifts by Google and Facebook.\n\\section{Experimental Setup}\n\\label{sec:experiments}\n\nWe evaluate our model on a set of intrinsic tasks centered around tracking entities and state changes in recipes to show that the model can simulate preliminary dynamics of the recipe task. Additionally, we provide a qualitative analysis of the internal components of our model. Finally, we evaluate the quality of the states encoded by our model on the extrinsic task of generating future steps in a recipe. \n\n\\subsection{Intrinsic Evaluation - Tracking}\n\\label{ssec:exps:intrinsic}\nIn the tracking task, we evaluate the model's ability to identify which entities are selected and what changes have been made to them in every step. We break the tracking task into two separate evaluations, entity selection and end state prediction, and also investigate whether the model learns internal representations that approximate recipe dynamics.\n\\vspace{-3mm}\n\\paragraph{Metrics}\nIn the entity selection test, we report the F1 score of choosing the correct entities in any step. A selected entity is defined as one whose attention weight $a_i$ is greater than 50\\% (\\S\\ref{ssec:model:aes}). Because entities may be harder to predict when they have been combined with other entities (e.g., the mixture may have a new name), we also report the recall for selecting combined (CR) and uncombined (UR) entities.\nIn the end state prediction test, we report how often the model correctly predicts the state change performed in a recipe step and the resultant end state. This score is then scaled by the accuracy of predicting which entities were changed in that same step. We report the average F1 and accuracy across the six state change types. \n\\vspace*{-3mm}\n\\paragraph{Baselines} \nWe compare our models against two baselines. First, we built a GRU model that is trained to predict entities and state changes independently. This can be viewed as a bare minimum network with no action representations or recurrent entity memory. \nThe second baseline is a Recurrent Entity Network \\citep{ren} with changes to fit our task. First, the model can tie memory cells to a subset of the full list of entities so that it only considers entities that are present in a particular recipe. Second, the entity distribution for writing to the memory cells is re-used when we query the memory cells. The normalized weighted average of the entity cells is used as the input to the state predictors. The unnormalized attention when writing to each cell is used to predict selected entities. Both baselines are trained with entity selection and state change losses (\\S\\ref{ssec:train:train}).\n\\vspace*{-3mm}\n\\paragraph{Ablations} We report results on six ablations. First, we remove the recurrent attention (Eq.~\\ref{eq:choice}). The model only predicts entities using the current encoder hidden state. In the second ablation, the model is trained with no coverage penalty (Eq.~\\ref{eq:train:usage}). The third ablation prunes the connection from the action selector $w_p$ to the entity selector (Eq.~\\ref{eq:bilinear}). We also explore not pretraining the action selector. Finally, we look at two ablations where we intialize the action embeddings with vectors from a skipgram model. In the first, the model operates normally, and in the second, we do not allow gradients to backpropagate to the action embeddings, updating only the mapping tensor $W_4$ instead (Eq.~\\ref{eq:comp_knowl}).\n\n\\begin{table}\n\\centering\n\\begin{tabular}{l | r  r  r | r  r  }\n\\multirow{2}{*}{Model} & \\multicolumn{3}{c}{Entity Selection} & \\multicolumn{2}{c}{State Change}  \\\\\n                       & {\\sc F1} & {\\sc UR} & {\\sc CR} & {\\sc F1} & {\\sc Acc}  \\\\\n\\hline\\hline\n2-layer LSTM Entity Recognizer                  & 50.98 & 74.03 & 13.33 & - & - \\\\\nAdapted Gated Recurrent Unit                   & 45.94 & 67.69 & 7.74 & 41.16  & 52.69    \\\\\nAdapted Recurrent Entity Network               & 48.57 & 71.88 & 9.87 & 42.32  & 53.47   \\\\\n\\hline\n- Recurrent Attention (Eq.~\\ref{eq:choice})   & 48.91 & 72.32 & 12.67 & 42.14  & 50.48   \\\\\n- Coverage Loss (Eq.~\\ref{eq:train:usage})    & 55.18 & 73.98 & 20.23 & 44.44  & \\textbf{55.20} \\\\\n- Action Connections (Eq.~\\ref{eq:bilinear})  & 54.85 & 73.54 & 20.03 & 44.05  & 54.81   \\\\\n- Action Selector Pretraining & 54.91 & 73.87 & 20.30 & 44.28 & 55.00 \\\\\n\\hline\n+ Pretrained Action Embeddings & 55.16 & 74.02 & 20.32 & 44.02 & 55.03 \\\\\n- Action Embedding Updates & 53.79 & 70.77 & 18.60 & 44.27 & 55.02 \\\\\n\\hline\nFull Model                                     & \\textbf{55.39} & \\textbf{74.88} & \\textbf{20.45} & \\textbf{44.65}  & 55.07  \\\\\n\\end{tabular}\n\\caption{Results for entity selection and state change selection}\n\\label{table:tracking}\n\\end{table}\n\n\\subsection{Extrinsic Evaluation - Generation} \nThe generation task tests whether our system can produce the next step in a recipe based on the previous steps that have been performed. The model is provided all of the previous steps as context.\n\\vspace*{-3mm}\n\\paragraph{Metrics} We report the combined BLEU score and ROUGE score of the generated sequence relative to the reference sequence. Each candidate sequence has one reference sentence. Both metrics are computed at the corpus-level. Also reported are ``VF1'', the F1 score for the overlap of the actions performed in the reference sequence and the verbs mentioned in the generated sequence, and ``SF1'', the F1 score for the overlap of end states annotated in the reference sequence and predicted by the generated sequences. End states for the generated sequences are extracted using the lexicon from Section~\\ref{ssec:states} based on the actions performed in the sentence.\n\\vspace*{-3mm}\n\\paragraph{Setup} To apply our model to the task of recipe step generation, we input the context sentences through the neural process network~and record the entity state vectors once the entire context has been read (\\S\\ref{ssec:model:pm}). These vectors can be viewed as a snapshot of the current state of the entities once the preceding context has been simulated inside the neural process network. We encode these vectors using a bidirectional GRU \\citep{cho2014learning} and take the final time step hidden state $\\mathbb{e}_I$. A different GRU encodes the context words in the same way (yielding $\\mathbb{h}_T$) and the first hidden state input to the decoder is computed using the projection function:\n\n\\begin{equation}\n{\\tilde h}_0 = W_{5}(\\mathbb{e}_I \\circ \\mathbb{h}_T)    \n\\end{equation}\n\nwhere $\\circ$ is the Hadamard product\nbetween the two encoder outputs. All models are trained by minimizing the negative loglikelihood of predicting the next word for the full sequence. Implementation details can be found in Appendix~\\ref{appsec:models}. \n\\vspace*{-3mm}\n\\paragraph{Baselines} For the generation task, we use three baselines: a seq2seq model with no attention \\citep{seq2seq}, an attentive seq2seq model \\citep{bahdanau2014neural},  and a similar variant as our NPN generator, except where the entity states have been computed by the Recurrent Entity Network (EntNet) baseline (\\S\\ref{ssec:exps:intrinsic}). Implementation details for baselines can be found in Appendix~\\ref{appsec:baseline}.\n\n\n\\section{Experimental Results}\n\n\\subsection{Extrinsic Evaluations}\n\\label{ssec:exps:gen}\n\n\\begin{table}\n    \\centering\n    \\begin{tabular}{l | r | r | r | r  }\n        Model & {\\sc BLEU} & {\\sc ROUGE-L} & {\\sc VF1} & {\\sc SF1} \\\\\n        \\hline\\hline\n        Vanilla Seq2Seq        & 2.81 & 33.00 & 16.17 & 40.21 \\\\\n        Attentive Seq2Seq      & 2.83 & 33.18 & 16.97 & 41.43 \\\\\n       \n        EntNet Generator          & 2.30 & 32.71 & 17.53 & 42.43 \\\\\n        \\hline\n        NPN Generator          & \\textbf{3.74} & \\textbf{35.64} & \\textbf{20.12} & \\textbf{43.40}  \\\\\n    \\end{tabular}\n    \\caption{Generation Results}\n    \\label{table:gen}\n\\end{table}\n\n\\paragraph{Recipe Step Generation} Our results in Table~\\ref{table:gen} indicate that sequences generated using the neural process network entity states as additional input yield higher scores than competitive baselines. The entity states allow the model to predict next steps conditioned on a representation of the world being simulated by the neural process network. \nAdditionally, the higher VF1 and SF1 scores indicate that the model is indeed using the extra information to better predict the actions that should follow the context provided. Example generations for each baselines\nfrom the dev set are provided in Table~\\ref{table:genex}, showing that the NPN generator can use information about ingredient states to reason about the most likely next step. The first and second examples are interesting as it shows that the NPN-aware model has learned to condition on entity state -- knowing that raw butter will likely be melted or that a cooked flan must be refrigerated.\nThe third example is also interesting because the model learns that cooked vegetables such as squash will sometimes be drained, even if it is not relevant to this recipe because the squash is steamed. The seq2seq and EntNet baselines, meanwhile, output reasonable sentences given the immediate context, but do not exhibit understanding of global patterns.\n\\subsection{Action Embeddings and Compositionality}\n\\vspace{-3mm}\n\\paragraph{Action Embeddings} In our model, each action is assigned its own embedding, but many actions induce similar changes in the physical world (e.g.,``cut\" and ``slice\"). After training, we compute the pairwise cosine similarity between each pair of action embeddings. In Table~\\ref{table:actions}, we see that actions that perform similar functions are neighbors in embedding space, indicating the model has captured certain semantic properties of these actions. Learning action representations through the state changes they induce has allowed the model to cluster actions by their transformation functions.\n\n\\begin{figure}\n\\small\n\\begin{floatrow}\n\\capbtabbox{\n  \\begin{tabular}{c | c} \\hline\n  Action & Nearest Neighbor Actions \\\\\n\\hline\\hline\ncut & slice, split, snap, slash, carve, slit, chop \\\\\nboil & cook, microwave, fry, steam, simmer\\\\\nadd & sprinkle, mix, reduce, splash, stir, dust \\\\\nwash & rinse, scrub, refresh, soak, wipe, scale \\\\\nmash & spread, puree, squeeze, liquefy, blend \\\\\nplace & ease, put, lace, arrange, leave \\\\\nrinse & wash, refresh, soak, wipe, scrub, clean\\\\ \nwarm & reheat, ignite, heat, light, crisp, preheat\\\\\nsteam & microwave, crisp, boil, parboil, heat \\\\\nsprinkle & top, pat, add, dip, salt, season \\\\\ngrease & coat, rub, dribble, spray, smear, line\\\\\n  \\end{tabular}\n}{\n  \\caption{Most similar actions based on cosine similarity of action embeddings}%\n  \\label{table:actions}\n}\n\\ffigbox{%\n  \\includegraphics[width=6.75cm]{composition_graph}\n}{%\n  \\caption{Change in cosine similarity of entity state embeddings}\n  \\label{fig:comp}\n}\n\\end{floatrow}\n\\end{figure}\n\n\\vspace{-3mm}\n\\paragraph{Entity Compositions} When individual entities are combined into new constructs, our model averages their state embeddings (Eq.~\\ref{eq:avg}), applies an action embedding to them (Eq.~\\ref{eq:comp_knowl}), and writes them to memory (Eq.~\\ref{eq:eu}). The state embeddings of entities that are combined should be overwritten by the same new embedding. In Figure~\\ref{fig:comp}, we present the percentage increase in cosine similarity for state embeddings of entities that are combined in a sentence (blue) as opposed to the percentage increase for those that are not (red bars). While the soft attention mechanism for entity selection allows similarities to leak between entity embeddings, our system is generally able to model the compositionality patterns that result from entities being combined into new constructs.\n\n\\section{Introduction}\n\\label{sec:intro}\n\nUnderstanding procedural text such as instructions or stories requires anticipating the implicit causal effects of actions on entities. For example,\ngiven instructions such as ``\\emph{add} blueberries to the muffin mix, then \\emph{bake} for one half hour,'' an intelligent agent must be able to anticipate a number of entailed facts (e.g., the blueberries are now in the oven; their \"temperature\" will increase). While this common sense reasoning is trivial for humans, most natural language understanding algorithms do not have the capacity to reason about causal effects not mentioned directly in the surface strings \\citep{lexinf,jia2017adversarial,distrepready}.\n\n\\begin{wrapfigure}{R}{0.48\\textwidth}\n\\centering\n\\includegraphics[width=\\textwidth]{Intro_Figure_v2.pdf}\n\\caption{The {\\it process} is a narrative of entity state changes induced by actions. In each sentence, these state changes are induced by simulated actions and must be remembered. }\n    \\label{fig:intro}\n\\end{wrapfigure}\n\nIn this paper, we introduce Neural Process Networks, a procedural language understanding system that tracks common sense attributes through neural simulation of action dynamics. \nOur network models interpretation of natural language instructions as a \\emph{process} of actions and their cumulative effects on entities. More concretely, reading one sentence at a time, our model attentively selects what actions to execute on which entities, and remembers the state changes induced with a recurrent memory structure. In Figure~\\ref{fig:intro}, for example, our model indexes the ``tomato\" embedding, selects the ``wash'' and ``cut'' functions and performs a computation that changes the ``tomato'' embedding so that it can reason about attributes such as its ``\\textsc{shape}'' and ``\\textsc{cleanliness}''.\n\nOur model contributes to a recent line of research that aims to model aspects of world state changes, such as language models and machine readers with explicit entity representations \\citep{ren,reflm,delm}, as well as other more general purpose memory network variants \\citep{memnets,e2ememnets,cbt,qrn}. This \\emph{world-centric} modeling of procedural language (i.e., understanding by simulation) abstracts away from the surface strings, complementing \\emph{text-centric} modeling of language, which focuses on syntactic and semantic labeling of surface words (i.e., understanding by labeling).\n\nUnlike previous approaches, however, our model also learns explicit action representations as functional operators\n(See Figure~\\ref{fig:intro}). \nWhile representations of action semantics could be acquired through an embodied agent that can see and interact with the world \\citep{actionenv}, we propose to learn these representations from text. In particular, we require the model to be able to \\emph{explain} the causal effects of actions by predicting natural language attributes about entities such as \\textsc{``location''} and \\textsc{``temperature''}. The model adjusts its representations of actions based on errors it makes in predicting the resultant state changes to attributes. This textual simulation allows us to model aspects of action causality that are not readily available in existing simulation environments. Indeed, most virtual environments offer limited aspects of the world -- with a primary focus on spatial relations \\citep{actionenv,renv2,envtorques}. They leave out various other dimensions of the world states that are implied by diverse everyday actions such as ``dissolve'' (change of \\textsc{``composition''}) and ``wash'' (change of \\textsc{``cleanliness''}). \n\n\nEmpirical results demonstrate that parametrizing explicit action embeddings provides an inductive bias that allows the neural process network~to learn more informative context representations for understanding and generating natural language procedural text. In addition, our model offers more interpretable internal representations and can reason about the unstated causal effects of actions explained through natural language descriptors.  \nFinally, we include a new dataset with fine-grained annotations on state changes, to be shared publicly, \nto encourage future research in this direction.\n\n\n\n\\section{\\MODELNAME}\nThe neural process network~is an interpreter that reads in natural language sentences, one at a time, and simulates the process of actions being applied to relevant entities through learned representations of actions and entities.\n\\vspace*{-1mm}\n\\subsection{Overview and Notation}\nThe main component of the neural process network~is the simulation module (\\S\\ref{ssec:model:pm}), a recurrent unit whose internals simulate the effects of actions being applied to entities. A set of $V$ actions is known a priori and an embedding is initialized for each one, $\\mathcal{F} = \\{f_1, ... f_V\\}$. Similarly, a set of $I$ entities is known and an embedding is initialized for each one: $\\mathcal{E} = \\{e_1, ... e_I\\}$. Each $e_i$ can be considered to encode information about state attributes of that entity, which can be extracted by a set of state predictors (\\S\\ref{ssec:model:sc}). As the model reads text, it ``applies\" action embeddings to the entity vectors, thereby changing the state information encoded about the entities.\n\nFor any document $d$, an initial list of entities $I_d$ is known and $\\mathcal{E}_d = \\{e_i \\vert i \\in I_d\\} \\subset \\mathcal{E}$ entity state embeddings are initialized. As the neural process network~reads a sentence from the document, it selects a subset of both $\\mathcal{F}$ (\\S\\ref{ssec:model:as}) and $\\mathcal{E}_d$ (\\S\\ref{ssec:model:aes}) based on the actions performed and entities affected in the sentence. The entity state embeddings are changed by the action and the new embeddings are used to predict end states for a set of state changes (\\S\\ref{ssec:model:sc}). The prediction error for end states is backpropagated  to the action embeddings, learning action representations that model the simulation of desired causal effects on entities. This process is broken down into five modules below. Unless explicitly defined, all $W$ and $b$ variables are parametrized linear projections and biases. We use the notation $\\{e_i\\}_t$ when referring to the values of the entity embeddings before processing sentence $s_t$.\n\\vspace*{-2mm}\n\\subsection{Sentence Encoder} \n\\label{ssec:model:se}\nGiven a sentence $s_t$, a Gated Recurrent Unit \\citep{cho2014learning} encodes each word and outputs its last hidden vector as a sentence encoding $h_t$ \\citep{seq2seq}.\n\\vspace*{-2mm}\n\\begin{figure*}[t!]\n    \\centering\n    \\includegraphics[width=0.9\\linewidth]{Model_Figure_key_v4.pdf}\n    \\caption{\\small Model Summary. The sentence encoder converts a sentence to a vector representation, $h_t$. The action selector and entity selector use the vector representation to choose the actions that are applied and the entities that are acted upon in the sentence. The simulation module indexes the action and entity state embeddings, and applies the transformation to the entities. The state predictors predict the new state of the entities if a state change has occurred. Equation references are provided in parentheses.}\n    \\label{fig:model_1}\n\\end{figure*}\n\\label{sec:model} \n\\subsection{Action Selector} \n\\label{ssec:model:as}\nGiven $h_t$ from the sentence encoder, the action selector (bottom left in Fig.~\\ref{fig:model_1}) contextually determines which action(s) from $\\mathcal{F}$ to execute. For example, if the input sentence is \\emph{``wash and cut beets''}, both $f_{wash}$ and $f_{cut}$  must be selected. To account for multiple actions, we make a soft selection over $\\mathcal{F}$, yielding a weighted sum of the selected action embeddings $\\bar{f_t}$:\n\\begin{equation}\n\\begin{aligned}\nw_p &= \\textrm{MLP}(h_t) \\\\\n\\bar w_p &= \\frac{w_p}{\\sum_{j=1}^V w_{p_j}}  \\\\\n\\bar f_t &= \\bar w_p^\\intercal \\mathcal{F} \\label{eq:as}\n\\end{aligned}\n\\end{equation}\n\\noindent where MLP is a parametrized feed-forward network with a sigmoid activation and $w_p \\in \\mathbb{R}^V$ is the attention distribution over $V$ possible actions (\\S\\ref{ssec:states}). We compose the action embedding by taking the weighted average of the selected actions. \n\\subsection{Entity Selector}\n\\label{ssec:model:aes}\n\\paragraph{Sentence Attention} Given $h_t$ from the sentence encoder, the entity selector chooses relevant entities using a soft attention mechanism:\n\\begin{equation}\n\\begin{aligned}\n\\tilde h_t &= \\text{ReLU}(W_1h_t + b_1)\\\\\nd_i &= \\sigma(e_{i_0}^\\intercal W_2 [\\tilde h_t; w_p]) \\label{eq:bilinear}\n\\end{aligned}\n\\end{equation}\nwhere  $W_2$ is a bilinear mapping, $e_{i_0}$ is a unique key for each entity (\\S\\ref{ssec:model:pm}), and $d_i$ is the attention weight for entity embedding $e_i$. For example, in \\emph{``wash and cut beets and carrots''}, the model should select $e_{beet}$ and $e_{carrot}$.\n\\vspace{-3mm}\n\\paragraph{Recurrent Attention} While sentence attention would suffice if entities were always explicitly mentioned, natural language often elides arguments or uses referent pronouns. As such, the module must be able to consider entities mentioned in previous sentences. Using $\\tilde h_t$, the model computes a soft choice over whether to choose affected entities from this step's attention $d_i$ or the previous step's attention distribution.\n\\begin{equation}\n\\begin{aligned}\nc = softmax(W_3 \\tilde h_t + b_3) \\\\\na_{i_{t}} = c_1 d_i + c_2 a_{i_{t-1}} + c_3 \\mathbf{0} \\label{eq:choice}\n\\end{aligned}\n\\end{equation}\n\\noindent where $c \\in \\mathbb{R}^3$ is the choice distribution, $a_{i_{t-1}}$ is the previous sentence's attention weight for each entity, $a_{i_{t}}$ is the final attention for each entity, and $\\mathbf{0}$ is a vector of zeroes (providing the option to not change any entity). Prior entity attentions can propagate forward for multiple steps.\n\\subsection{Simulation Module} \n\\label{ssec:model:pm}\n\\paragraph{Entity Memory} A unique state embedding $e_{i}$ is initialized for every entity $i$ in the document. A unique key to index each embedding $e_{i_0}$ is set as the initial value of the embedding \\citep{ren,kvnet}. After the model reads $s_t$, it modifies $\\{e_{i}\\}_t$ to reflect changes influenced by actions. At every time step, the entity memory receives the attention weights from the entity selector, normalizes them and computes a weighted average of the relevant entity state embeddings:\n\n\\begin{minipage}{.5\\linewidth}\n\\begin{equation}\n \\alpha_{i} = \\frac{a_i}{\\sum_{j=1}^{I_d} a_j} \\label{eq:attn}\n\\end{equation}\n\\end{minipage}%\n\\begin{minipage}{.5\\linewidth}\n\\begin{equation}\n  \\bar e_t = \\sum_{j=1}^{I_d} \\alpha_i e_i \\label{eq:avg}\n\\end{equation}\n\\end{minipage}\n\\vspace{-1mm}\n\\paragraph{Applicator} \nGiven the action summary embedding $\\bar{f_t}$ and the entity summary embedding $\\bar{e_t}$, the applicator (middle right in Fig.~\\ref{fig:model_1}) applies the selected actions to the selected entities, and outputs the new proposal entity embedding $k_t$.\n\\begin{equation}\nk_t = \\text{ReLU}(\\bar f_t W_4\\bar e_t + b_4) \\label{eq:comp_knowl}\n\\end{equation}\nwhere $W_4$ is a third order tensor projection. The vector $k_t$ is the new representation of the entity $\\bar e_t$ after the applicator simulates the action being applied to it. \n\\vspace{-3mm}\n\\paragraph{Entity Updater} \nThe entity updater interpolates the new proposal entity embedding $k_t$ and the set of current entity embeddings $\\{e_i\\}_t$:\n\\begin{equation}\ne_{i_{t+1}} = a_{i_t} k_t + ( 1 - a_{i_t}) e_{i_t} \\label{eq:eu}\n\\end{equation}\n\\noindent yielding an updated set of entity embeddings $\\{e_i\\}_{t+1}$. Each embedding is updated proportional to its entity's unnormalized attention $a_i$, allowing the model to completely overwrite the state embedding for any entity. For example, in the sentence \\textit{``mix the flour and water,''} the embeddings for $e_{flour}$ and $e_{water}$ must both be overwritten by $k_t$ because they no longer exist outside of this new composition.\n\\subsection{State Predictors} \n\\label{ssec:model:sc}\nGiven the new proposal entity embedding $k_t$, the state predictor (bottom right in Fig.~\\ref{fig:model_1}) predicts changes to the resulting entity embedding $k_t$ along the following six dimensions: location, cookedness, temperature, composition, shape, and cleanliness. Discrete multi-class classifiers, one for each dimension, take in $k_t$ and predict a unique end state for their corresponding state change type:\n\\begin{equation}\nP(Y_s | k_t) = softmax(W_{s}k_t + b_s)\n\\end{equation}\n\\noindent  For location changes, which require contextual information to predict the end state, $k_t$ is concatenated with the original sentence representation $h_t$ to predict the final state.\n\\section{Related Work}\n\\label{sec:related}\n\\vspace{-1mm}\nRecent studies in machine comprehension have used a neural memory component to store a running representation of processed text \\citep{memnets,e2ememnets,cbt,qrn}. While these approaches map text to memory vectors using standard neural encoder approaches, our model, in contrast, directly interprets text in terms of the effects actions induce in entities, providing an inductive bias for learning how to represent stored memories. More recent work in machine comprehension also sought to couple the memory representation with tracking entity states \\citep{ren}. Our work seeks to provide a relatively more structured representation of domain-specific action knowledge to provide an inductive bias to the reasoning process.\n\nNeural Programmers \\citep{neuralprogrammer, neuralprogrammer2} have also used functions to simulate reasoning, by building a model to select rows in a database and applying operation on those selected rows. While their work explicitly defined the effect of a number of operations for those rows, we provide a framework for learning representations for a more expansive set of actions, allowing the model to learn representations for how actions change the state space.\n\n\\indent Works on instructional language studied the task of building discrete graph representations of recipes using probabilistic models \\citep{actiongraph, mori2014flow, mori2012machine}. We propose a complementary new model by integrating action and entity relations into the neural network architecture and also address the additional challenge of tracking the state changes of the entities.\n\nAdditional work in tracking states with visual or multimodal context has focused on 1) building graph representations for how entities change in goal-oriented domains \\citep{chaiphysical,chaijointly,andor} or 2) tracking visual state changes based on decisions taken by agents in environment simulators such as videos or games \\citep{renv2,envtorques,actionenv}. Our work, in contrast, models state changes in embedding space using only text-based signals to map real-world actions to algebraic transformations. \n\\subsection{State Change Knowledge}\n\\label{ssec:states}\n\n\\begin{table}\n\\small\n\\centering\n\\caption{Example actions, the state changes they induce, and the possible end states}\n\\begin{tabular}{l | l | l}\nAction & State Change Types & End States  \\\\\n\\hline\n\\hline\nbraise & \\textsc{cookedness}; \\textsc{temperature} & \\textsc{cooked}; \\textsc{hot}\\\\\nchill & \\textsc{temperature} & \\textsc{cold} \\\\\nknead & \\textsc{shape} & \\textsc{molded}\\\\\nwash & \\textsc{cleanliness} & \\textsc{clean}\\\\\ndissolve & \\textsc{composition} & \\textsc{composed}\\\\\nrefrigerate & \\textsc{temperature}; \\textsc{location} & \\textsc{cold}; \\textsc{refrigerator} \\\\\nslice & \\textsc{shape} & \\textsc{separated}\n\\end{tabular}\n\\label{table:state_types}\n\\end{table}\n\nIn this work we focus on physical action verbs in cooking recipes. We manually collect a set of 384 actions such as \\textit{cut, bake, boil, arrange}, and {\\it place}, \norganizing their causal effects along the following predefined dimensions: \\textsc{location, cookedness, temperature, shape, cleanliness} and \\textsc{composition}. The textual simulation operated by the model induces state changes along these dimensions by applying actions functions from the above set of 384. For example, \\textit{cut} entails a change in \\textsc{shape}, while {\\it bake} entails a change in \\textsc{temperature, cookedness}, and even \\textsc{location}. We annotate the state changes each action induces, as well as the end state of the action, using Amazon Mechanical Turk. The set of possible end states for a state change can range from 2 for binary state changes to more than 200 (See Appendix~\\ref{appsec:ann} for details). Table~\\ref{table:state_types} provides examples of annotations in this action lexicon.\n\\subsection{Intrinsic Evaluations}\n\\paragraph{Entity Selection} As shown in Table~\\ref{table:tracking}, our full model outperforms all baselines at selecting entities, with an F1 score of 55.39\\%. The ablation study shows that the recurrent attention, coverage loss, action connections and action selector pretraining improve performance. Our success at predicting entities extends to both uncomposed entities, which are still in their raw forms (e.g., melt the {\\it butter} $\\rightarrow$ butter), and composed entities, in which all of the entities that make up a composition must be selected. For example, in a \\emph{Cooking lasagna} recipe, if the final step involves baking the prepared lasagna, the model must select all the entities that make up the lasagna (e.g., lasagna sheets, beef, tomato sauce). In Table~\\ref{table:examples}, we provide examples of our model's ability to handle complex cases such as compositional entities (Ex. 1, 3), and elided arguments over long time windows (Ex. 2). We also provide examples where the model fails to select the correct entities because it does not identify the mapping between a reference construct such as ``pizza\" (Ex. 4) or ``dough\" (Ex. 5) and the set of entities that composes it, showcasing the difficulty of selecting the full set for a composed entity.\n\\vspace*{-3mm}\n\n\n\n\\paragraph{State Change Tracking} In Table~\\ref{table:tracking}, we show that our full model outperforms competitive baselines such as Recurrent Entity Networks \\citep{ren} and jointly trained GRUs. While the ablation without the coverage loss shows higher accuracy, we attribute this to the fact that it predicts a smaller number of total state changes. Interestingly, initializing action embeddings with skipgram vectors and locking their values shows relatively high performance, indicating the potential gains in using powerful pretrained representations to represent actions.\n\\section{Training}\n\\label{sec:train}\n\\input{states}\n\\subsection{Dataset}\n\\label{ssec:data}\nFor learning and evaluation, we use a subset of the Now You're Cooking dataset \\citep{checklist}.\nWe chose 65816 recipes for training, 175 recipes for development, and 700 recipes for testing. For the development and test sets, crowdsourced workers densely annotate actions, entities and state changes that occur in each sentence so that we can tune hyperparameters and evaluate on gold evaluation sets. Annotation details are provided in Appendix~\\ref{appssec:devtest}. \n\\vspace*{-1mm}\n\\subsection{Component-wise Training}\n\\label{ssec:train:train}\nThe neural process network~is trained by jointly optimizing multiple losses for the action selector, entity selector, and state change predictors. Importantly, our training scheme uses weak supervision because dense annotations are prohibitively expensive to acquire at a very large scale. Thus, we heuristically extract verb mentions from each recipe step and assign a state change label based on the state changes induced by that action (\\S\\ref{ssec:states}). Entities are extracted similarly based on string matching between the instructions and the ingredient list.\nWe use the following losses for training:\n\\vspace*{-3mm}\n\\paragraph{Action Selection Loss}\nUsing noisy supervision, the action selector is trained to minimize the cross-entropy loss for each possible action, allowing multiple actions to be chosen at each step if multiple actions are mentioned in a sentence. The MLP in the action selector (Eq.~\\ref{eq:as}) is pretrained.\n\\vspace*{-3mm}\n\\paragraph{Entity Selection Loss} Similarly, to train the attentive entity selector, we minimize the binary cross-entropy loss of predicting whether each entity is affected in the sentence.\n\\vspace*{-3mm}\n\\paragraph{State Change Loss} For each state change predictor, we minimize the negative loglikelihood of predicting the correct end state for each state change.\n\n\\vspace*{-3mm}\n\\paragraph{Coverage Loss}\n\\label{ssec:train:use}\n\nAn underlying assumption in many narratives is that all entities that are mentioned should be important to the narrative.\nWe add a loss term that penalizes narratives whose combined attention weights for each entity does not sum to more than 1.\n\\begin{equation}\n\\mathcal{L}_{cover} = - \\frac{1}{I_d} \\sum_{i=1}^{I_d} \\log\\sum_{t=1}^S a_{i_t} \\label{eq:train:usage}\n\\end{equation}\n\n\\noindent where $a_{i_t}$ is the attention weight for a particular entity at sentence $t$ and $I_d$ is the number of entities in a document. $\\sum_{t=1}^S a_{i_t}$ is upper bounded by 1. This is similar to the coverage penalty used in neural machine translation \\citep{coverage}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\@startsection{section}{1}{\\z@}{3.5ex plus 1ex minus\n   .2ex}{2.3ex plus .2ex}{\\large\\bf}}\n   \n   \n\\renewcommand{\\thefootnote}{\\fnsymbol{footnote}}\n\\long\\def\\@makefntext#1{\\parindent 0cm\\noindent\n\\hbox to 1em{\\hss$^{\\@thefnmark}$}#1}\n\n\n\\newcommand{\\small}{\\small}\n\\makeatletter \n\\long\\def\\@makecaption#1#2{%\n  \\vskip\\abovecaptionskip\n  \\sbox\\@tempboxa{{\\small #1: #2}}%\n  \\ifdim \\wd\\@tempboxa >\\hsize\n    {\\small #1: #2\\par}\n  \\else\n    \\hbox to\\hsize{\\hfil\\box\\@tempboxa\\hfil}%\n  \\fi\n  \\vskip\\belowcaptionskip}\n\\makeatother  \n\n\n\\begin{document}\n\\begin{titlepage}\n\\vspace{.5in}\n\\begin{flushright}\nSeptember 2008\\\\\n\\end{flushright}\n\\vspace{.5in}\n\\begin{center}\n{\\Large\\bf\nThe Constraint Algebra\\\\[1ex]\nof Topologically Massive AdS Gravity} \n\n\\vspace*{.4in}\n{S.~C{\\sc arlip}\\footnote{\\it email: carlip@physics.ucdavis.edu}\\\\\n       {\\small\\it Department of Physics}\\\\\n       {\\small\\it University of California}\\\\\n       {\\small\\it Davis, CA 95616}\\\\{\\small\\it USA}}\n\\end{center}\n\n\\vspace{.5in}\n\\begin{center}\n{\\large\\bf Abstract}\n\\end{center}\n\\begin{center}\n\\begin{minipage}{4.5in}\n{\\small\nThree-dimensional topologically massive AdS gravity has a complicated \nconstraint algebra, making it difficult to count nonperturbative degrees \nof freedom.  I show that a new choice of variables greatly simplifies this \nalgebra, and confirm that the theory contains a single propagating mode \nfor all values of the mass parameter and the cosmological constant.  As an \nadded benefit, I rederive the central charges and conformal weights \nof the boundary conformal field theory from an explicit analysis of the \nasymptotic algebra of constraints.\n}\n\\end{minipage}\n\\end{center}\n\\end{titlepage}\n\\addtocounter{footnote}{-1}\n\n\\section{Introduction} \\label{intro}\n\nTopologically massive gravity \\cite{DJT}---(2+1)-dimensional Einstein \ngravity supplemented with a Chern-Simons term for the spin connection%\n---provides a fascinating playground for exploring higher-derivative \ngravity.   In contrast to the topological character of ordinary Einstein \ngravity in three dimensions, topologically massive gravity has a local \ndegree of freedom, a parity-violating massive spin two graviton that can \nbe described by a single indexless ``scalar'' field.  With the addition of a \nnegative cosmological constant $\\Lambda=-1\/\\ell^2$, surprising new \nfeatures emerge \\cite{CDWW}: for instance, the components of curvature \nperturbations propagate with different, chirality-dependent masses.\n\nTopologically massive AdS gravity may also provide a useful model\nin which to explore the AdS\/CFT correspondence.  The conformal\nboundary of a three-dimensional asymptotically anti-de Sitter spacetime \nis a flat two-dimensional cylinder, and the asymptotic symmetries are\ndescribed by a pair of Virasoro algebras \\cite{BH}.  The resulting\ntwo-dimensional conformal symmetry can be very powerful.  In pure Einstein\ngravity, for example, although the boundary conformal field theory is \nnot known \\cite{Carlip,Witten}, the classical central charges and\nconformal weights are sufficient to determine the BTZ black hole\nentropy \\cite{Strominger,BSS} and even the spectrum of Hawking \nradiation \\cite{Emparan}.  In topologically massive gravity, it was\nshown several years ago that the central charges of the ``left'' and \n``right''  Virasoro algebras split \\cite{KL,Solodukhin,Hotta}:\n\\begin{equation}\nc_\\pm = \\frac{3\\ell}{2G}\\left( 1 \\pm \\frac{1}{\\mu\\ell} \\right),\n\\label{intro1}\n\\end{equation}\nwhere the mass parameter $\\mu$ is fixed by the Chern-Simons coupling.\nThe classical contributions to conformal weights are also shifted, \nleading to interesting modifications of black hole thermodynamics \n\\cite{KL,Solodukhin,Park}.  Moreover, at $\\mu\\ell=\\pm1$ the boundary\ntheory becomes chiral.  Since the sum over topologies in three-dimensional\ngravity may require a chiral splitting \\cite{Maloney}, such a theory could \nbe of considerable interest.\n\nUnfortunately, this model appears to have a fundamental sickness.  With \nthe usual sign for the gravitational constant, the massive excitations of \ntopologically massive gravity carry negative energy \\cite{DJT}.  In the \nabsence of a cosmological constant, one can simply flip the sign of $G$,\nbut if $\\Lambda<0$, this will give a negative mass to the BTZ black hole\n\\cite{Moussa}.  The existence of a stable ground state is thus in doubt.\nThe possibility of a supersymmetric extension of the theory \\cite{Deser}\nsuggests the existence of a stable superselection sector, but this sector\nappears to exclude black holes.\n\nRecently, Li, Song, and Strominger proposed a possible cure \\cite{LSS}.\nAt the chiral point, a family of eigenstates of the Virasoro generator $L_0$\nrepresenting massive excitations disappears, and Li et al.\\ suggested \nthat the massive gravitons might no longer be present.  Unfortunately, a\ndifferent family of finite-energy eigenstates of $L_0$ has been found \n\\cite{GJ}, which violate the standard Fefferman-Graham asymptotic \nconditions \\cite{FG} but are still asymptotically anti-de Sitter; and worse, \na complete set of finite-energy asymptotically AdS wave packets also \nexists, even at the chiral coupling \\cite{CDWW}.  \n\nA loophole remains, however.  The computations of \\cite{LSS,GJ,CDWW}%\n---and, indeed, those of virtually every paper discussing this model---are based on \nclassical perturbation theory, expanding the metric in small fluctuations \naround AdS and keeping only lowest order terms.   The full field equations, \non the other hand, are highly nonlinear, and it is conceivable that new \nfeatures could emerge nonperturbatively.\n\nA general nonperturbative solution of the field equations of topologically\nmassive gravity seems distant, but we can learn a great deal by\nanalyzing the constraints.  For the case of a vanishing cosmological\nconstant, such an analysis was first performed by Deser and Xiang \\cite{DX}, \nand further amplified by Buchbinder et al.\\ \\cite{Buchbinder}.  The\nformalism is very complicated, in part because of the presence of third\nderivatives and second class constraints, but the results ultimately confirm \nthe existence of a single propagating degree of freedom.  For the asymptotically \nanti-de Sitter case, the literature is currently inconsistent: Park \\cite{Parkb}\nappears to find more than one degree of freedom (one ``for each internal\nindex''), while Grumiller et al.\\ \\cite{GJJ} find one configuration space \ndegree of freedom, but consider only the chiral coupling.\n\nIn this paper, I will show that a new choice of variables greatly simplifies the\nconstraint analysis, allowing an elegant expression of the constraint algebra\nand a simple counting of degrees of freedom.  I confirm the existence of a\nsingle propagating degree of freedom at all values of the couplings, and  \nrederive the central charges (\\ref{intro1}) from an explicit computation of\nthe algebra of asymptotic symmetries.\n\n\\section{Topologically massive AdS gravity}\n\nIn the first order formalism of (2+1)-dimensional gravity, the fundamental\nvariables are the triad $e^a = e^a{}_\\mu dx^\\mu$ and the spin connection \n$\\omega_a = \\frac{1}{2}\\epsilon_{abc}\\omega^{bc}{}_\\mu dx^\\mu$.\nThe Einstein-Hilbert action takes the form\\footnote{I choose units \n$16\\pi G=1$, a metric of signature $(-++)$, and a cosmological constant\n$\\Lambda=-1\/\\ell^2$, and set $\\epsilon_{012}=1$ (so $\\epsilon^{012}=-1$).}\n\\begin{equation}\nI_{EH}[e,\\omega] = 2 \\int \\left[ \n    e^a\\wedge\\left( d\\omega_a \n    + \\frac{1}{2}\\epsilon_{abc}\\omega^b\\wedge\\omega^c\\right)\n    + \\frac{1}{6}\\frac{1}{\\ell^2}\\epsilon_{abc}e^a\\wedge e^b\\wedge e^c\\right] ,\n\\label{b1}\n\\end{equation}\nwhere $e$ and $\\omega$ can be treated as independent variables.  The variation \nof $\\omega$ yields the torsion constraint\n\\begin{equation}\nT_a = D_\\omega e_a = de_a + \\epsilon_{abc}\\omega^b\\wedge e^c  = 0 ,\n\\label{b2}\n\\end{equation}\nwhile the variation of $e$ gives the Einstein field equations.   \n\nAn additional Chern-Simons term can be written for the spin connection,\n\\begin{equation}\nI_{CS}[\\omega] =  \\int\\left[\n    \\omega^a\\wedge\\left( d\\omega_a \n    + \\frac{1}{3}\\epsilon_{abc}\\omega^b\\wedge\\omega^c\\right)\\right] .\n\\label{b3}\n\\end{equation}\nIf $e$ and $\\omega$ are varied independently, the sum of the Einstein-Hilbert\nand Chern-Simons actions gives a model whose solutions are identical to those of\nordinary Einstein gravity \\cite{Wittenb}, although with a different symplectic \nstructure that may have implications for the quantum theory \\cite{Meusburger}.  \nIf the torsion constraint (\\ref{b2}) is imposed, however, one obtains a higher-derivative\ntheory, topologically massive gravity, with  \n\\begin{equation}\nI_{TMG}[e] = I_{EH}[e,\\omega(e)] + \\frac{1}{\\mu}I_{CS}[\\omega(e)] .\n\\label{b4}\n\\end{equation}\n\nTo simplify this action, let us define a new connection\n\\begin{equation}\nA^a = \\omega^a + \\mu e^a  ,\n\\label{b5}\n\\end{equation}\nwhose Chern-Simons action is\n$$\n\\frac{1}{\\mu} I_{CS}[A] = \\frac{1}{\\mu} I_{CS}[\\omega] + I_{EH}\n    + \\int \\left[ \\mu e^a\\wedge T_a + \\frac{1}{3}\\left(\\mu^2 - \\frac{1}{\\ell^2}\\right)\n    \\epsilon_{abc}e^a\\wedge e^b\\wedge e^c\\right] .\n$$\nRather than explicitly writing $\\omega$ as a function of $e$, we can\nimpose the torsion constraint with a Lagrange multiplier, as suggested in\n\\cite{DX,Carlipb}.  Then\n\\begin{equation}\nI_{TMG} = \\frac{1}{\\mu} I_{CS}[A] \n    + \\int \\left[ \\beta^a\\left( D_A e_a - \\mu \\epsilon_{abc}e^b\\wedge e^c\\right)\n    - \\alpha\\epsilon_{abc}e^a\\wedge e^b\\wedge e^c\\right] ,\n\\label{b6}\n\\end{equation}\nwhere  $\\beta^a = \\beta^a{}_\\mu dx^\\mu$ is a Lagrange multiplier for the\ntorsion constraint and \n$$\n\\alpha = \\frac{1}{3}\\left(\\mu^2 - \\frac{1}{\\ell^2}\\right) .\n$$ \nNote that the chiral coupling occurs at $\\alpha=0$; this is the only place in which the \nrelationship of $\\mu$ and $\\ell$ appears in this formulation.\n\nThe classical equations of motion may be obtained by varying $A$, $\\beta$, and $e$:\n\\begin{align}\n\\delta A: \\qquad& F_a + \\frac{\\mu}{2}\\epsilon_{abc}\\beta^b\\wedge e^c = 0 \n    \\qquad \\hbox{with}\\  F_a = dA_a + \\frac{1}{2}\\epsilon_{abc}A^b\\wedge A^c \n    \\nonumber\\\\\n\\delta\\beta: \\qquad& T_a = D_A e_a - \\mu \\epsilon_{abc}e^b\\wedge e^c = 0 \n     \\nonumber\\\\\n\\delta e: \\qquad&B_a \n    = D_A\\beta_a - 2\\mu\\epsilon_{abc}\\beta^be^c - 3\\alpha\\epsilon_{abc}e^be^c = 0 .\n    \\label{b7} \n\\end{align}\nI show in the Appendix that these are equivalent to the standard field equations for\ntopologically massive AdS gravity, and that they determine the Lagrange multiplier\n$\\beta$ to be \n\\begin{align}\n&\\beta^a{}_\\mu =\n  -\\frac{2}{\\mu}\\left(R^a{}_\\mu + \\frac{2}{\\ell^2}e^a{}_\\mu \n  + \\frac{3\\alpha}{2} e^a{}_\\mu\\right) \n   \\nonumber\\\\\n&\\beta = \\beta^a{}_\\mu e_a{}^\\mu = -\\frac{9\\alpha}{\\mu}  . \\label{b9}\n\\end{align}\n\n\\section{Poisson brackets and constraints}\\label{constraints}\n\nFrom the action (\\ref{b6}), we can now read off the terms involving time derivatives:\n\\begin{equation}\nI_{TMG} = \\int d^3x \\epsilon^{ij}\\left[ -\\frac{1}{\\mu}A^a{}_i\\partial_t A_{aj} \n    - \\beta^a{}_i\\partial_t e_{aj}\\right] + \\dots\n\\label{c1}\n\\end{equation}\nThe canonical Poisson brackets are thus\n\\begin{align}\n&\\left\\{ A^a{}_i, A^b{}_j\\right\\} = \\frac{\\mu}{2}\\eta^{ab}\\epsilon_{ij} \\nonumber\\\\\n&\\left\\{ e^a{}_i, \\beta^b{}_j\\right\\} = \\eta^{ab}\\epsilon_{ij} .\\label{c2}\n\\end{align}\nThe factor of $1\/2$ in the first bracket can be obtained from Dirac brackets, or more\nsimply by recognizing that $A_x$ and $A_y$ are conjugate and integrating by parts.  This\ndiagonalization of the Poisson brackets is one of the main simplifications coming from our \nchoice of variables.\n\nThe time components $A^a{}_t$, $e^a{}_t$, and $\\beta^a{}_t$ appear in the action \nwithout time derivatives, and can be considered Lagrange multipliers.  The corresponding \nconstraints are just the relevant components of the classical equations of motion:\n\\begin{align}\n&J_a = -\\frac{2}{\\mu}\\epsilon^{ij}\\left(\n    F_{aij} + \\frac{\\mu}{2}\\epsilon_{abc}\\beta^b{}_i e^c{}_j\\right)  \n   \\nonumber\\\\\n&T_a = -\\epsilon^{ij}\\left(D_i e_{aj} - \\mu \\epsilon_{abc}e^b{}_i e^c{}_j\\right) \n    \\nonumber \\\\\n&B_a \n    = -\\epsilon^{ij}\\left(D_i\\beta_{aj} - 2\\mu\\epsilon_{abc}\\beta^b{}_ie^c{}_j \n    - 3\\alpha\\epsilon_{abc}e^b{}_ie^c{}_j\\right) . \n\\label{c3} \n\\end{align}\nIt is  convenient to ``smear'' the constraints, integrating them against vectors.  More\nprecisely, let us define\n\\begin{align}\n&J[\\xi] = \\int_\\Sigma d^2x\\, \\xi^aJ_a + Q_J[\\xi]  \\nonumber\\\\\n&T[\\xi] = \\int_\\Sigma d^2x\\, \\xi^aT_a + Q_T[\\xi]  \\nonumber\\\\\n&B[\\xi]  =  \\int_\\Sigma d^2x\\, \\xi^aB_a + Q_B[\\xi] ,\n\\label{c4}\n\\end{align}\nwhere $Q_J$, $Q_T$, and $Q_B$ are the boundary terms needed to make the constraints\ndifferentiable \\cite{BH}.   For now, we shall assume that the parameters $\\xi^a$ fall \noff rapidly enough at infinity that we can freely integrate by parts and  ignore\nboundary terms; these will be restored below in section \\ref{BT}.  The Poisson brackets \nof these generators with the canonical variables $\\{A,e,\\beta\\}$ are now easy to compute,\nand are displayed explicitly in the Appendix.\n \nThe constraints $\\{J,T,B\\}$ should generate the full group of symmetries of the\naction (\\ref{b6}), including diffeomorphism invariance.  For (2+1)-dimensional \nEinstein gravity, it is known that an appropriate combination of the ``gauge''\nconstraints with field-dependent parameters does, indeed, generate\ndiffeomorphisms on shell \\cite{Wittenb,Banados0,Banados}.  The same is true here.  Let \n$\\xi^\\mu$ be an arbitrary three-vector, and define\n\\begin{equation}\n \\xi^a = e^a{}_\\mu\\xi^\\mu, \\qquad {\\hat\\xi}^a = \\beta^a{}_\\mu\\xi^\\mu, \\qquad \n    {\\tilde\\xi}^a = A^a{}_\\mu\\xi^\\mu .\n\\label{c6}\n\\end{equation}\nDefine a new combination $H$ of the constraints by\n\\begin{equation}\nH[\\xi] = B[\\xi] + T[{\\hat\\xi}] + J[{\\tilde\\xi}] .\n\\label{c7}\n\\end{equation}\nA simple computation then shows that\n\\begin{equation}\n\\{H[\\xi],X\\} = -{\\cal L}_\\xi X + \\hbox{\\it terms proportional to the equations of motion,}\n\\label{c8}\n\\end{equation}\nwhere $X$ is any of $\\{A,e,\\beta\\}$ and $\\cal L$ denotes the Lie derivative.  \n\n\\section{The algebra of constraints}\\label{algebra}\n\nWe next turn to the algebra of the constraints.  We may first check that $J[\\xi]$ generates \nlocal Lorentz transformations: a straightforward calculation shows that for any constraint \n$C[\\eta]$,\n\\begin{equation}\n\\left\\{ J[\\xi],C[\\eta]\\right\\} \n    = -C[\\xi\\times\\eta] \\quad \\hbox{with $(\\xi\\times\\eta)^a=\\epsilon^{abc}\\xi_b\\eta_c$} .\n\\label{d1}\n\\end{equation}\nThe $J[\\xi]$ are thus first class constraints.\n\nThe remaining constraints are more complicated.  We find that\n\\begin{align}\n\\left\\{ T[\\xi],T[\\eta]\\right\\} &=\n    -\\frac{\\mu}{2}\\int d^2x\\,\\xi^a\\eta^b\\left( \\epsilon^{ij}e_{ai}e_{bj}\\right) \\nonumber\\\\\n\\left\\{ B[\\xi],T[\\eta]\\right\\} &= -\\frac{\\mu}{2}J[\\xi\\times\\eta] + 2\\mu T[\\xi\\times\\eta]\n    + \\frac{\\mu}{2}\\int d^2x\\,\\xi^a\\eta^b\\left(  \\epsilon^{ij}\\beta_{ai}e_{bj}  \n    -  \\eta_{ab}\\epsilon^{ij}\\beta^c{}_ie_{cj}\\right)   \\nonumber\\\\\n\\left\\{ B[\\xi],B[\\eta]\\right\\} &= 2\\mu B[\\xi\\times\\eta] + 6\\alpha T[\\xi\\times\\eta]\n   -  \\frac{\\mu}{2}\\int d^2x\\,\\xi^a\\eta^b\\left(\\epsilon^{ij}\\beta_{ai}\\beta_{bj}\\right) .\n   \\label{d2}\n\\end{align}\nThe appearance on the right-hand side of terms that are not proportional to the \nconstraints can mean two things: either there are secondary constraints, or some of our\nconstraints are second class \\cite{HT}.  To distinguish the two possibilities, note first that\n$\\epsilon^{ij}e_{ai}e_{bj}$ is zero only if the triad is noninvertible, certainly\nnot a restriction we wish to impose.  Similarly, $\\epsilon^{ij}\\beta_{ai}\\beta_{bj}$\nand $\\epsilon^{ij}e_{ai}\\beta_{bj}$ vanishes only if some components of $\\beta_{ai}$ \nare linearly dependent.  From (\\ref{b9}), this is not generically true, holding only for \n``pure Einstein gravity'' solutions.  \n\nOn the other hand, the quantity\n$$\n\\Delta = \\epsilon^{ij}\\beta^c{}_i e_{cj} = \\epsilon^{ij}\\beta_{ij}\n$$\nalways vanishes classically, by virtue of the symmetry of $\\beta_{\\mu\\nu}$.   We can \ntherefore treat $\\Delta$ as a secondary constraint,\\footnote{Alternatively, we could  \ntreat the Hamiltonian as a second class constraint; as discussed in \\cite{BGH}, the rank \nof the brackets of the primary constraints would then no longer be constant, and the Dirac \nbrackets would become singular in some regions of phase space.}  and obtain the further \ncommutators\n\\begin{align}\n&\\left\\{ T[\\xi],\\Delta\\right\\} = -\\epsilon^{ij}D_i\\left(\\xi^ae_{aj}\\right) \n     - \\mu\\epsilon^{abc}\\left(\\epsilon^{ij}e_{ai}e_{bj}\\right)\\xi_c  - \\xi^aT_a  \n    \\nonumber\\\\\n&\\left\\{ B[\\xi],\\Delta\\right\\} = \\epsilon^{ij}D_i\\left(\\xi^a\\beta_{aj}\\right) \n     - 2\\mu\\epsilon^{abc}\\left(\\epsilon^{ij}\\beta_{ai}e_{bj}\\right)\\xi_c\n     - 9\\alpha\\epsilon^{abc}\\left(\\epsilon^{ij}e_{ai}e_{bj}\\right)\\xi_c + \\xi^aB_a \n    \\nonumber\\\\\n&\\left\\{ \\Delta(x),\\Delta(x')\\right\\} = 0 ,\\label{d3}\n\\end{align}\nalong with the relation $\\{J[\\xi],\\Delta\\}=0$ that we would expect from the role of\n$J$ as a generator of local Lorentz transformations.\n\nOur task is now to diagonalize the Poisson brackets (\\ref{d2})--(\\ref{d3}), to determine\nthe first and second class constraints.  To do so, let us define ${\\hat\\xi}^a$ \nto be such that \n$$\ne_{ai}{\\hat\\xi}^a = \\beta_{ai}\\xi^a ,\n$$  \nand let\n\\begin{equation}\n{\\hat B}[\\xi] = B[\\xi] + T[{\\hat\\xi}] .\n\\label{d4}\n\\end{equation}\nThe existence of $\\hat\\xi$ is guaranteed by the invertibility of the triad; we will take\nadvantage of its nonuniqueness below.   A straightforward computation then gives\n\\begin{align}\n\\left\\{ {\\hat B}[\\xi],T[\\eta]\\right\\} &=  \n    -\\frac{\\mu}{2}J[\\xi\\times\\eta] \n    + 2\\mu T[\\xi\\times\\eta]   \\nonumber\\\\\n    &\\quad - \\frac{\\mu}{2}\\int_\\Sigma d^2x\\,\\xi\\cdot\\eta\\,\\Delta \n    - T\\bigl[\\{T[\\eta],{\\hat\\xi}\\}\\bigr] \\approx0\\nonumber\\\\\n\\left\\{ {\\hat B}[\\xi],B[\\eta]\\right\\} &= \n    2\\mu B[\\xi\\times\\eta] -\\frac{\\mu}{2}J[{\\hat\\xi}\\times\\eta]\n    + 2\\mu T[{\\hat\\xi}\\times\\eta] +6\\alpha T[\\xi\\times\\eta]  \\nonumber\\\\\n    &\\quad + \\frac{\\mu}{2}\\int_\\Sigma d^2x\\,{\\hat\\xi}\\cdot\\eta\\,\\Delta \n    - T\\bigl[ \\{B[\\eta],{\\hat\\xi}\\}\\bigr] \\approx0\\nonumber\\\\\n\\left\\{{\\hat  B}[\\xi],\\Delta\\right\\} \n     = &- \\epsilon^{abc}\\left(\\epsilon^{ij}e_{ai}e_{bj}\\right)\n     \\left(\\mu{\\hat\\xi}_c + 9\\alpha\\xi_c\\right)\n     -2\\mu\\epsilon^{abc}\\left(\\epsilon^{ij}\\beta_{ai}e_{bj}\\right)\\xi_c\\nonumber\\\\\n     &- {\\hat\\xi}^aT_a + \\xi^aB_a  - \\int_\\Sigma d^2x'\\, T_a(x')\\{\\Delta(x),{\\hat\\xi}^a(x')\\} ,\n\\label{d5}\n\\end{align}\nwhere $\\approx$ means ``weakly equal,'' that is, ``equal up to constraints.''  The first \ntwo brackets are weakly zero; if $\\hat\\xi$ can be chosen so that the third is as well, then \nthe ${\\hat B}[\\xi]$ will be first class constraints.\n\nTo see that this is possible, we first use the invertibility of $e^a{}_\\mu$ to write\n\\begin{equation}\n{\\hat\\xi}^a = e^{a\\mu}\\beta_{b\\mu}\\xi^b + e^{at}\\eta ,\n\\label{d6}\n\\end{equation}\nwhere $\\eta$ is arbitrary.  It is already evident that the last bracket in (\\ref{d5}) \ncan be made weakly zero, since the right-hand side is linear in $\\eta$.  More explicitly,\nnote that\n$$\n\\epsilon^{\\mu\\nu\\rho}e^a{}_\\mu e^b{}_\\nu e^c{}_\\rho = e\\epsilon^{abc} \n$$\nwith $e = \\det|e^a{}_\\mu|$, and therefore\n$$\n\\epsilon^{ij}e^a{}_ie^b{}_j = -e\\epsilon^{abc}e_c{}^t, \\qquad\n     \\epsilon^{ij}e^b{}_j = -e\\epsilon^{abc}e_a{}^ie_c{}^t .\n$$\nSome simple manipulation then yields\n$$\n\\left\\{{\\hat  B}[\\xi],\\Delta\\right\\} \n     \\approx -2\\mu eg^{tt}{\\eta} + 2\\mu e\\left(\\beta^{tc} - \\beta^{ct}\\right)\\xi_c\n     - 2\\mu e\\left(\\beta + \\frac{9\\alpha}{\\mu} \\right)e^{ct}\\xi_c ,\n$$\nand we can clearly choose $\\eta$ so that the right-hand side vanishes.  More than that, \nit is evident from (\\ref{b9}) that $\\eta = 0$ on shell.  In that case, ${\\hat\\xi}^a$ is\nidentical to the parameter appearing in (\\ref{c7}), and ${\\hat B}[\\xi]$ is essentially the \ngenerator of diffeomorphisms.\n\nWe now consider the remaining constraints, which we can take to be $T^a$ and \n$\\Delta$.  It is convenient to write $T^\\mu = T^ae_a{}^\\mu$.  The Poisson brackets \n(\\ref{d2})--(\\ref{d3}) then give\n\\begin{align}\n&\\left\\{ T^i(x),T^j(x')\\right\\} \\approx \n   -\\frac{\\mu}{2}\\epsilon^{ij}\\delta^2(x-x') \\nonumber\\\\\n&\\left\\{ T^i(x),T^t(x')\\right\\} \\approx \n   \\left\\{ T^t(x),T^t(x')\\right\\} \\approx 0 \\nonumber\\\\\n&\\left\\{ T^i(x),\\Delta(x')\\right\\} \\approx \n   -\\left(\\epsilon^{ij}D_j + 2\\mu eg^{ti}\\right)\\delta^2(x-x') \\nonumber\\\\\n&\\left\\{ T^t(x),\\Delta(x')\\right\\} \\approx -2\\mu eg^{tt}\\delta^2(x-x') . \\label{d7}\n\\end{align}\nIt is clear upon inspection that for any values of $\\mu$ and $\\alpha$ (except for the\nconformal limit $\\mu=0$), the matrix of Poisson brackets has a nonzero determinant.\nIn fact, as I describe in the Appendix, it is not too hard to compute its inverse explicitly.  \nHence no further combination of the $\\{T^a,\\Delta\\}$ gives an additional first class \nconstraint.   \n\nWe thus have nine canonical pairs of variables ($A^a{}_i$, $e^a{}_i$, and $\\beta^a{}_i$), \nsix first class constraints ($J^a$ and ${\\tilde B}^a$) , and four second class constraints \n($T^a$ and $\\Delta$).  Each first class constraint eliminates two phase space degrees\nof freedom, while each second class constraint eliminates one \\cite{HT}; we therefore have\n$18-12-4 = 2$ degrees of freedom left, that is, one canonical pair of free data, describing \na single local excitation.  While the values of $\\mu$ and $\\ell$, in the combination $\\alpha$, \naffect the algebra of constraints, no choice leads to a change in the types of the constraints \nor a jump in the number of degrees of freedom.  In particular, for the chiral values \n$\\mu\\ell=\\pm1$, these results agree with \\cite{GJJ}, while in the asymptotically flat limit \n$\\ell\\rightarrow\\infty$ they go smoothly to the results of \\cite{DX}.\n\n\\section{Asymptotic symmetries} \n\nLet us briefly recall a few features of the first-order formulation of (2+1)-dimensional\nEinstein gravity with a negative cosmological constant \\cite{Wittenb,Banados0,Banados}.\nThe theory has six first class constraints, which give a canonical representation \nof the underlying symmetries of the theory,  local Lorentz invariance and \ndiffeomorphism invariance.  The constraints can be combined to form two mutually \ncommuting sets of three generators.  Each set forms a Virasoro algebra, and when \nevaluated at the asymptotic symmetries of anti-de Sitter space, the algebras have\nclassical central charges.  These central charges, along with the classical conformal \nweights, provide a powerful tool for investigating the boundary conformal field theory.\n\nFor topologically massive AdS gravity, we also have two sets of first class \nconstraints, $J^a$ and ${\\hat B}^a$, which again reflect local Lorentz invariance \nand diffeomorphism invariance.  In general, though, we should not expect these to\nsplit into commuting ``left'' and ``right'' sectors; interactions are likely to\ncouple the left- and right-movers.  Indeed, from (\\ref{d2}),  the symmetry \ngenerators do not commute: the Poisson brackets of $\\hat B$ include a term \nproportional to $J$.   \n\nTo understand the central charges and conformal weights, though, it is enough to \nlook at a neighborhood of the AdS boundary.  There, from (\\ref{b9}) and (\\ref{d6}), \n\\begin{equation}\n{\\hat\\xi}^a = -\\frac{3\\alpha}{\\mu}\\xi^a .\n\\label{e0}\n\\end{equation}\nIf we define\n$$\nL_\\pm[\\xi] =  {\\hat B}[\\xi] + a_\\pm J[\\xi] ,\n$$\nit is easy to check that\n\\begin{align}\n\\left\\{ L_+[\\xi],L_-[\\eta]\\right\\}\n&= \\left\\{ {\\hat B}[\\xi] + a_+J[\\xi], {\\hat B}[\\eta] + a_-J[\\eta] \\right\\}\n\\nonumber\\\\\n  &= (2\\mu - a_+ - a_-) {\\hat B}[\\xi\\times\\eta] \n  + (3\\alpha -  a_+a_-) J[\\xi\\times\\eta] .\n\\label{e1}\n\\end{align}\nThe right-hand side of (\\ref{e1}) will vanish if\n$$\na_\\pm = \\mu\\pm\\frac{1}{\\ell} ,\n$$\nthat is,\n\\begin{equation}\nL_\\pm[\\xi] = {\\hat B}[\\xi] \n   + \\left(\\mu\\pm\\frac{1}{\\ell}\\right)  J[\\xi] .\n\\label{e2}\n\\end{equation}\nThe remaining Poisson brackets are then\n\\begin{align}\n&\\left\\{ L_\\pm[\\xi],L_\\pm[\\eta]\\right\\} \n   = \\mp {\\textstyle\\frac{2}{\\ell}}L_\\pm[\\xi\\times\\eta]  \\nonumber\\\\\n&\\left\\{ L_+[\\xi],L_-[\\eta]\\right\\} = 0 . \\label{e3a}\n\\end{align}\n\nAn added complication can arise if the parameters  $\\xi^a$ are field-dependent:\nthey may then have nontrivial Poisson brackets with the $L_\\pm$, leading to additional \nterms in the algebra (\\ref{e3a}).  In particular, we saw earlier that the parameters \ncharacterizing diffeomorphisms are of the form $\\xi^a = e^a{}_\\mu\\xi^\\mu$.   \nThe algebra thus becomes\n\\begin{align}\n&\\left\\{ L_\\pm[\\xi],L_\\pm[\\eta]\\right\\} =\n    L_\\pm\\left[ \\{L_\\pm[\\xi],e^a{}_i\\}\\eta^i \n    - \\{L_\\pm[\\eta],e^a{}_i\\}\\xi^i \\mp {\\textstyle \\frac{2}{\\ell}}(\\xi\\times\\eta)^a\\right] \n   \\nonumber\\\\\n&\\left\\{ L_+[\\xi],L_-[\\eta]\\right\\} \n   = L_-\\left[ \\{L_+[\\xi],e^a{}_i\\}\\eta^i\\right] - L_+\\left[\\{L_-[\\eta],e^a{}_i\\}\\xi^i \\right] .\n\\label{e4}\n\\end{align}\nAgain, though, matters simplify when we consider only a small neighborhood of \nthe AdS boundary.  It is clear that if we could find parameters $\\xi$ and $\\bar\\xi$\nsuch that $\\{L_+[\\xi],e^a{}_i\\} = \\{L_-[{\\bar\\xi}],e^a{}_i\\}=0$, the extra terms  \nin (\\ref{e4}) would vanish.  Globally, this is rarely possible, but we can define asymptotic \nsymmetries for which \n\\begin{align}\n&\\left\\{ L_+[\\xi], e^a{}_i\\right\\} = - \\left(\\partial_i\\xi^a \n   + \\epsilon^{abc}(\\omega_{bi} + \\frac{1}{\\ell}e_{bi})\\xi_c\\right) \\sim 0 \\nonumber\\\\\n&\\left\\{ L_-[{\\bar\\xi}], e^a{}_i\\right\\} =  - \\left(\\partial_i{\\bar\\xi}^a \n   + \\epsilon^{abc}(\\omega_{bi} - \\frac{1}{\\ell}e_{bi}){\\bar\\xi}_c\\right) \\sim 0\n\\label{e6}\n\\end{align}\nat the AdS boundary.\\footnote{It is interesting to note that the covariant derivatives \nhere are identical to those in the gauge formulation of ordinary Einstein gravity \n\\cite{Wittenb,Banados0,Banados}. } We shall see in the next section that these eliminate \nthe extra terms in the commutator of $L_+$ and $L_-$ at the boundary.\n\nEquation (\\ref{e6}) is easy to solve.  If we choose coordinates such that the leading terms \nin the metric take the form  \n$$\nds^2 = \\ell^2d\\rho^2 + e^{2\\rho}\\left(\\ell^2d\\varphi^2 - dt^2\\right) ,\n$$\nwe find two families, labeled by functions \n$f(\\varphi+t\/\\ell)$ and ${\\bar f}(\\varphi-t\/\\ell)$:\n\\begin{alignat}{2}\n&\\xi^0_f = \\frac{\\ell}{2}e^\\rho f   \\qquad\\qquad\n&&{\\bar\\xi}^0_{\\bar f} =  \\frac{\\ell}{2} e^\\rho{\\bar f} \\nonumber\\\\\n&\\xi^1_f = - \\frac{\\ell}{2}\\partial_\\varphi f   \\qquad\\qquad\n&&{\\bar\\xi}^1_{\\bar f} =  \\frac{\\ell}{2} \\partial_\\varphi {\\bar f} \\nonumber\\\\\n&\\xi^2_f =   \\frac{\\ell}{2} e^\\rho f   \\qquad\\qquad\n&&{\\bar\\xi}^2_{\\bar f}  = - \\frac{\\ell}{2} e^\\rho{\\bar f} .\n\\label{e5}\n\\end{alignat}\nI have chosen a normalization such that the zero-modes of $\\xi^t$ and ${\\bar\\xi}^t$ are\npositive and such that\n$$\n\\left[\\xi_f,\\xi_g\\right]^a = \\xi_{\\{f,g\\}}^a, \\qquad\n\\left[{\\bar\\xi}_{\\bar f},{\\bar\\xi}_{\\bar g}\\right]^a = -{\\bar\\xi}_{\\{{\\bar f},{\\bar g}\\}}^a, \n$$\nwhere $[\\xi,\\eta]^\\mu = \\xi^\\nu\\partial_\\nu\\eta^\\mu - \\eta^\\nu\\partial_\\nu\\xi^\\mu$ is\nthe ordinary commutator of (2+1)-dimensional vector fields and \n$\\{f,g\\} = f\\partial_\\varphi g - g\\partial_\\varphi f$ is the commutator of $f$ and $g$ \nviewed as one-dimensional vector fields on the circle.  Not surprisingly, the parameters (\\ref{e5})\nmatch those found in ordinary Einstein gravity \\cite{Banados0,Banados}, and agree to \nlowest order with the asymptotic AdS Killing  vectors found long ago by Brown and Henneaux \n\\cite{BH}.\n\nRestricted to such transformations, the algebra  (\\ref{e4}) now becomes\n\\begin{align}\n&\\left\\{ L_+[\\xi_f],L_+[\\xi_g]\\right\\} =  L_+\\left[[\\xi_f,\\xi_g]\\right]  + L_+[\\chi(f,g)]\n   \\nonumber\\\\\n&\\left\\{ L_-[{\\bar\\xi}_{{\\bar f}}], L_-[{\\bar\\xi}_{{\\bar g}}]\\right\\}\n   =  L_-\\left[[{\\bar \\xi}_{\\bar f},{\\bar \\xi}_{\\bar g}]\\right] \n   + L_-[\\chi({\\bar f},{\\bar g})]\n   \\nonumber\\\\\n&\\left\\{ L_+[\\xi_f],L_-[{\\bar\\xi}_{\\bar g}]\\right\\} = \n   -({\\hat B} + \\mu J)[\\chi(f,{\\bar g})] - J[{\\tilde\\chi}(f,{\\bar g})] , \n\\label{e3b}\n\\end{align}\nwith\n\\begin{alignat}{2}\n&\\chi^1(f,g) = \\frac{\\ell}{4}\\partial_\\varphi\\left(f\\partial_\\varphi g - g\\partial_\\varphi f\\right),\n\\quad &&\\chi^0(f,g) =\\chi^2(f,g) = 0  \\nonumber\\\\\n&{\\tilde\\chi}^1(f,g) = \\frac{1}{4}\\left(f\\partial_\\varphi^2g + g\\partial_\\varphi^2f\\right) ,\n\\quad &&{\\tilde\\chi}^0(f,g) ={\\tilde\\chi}^2(f,g) = 0 .\n\\label{ex}\n\\end{alignat}\nWe shall see in the next section that the terms involving $\\chi$ and $\\tilde\\chi$ give no\ncontribution at the AdS boundary.\n\n\\section{Boundary terms and central charges}\\label{BT}\n\nUp to now, we have focused on the ``bulk'' contributions to the constraints.  We must now  \nrestore the boundary terms.  Let us first recall a few general features \\cite{BH,BHb}.  Consider\na theory of fields $\\{\\phi_i\\}$ in $n+1$ dimensions, with gauge transformations labeled\nby parameters $\\xi$ and generated by\n$$\nG[\\xi,\\phi] = \\int_\\Sigma d^n\\!x\\,{\\cal G}[\\xi,\\phi] .\n$$ \nUp to boundary terms, these generators should satisfy the appropriate gauge algebra\n$$\n\\left\\{ G[\\xi,\\phi], G[\\eta,\\phi] \\right\\} = G[\\{\\xi,\\eta\\},\\phi] ,\n$$\nwhere $\\{\\xi,\\eta\\}$ is the Lie bracket for the gauge group.\n  \nNow let us restore the boundary terms.  Under a general variation of the fields, \n$$\n\\delta G[\\xi,\\phi] = \\int_\\Sigma d^n\\!x\\, \\frac{\\delta{\\cal G}}{\\delta\\phi_i}\\delta\\phi_i\n    + \\int_{\\partial\\Sigma}d^{n-1}\\!x\\, B[\\xi,\\phi,\\delta\\phi] .\n$$\nIf the boundary term $B$ is nonzero, $G$ is said to not be ``differentiable.''  In particular, the \npresence of $B$ will lead to delta-function singularities in the Poisson brackets.  It may be\npossible to generalize the algebra to include such boundary singularities \\cite{Solo,Bering},\nbut it is normally simpler to choose boundary conditions such that $B$ is itself a total \nvariation,\n$$\nB[\\xi,\\phi,\\delta\\phi] = -\\delta Q[\\xi,\\phi]  .\n$$\nThe combination ${\\bar G}[\\xi,\\phi] = G[\\xi,\\phi]  + Q[\\xi,\\phi] $ will then have a well-defined\nvariation, with no boundary terms, and it is easy to show that\n\\begin{align}\n\\left\\{{\\bar G}[\\xi,\\phi],{\\bar G}[\\eta,\\phi]\\right\\}\n   &=  \\iint d^n\\!x'\\, d^n\\!x\\,  \n  \\frac{\\delta{\\cal G}[\\xi,\\phi]}{\\delta\\phi_i(x)}\n   \\frac{\\delta{\\cal G}[\\eta,\\phi]}{\\delta\\phi_j(x')}\\{\\phi_i(x),\\phi_j(x')\\} \\nonumber\\\\\n   &= {\\bar G}[\\{\\xi,\\eta\\},\\phi] + K(\\xi,\\eta) .\n\\label{k1}\n\\end{align}\n\nThe central term $K(\\xi,\\eta)$ arises from boundary terms in the integrals, and need not vanish.  It\nis most easily evaluated by considering the algebra (\\ref{k1}) for the ``vacuum'' configuration, \nfor which the boundary charges $Q$ vanish; the right-hand side of (\\ref{k1}) then consists solely \nof the central term.\n\nTo apply this general formalism to our case, we must first return to (\\ref{d1}) and (\\ref{d2}) \nand keep track of any boundary terms.  A straightforward calculation yields\n\\begin{align}\n&\\left\\{J[\\xi],J[\\eta]\\right\\} \n    = \\dots + \\frac{2}{\\mu}\\int_{\\partial\\Sigma}\\xi^aD_\\varphi\\eta_a d\\varphi \\nonumber\\\\\n&\\left\\{J[\\xi],T[\\eta]\\right\\} \n   = \\dots - \\int_{\\partial\\Sigma}(\\xi\\times\\eta)_ae^a{}_\\varphi d\\varphi \\nonumber\\\\\n&\\left\\{J[\\xi],B[\\eta]\\right\\} \n   = \\dots - \\int_{\\partial\\Sigma}(\\xi\\times\\eta)_a\\beta^a{}_\\varphi d\\varphi \\nonumber\\\\\n&\\left\\{T[\\xi],T[\\eta]\\right\\} = \\dots \\nonumber\\\\\n&\\left\\{B[\\xi],T[\\eta]\\right\\} \n   = \\dots + \\int_{\\partial\\Sigma}\\left[ \\xi^aD_\\varphi\\eta_a \n   + 2\\mu(\\xi\\times\\eta)_ae^a{}_\\varphi\\right] d\\varphi \\nonumber\\\\\n&\\left\\{B[\\xi],B[\\eta]\\right\\} \n   = \\dots + \\int_{\\partial\\Sigma}(\\xi\\times\\eta)_a\\left(2\\mu\\beta^a{}_\\varphi \n   + 6\\alpha e^a{}_\\varphi\\right)d\\varphi ,\n\\label{k2}\n\\end{align}\nwhere the omitted bulk terms are all proportional to the constraints, and vanish weakly. \nFor asymptotically anti-de Sitter boundary conditions, we see from (\\ref{b9}) that   \n$\\beta^a = -\\frac{3\\alpha}{\\mu}e^a$ and ${\\hat\\xi}^a = -\\frac{3\\alpha}{\\mu}\\xi^a$\nat the boundary.  Some simple algebra then gives\n\\begin{alignat}{2}\n\\left\\{ L_\\pm[\\xi],L_\\pm[\\eta]\\right\\}& &&= \\dots \n   \\pm\\frac{4}{\\ell} \\int_{\\partial\\Sigma}\\left[ \n  \\left(1 \\pm \\frac{1}{\\mu\\ell}\\right)\\xi^a D_\\varphi \\eta_a\n  + \\frac{3\\alpha}{\\mu}(\\xi\\times\\eta)^a e_{a\\varphi}\\right]\n  d\\varphi \\nonumber\\\\\n&&&= \\dots \\pm\\frac{4}{\\ell}\\left( 1 \\pm \\frac{1}{\\mu\\ell}\\right)\n    \\int_{\\partial\\Sigma}\\xi^a\\left[ \\partial_\\varphi\\eta_a\n    + \\epsilon_{abc}\\left(\\omega^b{}_\\varphi \\pm \\frac{1}{\\ell}e^b{}_\\varphi\\right) \\eta^c\n    \\right] d\\varphi \\nonumber\\\\\n\\left\\{ L_+[\\xi],L_-[\\eta]\\right\\}& &&= \\dots .\n\\label{f3}\n\\end{alignat}\nEvaluated at the AdS ``vacuum'' state, the right-hand sides of these\nexpressions are the central terms $K_\\pm$.\n\nIf our asymptotic symmetries (\\ref{e5}) were exact---that is, if (\\ref{e6}) were \nsatisfied exactly, and not just asymptotically---then the integrands on the right-hand \nside of (\\ref{f3}) would vanish.  But the symmetries are not quite exact, and a simple \ncalculation shows that\n\\begin{align}\n&\\left\\{ L_+[\\xi_f],L_+[\\xi_g]\\right\\} =  \\dots\n   + \\frac{\\ell}{32\\pi G}\\left( 1 + \\frac{1}{\\mu\\ell}\\right)\n    \\int_{\\partial\\Sigma}\\left( \\partial_\\varphi f \\partial^2_\\varphi g \n    - \\partial_\\varphi g \\partial^2_\\varphi f \\right) d\\varphi \\nonumber\\\\\n   \\nonumber\\\\\n&\\left\\{ L_-[{\\bar\\xi}_{{\\bar f}}], L_-[{\\bar\\xi}_{{\\bar g}}]\\right\\} = \\dots\n   - \\frac{\\ell}{32\\pi G}\\left( 1 - \\frac{1}{\\mu\\ell}\\right)\n   \\int_{\\partial\\Sigma}\\left( \\partial_\\varphi {\\bar f} \\partial^2_\\varphi {\\bar g} \n    - \\partial_\\varphi {\\bar g} \\partial^2_\\varphi {\\bar f} \\right) d\\varphi ,\n\\label{fx}\n\\end{align}\nwhere I have restored the factors of $16\\pi G$.  These are precisely the central terms\nfor two Virasoro algebras with central charges\n$$\nc_\\pm = \\frac{3\\ell}{2G}\\left(1\\pm\\frac{1}{\\mu\\ell}\\right),\n$$\nmatching the results (\\ref{intro1}) that had been previously obtained using very \ndifferent methods \\cite{KL,Solodukhin,Hotta}.\n\nFinally, let us directly evaluate the boundary terms $Q_{L_\\pm}$.  Here we can use\nsome results from pure Einstein gravity, where the same problem was discussed\nin \\cite{Banados0,Banados}.  Note first that from (\\ref{c3}) and (\\ref{e2}), the\nboundary terms in the variation of $L_\\pm$ are\n\\begin{align}\n\\delta L_\\pm[\\xi] \n&= \\dots - \\int_{\\partial\\Sigma} \\left[\n    \\xi^a\\delta\\beta_{a\\varphi} + {\\hat\\xi}^a\\delta e_{a\\varphi}\n   + \\frac{2}{\\mu}\\left(\\mu\\pm\\frac{1}{\\ell}\\right)\\xi^a\\delta A_{a\\varphi}\n   \\right]d\\varphi \\nonumber\\\\\n&= \\dots - \\int_{\\partial\\Sigma} \\xi^\\mu\\left[\n   e^a{}_\\mu\\delta\\beta_{a\\varphi} + \\beta^a{}_\\mu\\delta e_{a\\varphi}\n   + \\frac{2}{\\mu}\\left(\\mu\\pm\\frac{1}{\\ell}\\right)e^a{}_\\mu\\delta A_{a\\varphi}\n   \\right]d\\varphi .\n\\label{l1}\n\\end{align}\nAs before, anti-de Sitter boundary conditions require that\n$\\beta^a = -\\frac{3\\alpha}{\\mu}e^a$, and a bit of algebra reduces (\\ref{l1}) to\n\\begin{align}\n\\delta L_\\pm[\\xi] \n&= \\dots - \\int_{\\partial\\Sigma} \\xi^\\mu\\left[\n    -\\frac{6\\alpha}{\\mu}e^a{}_\\mu\\delta e_{a\\varphi} \n    + 2 \\left( 1 \\pm \\frac{1}{\\mu\\ell}\\right)e^a{}_\\mu\\delta A_{a\\varphi} \n     \\right]d\\varphi  \\nonumber\\\\\n&= \\dots - 2\\left( 1 \\pm \\frac{1}{\\mu\\ell}\\right) \\int_{\\partial\\Sigma}\n    \\xi^\\mu e^a{}_\\mu \\,\\delta\\left(\n    \\omega_{a\\varphi}\\pm\\frac{1}{\\ell}e_{a\\varphi}\\right)d\\varphi .\n\\label{l2}\n\\end{align}\nWe now adopt the boundary conditions of \\cite{Banados0,Banados}, which translate\nto\n\\begin{equation*}\n\\omega^a{}_t =  \\frac{1}{\\ell^2}e^a{}_\\varphi, \\qquad\n\\omega^a{}_\\varphi =  e^a{}_t, \\qquad \\delta e^a{}_\\rho = 0 ,\n\\label{l3}\n\\end{equation*} \nand note that for our asymptotic symmetries, $\\xi^\\varphi = \\pm\\frac{1}{\\ell}\\xi^t$.  \nThe variation (\\ref{l2}) is thus\n\\begin{align*}\n\\delta L_\\pm[\\xi] \n&= \\dots - 2\\left( 1 \\pm \\frac{1}{\\mu\\ell}\\right) \\int_{\\partial\\Sigma}\n    \\left[\\xi^t\\left(e^a{}_t\\pm\\frac{1}{\\ell}e^a{}_\\varphi\\right) + \\xi^\\rho e^a{}_\\rho\\right]\n    \\delta \\left(e_{at}\\pm \\frac{1}{\\ell}e_{a\\varphi}\\right) d\\varphi \\nonumber\\\\\n&= \\dots -\\left( 1 \\pm \\frac{1}{\\mu\\ell}\\right) \\delta \\int_{\\partial\\Sigma}\n    \\left[\\xi^t \\left(e^a{}_t\\pm \\frac{1}{\\ell}e^a{}_\\varphi \\right)\n    + 2 \\xi^\\rho\\, e^a{}_\\rho\\right]\n    \\left(e_{at}\\pm \\frac{1}{\\ell}e_{a\\varphi}\\right) d\\varphi ,\n\\label{l4}\n\\end{align*}\nand the $L_\\pm$ are thus differentiable if we add boundary terms\n\\begin{equation}\nQ_\\pm[\\xi] =  \\frac{1}{16\\pi G}\\left( 1 \\pm \\frac{1}{\\mu\\ell}\\right) \\int_{\\partial\\Sigma}\n    \\left[\\xi^t \\left(e^a{}_t\\pm \\frac{1}{\\ell}e^a{}_\\varphi \\right)\n    + 2 \\xi^\\rho\\, e^a{}_\\rho\\right]\n    \\left(e_{at}\\pm \\frac{1}{\\ell}e_{a\\varphi}\\right) d\\varphi .\n\\label{l5}\n\\end{equation}\n\nThese boundary terms are identical to those of ordinary Einstein gravity, except\nfor the prefactors of $1 \\pm \\frac{1}{\\mu\\ell}$.   That is,\n\\begin{equation}\nQ_\\pm^{\\hbox{\\tiny TMG}}[\\xi] \n    = \\left( 1 \\pm \\frac{1}{\\mu\\ell}\\right) Q_\\pm^{\\hbox{\\tiny Einstein}}[\\xi] ,\n\\label{l6}\n\\end{equation}\nin agreement with \\cite{KL,Solodukhin}.  Further, we can now verify the claim in \nthe preceding section that the $\\chi$ and $\\tilde\\chi$ terms in (\\ref{ex})\nare irrelevant at the boundary.  Indeed, these terms only appear in (\\ref{l5}) in\nthe form $\\chi^\\rho(g_{\\rho t}\\pm \\frac{1}{\\ell}g_{\\rho\\varphi})$, and\nvanish by virtue of our boundary conditions.\n\n\\section{Chirality}\n\nIt has recently been argued that topologically massive AdS gravity is chiral at \nthe critical coupling $\\mu\\ell=\\pm1$ \\cite{Stromingerb}.  In the present context, \nthis feature can be understood as follows.  \n\nConsider first a generic coupling, and let $\\xi^\\mu$ be a vector field that satisfies \nthe fall-off conditions (\\ref{e5}) but is nonzero at the boundary.  From (\\ref{fx}), \nthe constraints $L_\\pm[\\xi]$ are no longer first class:  their Poisson brackets \nare not weakly zero.  Constraints that are not first class do not generate gauge\ntransformations, but rather determine asymptotic symmetries \\cite{Benguria}.  \nHence some configurations that are formally diffeomorphic will nevertheless be\nphysically inequivalent---they will differ by a symmetry rather than a gauge \nequivalence.   As a consequence,  new ``would-be pure gauge'' degrees of freedom \nappear at the boundary, which are conjecturally the source of the degrees of \nfreedom of the black hole \\cite{Carlip,Carlipc,Carlipd}.\n\nIf $\\mu\\ell=1$, on the other hand---or, by an obvious extension, $\\mu\\ell=-1$---it \nis apparent from (\\ref{fx}) and (\\ref{l5}) that $c_-$ and $Q_-$ vanish.  Thus \n$L_-[\\xi]$ remains first class even at the boundary, and one chirality of diffeomorphisms \nextends to the boundary as a true gauge invariance.  This eliminates one chiral sector \nof the ``massless gravitons'' discussed in \\cite{LSS}.  The remaining asymptotic \nsymmetry group consists of only one copy of the Virasoro algebra, and the boundary \ntheory is thus chiral.\n\nNote, however, that this argument does not eliminate bulk excitations that are not\ndiffeomorphic to zero in the interior.  In particular, the linearized excitations of\n\\cite{CDWW} and \\cite{GKP} yield solutions with nonconstant curvature.  No \ndiffeomorphism, whether or not it extends to the boundary, can remove such \nexcitations.  \n\n \n\\section{Conclusions}\n\nThis work has, first of all, established the existence of a local degree of  freedom \nin topologically massive AdS gravity at all values of the couplings.  In \nparticular, I confirm the results of  \\cite{GJJ} for the chiral coupling $\\mu\\ell=\\pm1$.  \nThe constraint analysis presented here is, in a sense, complementary to the \nperturbative analysis of \\cite{CDWW,GKP}.  Those papers show that weak \nfield solutions exist and remain well-behaved at the AdS boundary, but cannot \naddress effects beyond the weak field approximation, while the present analysis \nis fully nonperturbative, but does not address boundary behavior.  Since the weak\nfield perturbations have negative energy (relative to the black hole), these results\ntogether provide a strong indication that the theory is unstable.\n\nOn the other hand, this work also confirms that the boundary central charges of\ntopologically massive AdS gravity are shifted, and that at the chiral coupling,\none of the two central charges vanishes.  This presents a bit of a puzzle for the\nAdS\/CFT correspondence: the central charge measures the number of states in\nthe dual conformal field theory, and the vanishing of a central charge should\nmean, in some sense, that some fields disappear.   \n\nNote, though, that the \\emph{total} central charge, $c_++c_-$, is independent\nof $\\mu$; the vanishing of $c_-$ at $\\mu\\ell=1$ is compensated by an increase\nin $c_+$.  The same behavior can be seen in the boundary conformal weights:  \nby (\\ref{l5}), when $c_-$ and $Q_-$ vanish, $Q_+$ doubles.  For the BTZ black \nhole, this is reflected in the fact that all solutions are extremal at the chiral \ncoupling \\cite{Moussa,LSS}, while the entropy nevertheless remains independent \nof $\\mu$.  How this feature is manifested in the bulk---where the constraint algebra, \nat least, shows no special behavior as couplings vary---remains a mystery.\n\nThe value of the total central charge also presents a second puzzle: it is the \nsame for topologically massive gravity as it is for ordinary Einstein gravity.  \nThe counting of states via the Cardy formula will thus match the results of \nEinstein gravity, which are already sufficient to account for for the BTZ black\nhole entropy; we will see no additional contribution from the ``massive \ngraviton'' at \\emph{any} value of the coupling constant.  This should not \nreally be such a surprise, though: the classical contribution (\\ref{intro1}) to \nthe central charge is really of order ${\\cal O}(1\/\\hbar)$, while an ordinary \npropagating field contributes ${\\cal O}(1)$.  This suggests that the classical \nPoisson bracket analysis of the boundary conformal field theory might not \ncapture enough information to tell us about the massive graviton degrees\nof freedom, which may only appear at higher orders in $\\hbar$.\n\nCan chiral topologically massive gravity be saved?  The negative-energy weak \nfield excitations of \\cite{CDWW} can be built from compactly supported initial \ndata---that is, they represent arbitrarily small and arbitrarily localized \nperturbations, which cannot be excluded by boundary conditions in any obvious \nway.  The constraint analysis developed here further shows that these perturbations\nrepresent the ``right amount'' of initial data, one free phase space degree of \nfreedom per point.  It remains conceivable, however, that higher order corrections\nto the weak field solutions violate Fefferman-Graham boundary conditions, or\nlead to a finite lower bound to the negative energies that appear perturbatively.\nUnfortunately, the one known positive energy theorem for topologically massive \nAdS gravity, which follows from the existence of a supersymmetric extension,   \ngoes in the wrong direction \\cite{Deser,Abbott}: with the sign choice for\nwhich the black hole has positive mass, the energy of local excitations is strictly \nnonpositive.  Nevertheless, a more detailed investigation of boundary conditions\nbeyond first order perturbation theory could be of interest.\n \n\n\\vspace{1.5ex}\n\\begin{flushleft}\n\\large\\bf Acknowledgments\n\\end{flushleft}\n\\noindent \nI would like to thank Stanley Deser, Marc Henneaux, Andrew Waldron, and Derek \nWise for helpful discussions.\nThis work was supported in part by the Department of Energy under grant\nDE-FG02-91ER40674.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}