916 \label{sec:gluing}% |
916 \label{sec:gluing}% |
917 |
917 |
918 \subsection{`Topological' $A_\infty$ $n$-categories} |
918 \subsection{`Topological' $A_\infty$ $n$-categories} |
919 \label{sec:topological-A-infty}% |
919 \label{sec:topological-A-infty}% |
920 |
920 |
921 This section prepares the ground for establishing Property \ref{property:gluing} by defining the notion of a \emph{topological $A_\infty$ |
921 This section prepares the ground for establishing Property \ref{property:gluing} by defining the notion of a \emph{topological $A_\infty$-$n$-category}. |
922 $n$-category}. The main result of this section is |
922 The main result of this section is |
923 |
923 |
924 \begin{thm} |
924 \begin{thm} |
925 Topological $A_\infty$ $1$-categories are equivalent to `standard' |
925 Topological $A_\infty$-$1$-categories are equivalent to `standard' |
926 $A_\infty$ $1$-categories. |
926 $A_\infty$-$1$-categories. |
927 \end{thm} |
927 \end{thm} |
928 |
928 |
929 |
929 Before proving this theorem, we embark upon a long string of definitions. For expository purposes, we begin with the $n=1$ special cases, and define |
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930 first topological $A_\infty$-algebras, then topological $A_\infty$-categories, and then topological $A_\infty$-modules over these. We then turn |
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931 to the general $n$ case, defining topological $A_\infty$-$n$-categories and their modules. |
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932 \nn{Something about duals?} |
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933 \todo{Explain that we're not making contact with any previous notions for the general $n$ case?} |
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934 |
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935 \begin{defn} |
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936 \label{defn:topological-algebra}% |
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937 A ``topological $A_\infty$-algebra'' $A$ consists of the data |
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938 \begin{enumerate} |
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939 \item for each $1$-manifold $J$ diffeomorphic to the standard interval $I=\left[0,1\right]$, a complex of vector spaces $A(J)$, |
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940 \item along with an `evaluation' chain map $\ev_J : \CD{J} \tensor A(J) \to A(J)$, |
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941 \item for each diffeomorphism $\phi : J \to J'$, an isomorphism $A(\phi) : A(J) \isoto A(J')$, |
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942 \item and whenever $\bdy J \cap \bdy J'$ is a single point, a gluing map $\gl_{J,J'} : A(J) \tensor A(J') \to A(J \cup J')$, |
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943 \end{enumerate} |
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944 satisfying the following conditions. |
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945 \begin{itemize} |
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946 \item If $\phi$ is a diffeomorphism from $J$ to itself, the maps $\ev_J(\phi, -)$ and $A(\phi)$ are the same. |
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947 \item The evaluation chain map is associative, in that the diagram |
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948 \begin{equation*} |
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949 \xymatrix{ |
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950 \CD{J} \tensor \CD{J} \tensor A(J) \ar[r]^{\Id \tensor \ev_J} \ar[d]_{\compose \tensor \Id} & |
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951 \CD{J} \tensor A(J) \ar[d]^{\ev_J} \\ |
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952 \CD{J} \tensor A(J) \ar[r]_{\ev_J} & |
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953 A(J) |
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954 } |
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955 \end{equation*} |
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956 commutes. (Here the map $\compose : \CD{J} \tensor \CD{J} \to \CD{J}$ is a composition: take products of singular chains first, then use the group multiplication in $\Diff(J)$.) |
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957 \item The gluing maps are \emph{strictly} associative. That is, given $J$, $J'$ and $J''$, the diagram |
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958 \begin{equation*} |
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959 \xymatrix{ |
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960 A(J) \tensor A(J') \tensor A(J'') \ar[rr]^{\gl_{J,J'} \tensor \Id} \ar[d]_{\Id \tensor \gl_{J',J''}} && |
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961 A(J \cup J') \tensor A(J'') \ar[d]^{\gl_{J \cup J', J''}} \\ |
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962 A(J) \tensor A(J' \cup J'') \ar[rr]_{\gl_{J, J' \cup J''}} && |
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963 A(J \cup J' \cup J'') |
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964 } |
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965 \end{equation*} |
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966 commutes. |
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967 \end{itemize} |
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968 \end{defn} |
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969 |
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970 \begin{rem} |
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971 Of course, the first and third pieces of data (the complexes, and the isomorphisms) together just constitute a functor from the category of |
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972 intervals and diffeomorphisms between them to the category of complexes of vector spaces. |
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973 Further, one can combine the second and third pieces of data, asking instead for a map |
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974 \begin{equation*} |
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975 \ev_{J,J'} : \CD{J \to J'} \tensor A(J) \to A(J'). |
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976 \end{equation*} |
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977 (Any $k$-parameter family of diffeomorphisms in $C_k(\Diff(J \to J'))$ factors into a single diffeomorphism $J \to J'$ and a $k$-parameter family of |
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978 diffeomorphisms in $\CD{J'}$.) |
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979 \end{rem} |
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980 |
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981 To generalise the definition to that of a category, we simply introduce a set of objects which we call $A(pt)$. Now we associate complexes to each |
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982 interval with boundary conditions $(J, c_-, c_+)$, with $c_-, c_+ \in A(pt)$, and only ask for gluing maps when the boundary conditions match up: |
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983 \begin{equation*} |
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984 \gl : A(J, c_-, c_0) \tensor A(J', c_0, c_+) \to A(J \cup J', c_-, c_+). |
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985 \end{equation*} |
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986 The action of diffeomorphisms, and $k$-parameter families of diffeomorphisms, ignore the boundary conditions. |
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987 |
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988 The definition of a module follows closely the definition of an algebra or category. |
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989 \begin{defn} |
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990 \label{defn:topological-module}% |
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991 A topological $A_\infty$-(left-)module $M$ over a topological $A_\infty$ category $A$ consists of the data |
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992 \begin{enumerate} |
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993 \item a functor $K \mapsto M(K)$ from $1$-manifolds diffeomorphic to the standard interval, with a marked point on a boundary, to complexes of vector spaces, |
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994 \item along with an `evaluation' map $\ev_K : \CD{K} \tensor M(K) \to M(K)$ |
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995 \item whenever $\bdy J \cap K$ is a single point, and isn't the marked point of $K$ \todo{ugh, that's so gross}, a gluing map |
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996 $\gl_{J,K} : A(J) \tensor M(K) \to M(J \cup K)$ |
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997 \end{enumerate} |
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998 satisfying the obvious analogous conditions as in Definition \ref{defn:topological-algebra}. |
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999 \end{defn} |
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1000 |
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1001 \todo{Bimodules, and gluing} |
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1002 |
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1003 \todo{the motivating example $C_*(\maps(X, M))$} |
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1004 |
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1005 \todo{higher $n$} |
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1006 |
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1007 |
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1008 \newcommand{\skel}[1]{\operatorname{skeleton}(#1)} |
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1009 |
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1010 Given a topological $A_\infty$-category $\cC$, we can construct an `algebraic' $A_\infty$ category $\skel{\cC}$. First, pick your |
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1011 favorite diffeomorphism $\phi: I \cup I \to I$. |
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1012 \begin{defn} |
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1013 We'll write $\skel{\cC} = (A, m_k)$. Define $A = \cC(I)$, and $m_2 : A \tensor A \to A$ by |
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1014 \begin{equation*} |
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1015 m_2 \cC(I) \tensor \cC(I) \xrightarrow{\gl_{I,I}} \cC(I \cup I) \xrightarrow{\cC(\phi)} \cC(I). |
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1016 \end{equation*} |
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1017 Next, we define all the `higher associators' $m_k$ by |
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1018 \todo{} |
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1019 \end{defn} |
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1020 |
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1021 Give an `algebraic' $A_\infty$ category $(A, m_k)$, we can construct a topological $A_\infty$-category, which we call $\bc_*^A$. You should |
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1022 think of this at the generalisation of the blob complex, although the construction we give will \emph{not} specialise to exactly the usual definition |
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1023 in the case the $A$ is actually an associative category. |
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1024 \begin{defn} |
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1025 \end{defn} |
930 |
1026 |
931 \nn{Need to let the input $n$-category $C$ be a graded thing (e.g. DG |
1027 \nn{Need to let the input $n$-category $C$ be a graded thing (e.g. DG |
932 $n$-category or $A_\infty$ $n$-category). DG $n$-category case is pretty |
1028 $n$-category or $A_\infty$ $n$-category). DG $n$-category case is pretty |
933 easy, I think, so maybe it should be done earlier??} |
1029 easy, I think, so maybe it should be done earlier??} |
934 |
1030 |