author | kevin@6e1638ff-ae45-0410-89bd-df963105f760 |
Fri, 05 Jun 2009 16:10:37 +0000 | |
changeset 70 | 5ab0e6f0d89e |
parent 65 | 15a79fb469e1 |
child 72 | ed2594ff5870 |
permissions | -rw-r--r-- |
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\section{Gluing - needs to be rewritten/replaced} |
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\label{sec:gluing}% |
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We now turn to establishing the gluing formula for blob homology, restated from Property \ref{property:gluing} in the Introduction |
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\begin{itemize} |
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%\mbox{}% <-- gets the indenting right |
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\item For any $(n-1)$-manifold $Y$, the blob homology of $Y \times I$ is |
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naturally an $A_\infty$ category. % We'll write $\bc_*(Y)$ for $\bc_*(Y \times I)$ below. |
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\item For any $n$-manifold $X$, with $Y$ a codimension $0$-submanifold of its boundary, the blob homology of $X$ is naturally an |
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$A_\infty$ module for $\bc_*(Y \times I)$. |
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\item For any $n$-manifold $X$, with $Y \cup Y^{\text{op}}$ a codimension |
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$0$-submanifold of its boundary, the blob homology of $X'$, obtained from |
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$X$ by gluing along $Y$, is the $A_\infty$ self-tensor product of |
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$\bc_*(X)$ as an $\bc_*(Y \times I)$-bimodule. |
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\begin{equation*} |
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\bc_*(X') \iso \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y \times I)}} \!\!\!\!\!\!\xymatrix{ \ar@(ru,rd)@<-1ex>[]} |
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\end{equation*} |
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\end{itemize} |
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Although this gluing formula is stated in terms of $A_\infty$ categories and their (bi-)modules, it will be more natural for us to give alternative |
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definitions of `topological' $A_\infty$-categories and their bimodules, explain how to translate between the `algebraic' and `topological' definitions, |
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and then prove the gluing formula in the topological langauge. Section \ref{sec:topological-A-infty} below explains these definitions, and establishes |
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the desired equivalence. This is quite involved, and in particular requires us to generalise the definition of blob homology to allow $A_\infty$ algebras |
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as inputs, and to re-establish many of the properties of blob homology in this generality. Many readers may prefer to read the |
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Definitions \ref{defn:topological-algebra} and \ref{defn:topological-module} of `topological' $A_\infty$-categories, and Definition \ref{???} of the |
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self-tensor product of a `topological' $A_\infty$-bimodule, then skip to \S \ref{sec:boundary-action} and \S \ref{sec:gluing-formula} for the proofs |
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of the gluing formula in the topological context. |
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\subsection{`Topological' $A_\infty$ $n$-categories} |
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\label{sec:topological-A-infty}% |
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This section prepares the ground for establishing Property \ref{property:gluing} by defining the notion of a \emph{topological $A_\infty$-$n$-category}. |
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The main result of this section is |
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\begin{thm} |
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Topological $A_\infty$-$1$-categories are equivalent to the usual notion of |
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$A_\infty$-$1$-categories. |
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\end{thm} |
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Before proving this theorem, we embark upon a long string of definitions. |
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For expository purposes, we begin with the $n=1$ special cases,\scott{Why are we treating the $n>1$ cases at all?} and define |
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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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to the general $n$ case, defining topological $A_\infty$-$n$-categories and their modules. |
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\nn{Something about duals?} |
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\todo{Explain that we're not making contact with any previous notions for the general $n$ case?} |
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\kevin{probably we should say something about the relation |
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to [framed] $E_\infty$ algebras |
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} |
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\todo{} |
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Various citations we might want to make: |
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\begin{itemize} |
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\item \cite{MR2061854} McClure and Smith's review article |
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\item \cite{MR0420610} May, (inter alia, definition of $E_\infty$ operad) |
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\item \cite{MR0236922,MR0420609} Boardman and Vogt |
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\item \cite{MR1256989} definition of framed little-discs operad |
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\end{itemize} |
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\begin{defn} |
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\label{defn:topological-algebra}% |
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A ``topological $A_\infty$-algebra'' $A$ consists of the following data. |
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\begin{enumerate} |
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\item For each $1$-manifold $J$ diffeomorphic to the standard interval |
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$I=\left[0,1\right]$, a complex of vector spaces $A(J)$. |
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% either roll functoriality into the evaluation map |
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\item For each pair of intervals $J,J'$ an `evaluation' chain map |
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$\ev_{J \to J'} : \CD{J \to J'} \tensor A(J) \to A(J')$. |
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\item For each decomposition of intervals $J = J'\cup J''$, |
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a gluing map $\gl_{J',J''} : A(J') \tensor A(J'') \to A(J)$. |
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% or do it as two separate pieces of data |
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%\item along with an `evaluation' chain map $\ev_J : \CD{J} \tensor A(J) \to A(J)$, |
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%\item for each diffeomorphism $\phi : J \to J'$, an isomorphism $A(\phi) : A(J) \isoto A(J')$, |
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%\item and for each pair of intervals $J,J'$ a gluing map $\gl_{J,J'} : A(J) \tensor A(J') \to A(J \cup J')$, |
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\end{enumerate} |
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This data is required to satisfy the following conditions. |
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\begin{itemize} |
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\item The evaluation chain map is associative, in that the diagram |
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\begin{equation*} |
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\xymatrix{ |
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& \quad \mathclap{\CD{J' \to J''} \tensor \CD{J \to J'} \tensor A(J)} \quad \ar[dr]^{\id \tensor \ev_{J \to J'}} \ar[dl]_{\compose \tensor \id} & \\ |
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\CD{J' \to J''} \tensor A(J') \ar[dr]^{\ev_{J' \to J''}} & & \CD{J \to J''} \tensor A(J) \ar[dl]_{\ev_{J \to J''}} \\ |
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& A(J'') & |
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} |
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\end{equation*} |
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commutes up to homotopy. |
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Here the map $$\compose : \CD{J' \to J''} \tensor \CD{J \to J'} \to \CD{J \to J''}$$ is a composition: take products of singular chains first, then compose diffeomorphisms. |
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%% or the version for separate pieces of data: |
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%\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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%\item The evaluation chain map is associative, in that the diagram |
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%\begin{equation*} |
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%\xymatrix{ |
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%\CD{J} \tensor \CD{J} \tensor A(J) \ar[r]^{\id \tensor \ev_J} \ar[d]_{\compose \tensor \id} & |
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%\CD{J} \tensor A(J) \ar[d]^{\ev_J} \\ |
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%\CD{J} \tensor A(J) \ar[r]_{\ev_J} & |
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%A(J) |
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%} |
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%\end{equation*} |
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%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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\item The gluing maps are \emph{strictly} associative. That is, given $J$, $J'$ and $J''$, the diagram |
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\begin{equation*} |
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\xymatrix{ |
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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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A(J \cup J') \tensor A(J'') \ar[d]^{\gl_{J \cup J', J''}} \\ |
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A(J) \tensor A(J' \cup J'') \ar[rr]_{\gl_{J, J' \cup J''}} && |
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A(J \cup J' \cup J'') |
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} |
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\end{equation*} |
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commutes. |
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\item The gluing and evaluation maps are compatible. |
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\nn{give diagram, or just say ``in the obvious way", or refer to diagram in blob eval map section?} |
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\end{itemize} |
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\end{defn} |
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\begin{rem} |
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We can restrict the evaluation map to $0$-chains, and see that $J \mapsto A(J)$ and $(\phi:J \to J') \mapsto \ev_{J \to J'}(\phi, \bullet)$ together |
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constitute a functor from the category of intervals and diffeomorphisms between them to the category of complexes of vector spaces. |
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Further, once this functor has been specified, we only need to know how the evaluation map acts when $J = J'$. |
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\end{rem} |
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%% if we do things separately, we should say this: |
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%\begin{rem} |
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%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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%intervals and diffeomorphisms between them to the category of complexes of vector spaces. |
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%Further, one can combine the second and third pieces of data, asking instead for a map |
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%\begin{equation*} |
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%\ev_{J,J'} : \CD{J \to J'} \tensor A(J) \to A(J'). |
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%\end{equation*} |
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%(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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%diffeomorphisms in $\CD{J'}$.) |
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%\end{rem} |
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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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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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\begin{equation*} |
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\gl : A(J, c_-, c_0) \tensor A(J', c_0, c_+) \to A(J \cup J', c_-, c_+). |
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\end{equation*} |
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The action of diffeomorphisms (and of $k$-parameter families of diffeomorphisms) ignores the boundary conditions. |
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\todo{we presumably need to say something about $\id_c \in A(J, c, c)$.} |
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At this point we can give two motivating examples. The first is `chains of maps to $M$' for some fixed target space $M$. |
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\begin{defn} |
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Define the topological $A_\infty$ category $C_*(\Maps(\bullet \to M))$ by |
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\begin{enumerate} |
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\item $A(J) = C_*(\Maps(J \to M))$, singular chains on the space of smooth maps from $J$ to $M$, |
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\item $\ev_{J,J'} : \CD{J \to J'} \tensor A(J) \to A(J')$ is the composition |
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\begin{align*} |
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\CD{J \to J'} \tensor C_*(\Maps(J \to M)) & \to C_*(\Diff(J \to J') \times \Maps(J \to M)) \\ & \to C_*(\Maps(J' \to M)), |
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\end{align*} |
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where the first map is the product of singular chains, and the second is precomposition by the inverse of a diffeomorphism, |
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\item $\gl_{J,J'} : A(J) \tensor A(J')$ takes the product of singular chains, then glues maps to $M$ together. |
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\end{enumerate} |
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The associativity conditions are trivially satisfied. |
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\end{defn} |
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The second example is simply the blob complex of $Y \times J$, for any $n-1$ manifold $Y$. We define $A(J) = \bc_*(Y \times J)$. |
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Observe $\Diff(J \to J')$ embeds into $\Diff(Y \times J \to Y \times J')$. The evaluation and gluing maps then come directly from Properties |
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\ref{property:evaluation} and \ref{property:gluing-map} respectively. We'll often write $bc_*(Y)$ for this algebra. |
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The definition of a module follows closely the definition of an algebra or category. |
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\begin{defn} |
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\label{defn:topological-module}% |
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A topological $A_\infty$-(left-)module $M$ over a topological $A_\infty$ category $A$ |
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consists of the following data. |
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\begin{enumerate} |
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\item A functor $K \mapsto M(K)$ from $1$-manifolds diffeomorphic to the standard interval, with the upper boundary point `marked', to complexes of vector spaces. |
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\item For each pair of such marked intervals, |
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an `evaluation' chain map $\ev_{K\to K'} : \CD{K \to K'} \tensor M(K) \to M(K')$. |
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\item For each decomposition $K = J\cup K'$ of the marked interval |
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$K$ into an unmarked interval $J$ and a marked interval $K'$, a gluing map |
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$\gl_{J,K'} : A(J) \tensor M(K') \to M(K)$. |
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\end{enumerate} |
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The above data is required to satisfy |
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conditions analogous to those in Definition \ref{defn:topological-algebra}. |
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\end{defn} |
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For any manifold $X$ with $\bdy X = Y$ (or indeed just with $Y$ a codimension $0$-submanifold of $\bdy X$) we can think of $\bc_*(X)$ as |
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a topological $A_\infty$ module over $\bc_*(Y)$, the topological $A_\infty$ category described above. |
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For each interval $K$, we have $M(K) = \bc_*((Y \times K) \cup_Y X)$. |
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(Here we glue $Y \times pt$ to $Y \subset \bdy X$, where $pt$ is the marked point of $K$.) Again, the evaluation and gluing maps come directly from Properties |
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\ref{property:evaluation} and \ref{property:gluing-map} respectively. |
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The definition of a bimodule is like the definition of a module, |
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except that we have two disjoint marked intervals $K$ and $L$, one with a marked point |
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on the upper boundary and the other with a marked point on the lower boundary. |
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There are evaluation maps corresponding to gluing unmarked intervals |
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to the unmarked ends of $K$ and $L$. |
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Let $X$ be an $n$-manifold with a copy of $Y \du -Y$ embedded as a |
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codimension-0 submanifold of $\bdy X$. |
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Then the the assignment $K,L \mapsto \bc_*(X \cup_Y (Y\times K) \cup_{-Y} (-Y\times L))$ has the |
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structure of a topological $A_\infty$ bimodule over $\bc_*(Y)$. |
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Next we define the coend |
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(or gluing or tensor product or self tensor product, depending on the context) |
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$\gl(M)$ of a topological $A_\infty$ bimodule $M$. This will be an `initial' or `universal' object satisfying various properties. |
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\begin{defn} |
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We define a category $\cG(M)$. Objects consist of the following data. |
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\begin{itemize} |
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\item For each interval $N$ with both endpoints marked, a complex of vector spaces C(N). |
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\item For each pair of intervals $N,N'$ an evaluation chain map |
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$\ev_{N \to N'} : \CD{N \to N'} \tensor C(N) \to C(N')$. |
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\item For each decomposition of intervals $N = K\cup L$, |
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a gluing map $\gl_{K,L} : M(K,L) \to C(N)$. |
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\end{itemize} |
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This data must satisfy the following conditions. |
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\begin{itemize} |
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\item The evaluation maps are associative. |
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\nn{up to homotopy?} |
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\item Gluing is strictly associative. |
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That is, given a decomposition $N = K\cup J\cup L$, the chain maps associated to |
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$K\du J\du L \to (K\cup J)\du L \to N$ and $K\du J\du L \to K\du (J\cup L) \to N$ |
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agree. |
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\item the gluing and evaluation maps are compatible. |
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\end{itemize} |
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A morphism $f$ between such objects $C$ and $C'$ is a chain map $f_N : C(N) \to C'(N)$ for each interval $N$ with both endpoints marked, |
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satisfying the following conditions. |
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\begin{itemize} |
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\item For each pair of intervals $N,N'$, the diagram |
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\begin{equation*} |
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\xymatrix{ |
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\CD{N \to N'} \tensor C(N) \ar[d]_{\ev} \ar[r]^{\id \tensor f_N} & \CD{N \to N'} \tensor C'(N) \ar[d]^{\ev} \\ |
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C(N) \ar[r]_{f_N} & C'(N) |
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} |
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\end{equation*} |
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commutes. |
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\item For each decomposition of intervals $N = K \cup L$, the gluing map for $C'$, $\gl'_{K,L} : M(K,L) \to C'(N)$ is the composition |
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$$M(K,L) \xto{\gl_{K,L}} C(N) \xto{f_N} C'(N).$$ |
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\end{itemize} |
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\end{defn} |
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We now define $\gl(M)$ to be an initial object in the category $\cG{M}$. This just says that for any other object $C'$ in $\cG{M}$, |
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there are chain maps $f_N: \gl(M)(N) \to C'(N)$, compatible with the action of families of diffeomorphisms, so that the gluing maps $M(K,L) \to C'(N)$ |
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factor through the gluing maps for $\gl(M)$. |
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We return to our two favourite examples. First, the coend of the topological $A_\infty$ category $C_*(\Maps(\bullet \to M))$ as a bimodule over itself |
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is essentially $C_*(\Maps(S^1 \to M))$. \todo{} |
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For the second example, given $X$ and $Y\du -Y \sub \bdy X$, the assignment |
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$$N \mapsto \bc_*(X \cup_{Y\du -Y} (N\times Y))$$ clearly gives an object in $\cG{M}$. |
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Showing that it is an initial object is the content of the gluing theorem proved below. |
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The definitions for a topological $A_\infty$-$n$-category are very similar to the above |
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$n=1$ case. |
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One replaces intervals with manifolds diffeomorphic to the ball $B^n$. |
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Marked points are replaced by copies of $B^{n-1}$ in $\bdy B^n$. |
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\nn{give examples: $A(J^n) = \bc_*(Z\times J)$ and $A(J^n) = C_*(\Maps(J \to M))$.} |
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65
15a79fb469e1
edits for "basic properties" section
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
55
diff
changeset
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\todo{the motivating example $C_*(\Maps(X, M))$} |
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\newcommand{\skel}[1]{\operatorname{skeleton}(#1)} |
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Given a topological $A_\infty$-category $\cC$, we can construct an `algebraic' $A_\infty$ category $\skel{\cC}$. First, pick your |
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favorite diffeomorphism $\phi: I \cup I \to I$. |
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\begin{defn} |
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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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\begin{equation*} |
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m_2 \cC(I) \tensor \cC(I) \xrightarrow{\gl_{I,I}} \cC(I \cup I) \xrightarrow{\cC(\phi)} \cC(I). |
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\end{equation*} |
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Next, we define all the `higher associators' $m_k$ by |
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\todo{} |
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\end{defn} |
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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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think of this as a generalisation of the blob complex, although the construction we give will \emph{not} specialise to exactly the usual definition |
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in the case the $A$ is actually an associative category. |
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We'll first define $\cT_{k,n}$ to be the set of planar forests consisting of $n-k$ trees, with a total of $n$ leaves. Thus |
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\todo{$\cT_{0,n}$ has 1 element, with $n$ vertical lines, $\cT_{1,n}$ has $n-1$ elements, each with a single trivalent vertex, $\cT_{2,n}$ etc...} |
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\begin{align*} |
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\end{align*} |
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\begin{defn} |
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The topological $A_\infty$ category $\bc_*^A$ is doubly graded, by `blob degree' and `internal degree'. We'll write $\bc_k^A$ for the blob degree $k$ piece. |
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The homological degree of an element $a \in \bc_*^A(J)$ |
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is the sum of the blob degree and the internal degree. |
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We first define $\bc_0^A(J)$ as a vector space by |
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\begin{equation*} |
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\bc_0^A(J) = \DirectSum_{\substack{\{J_i\}_{i=1}^n \\ \mathclap{\bigcup_i J_i = J}}} \Tensor_{i=1}^n (\CD{J_i \to I} \tensor A). |
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\end{equation*} |
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(That is, for each division of $J$ into finitely many subintervals, |
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we have the tensor product of chains of diffeomorphisms from each subinterval to the standard interval, |
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and a copy of $A$ for each subinterval.) |
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The internal degree of an element $(f_1 \tensor a_1, \ldots, f_n \tensor a_n)$ is the sum of the dimensions of the singular chains |
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plus the sum of the homological degrees of the elements of $A$. |
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The differential is defined just by the graded Leibniz rule and the differentials on $\CD{J_i \to I}$ and on $A$. |
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Next, |
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\begin{equation*} |
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\bc_1^A(J) = \DirectSum_{\substack{\{J_i\}_{i=1}^n \\ \mathclap{\bigcup_i J_i = J}}} \DirectSum_{T \in \cT_{1,n}} \Tensor_{i=1}^n (\CD{J_i \to I} \tensor A). |
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\end{equation*} |
|
298 |
\end{defn} |
|
299 |
||
300 |
\begin{figure}[!ht] |
|
301 |
\begin{equation*} |
|
302 |
\mathfig{0.7}{associahedron/A4-vertices} |
|
303 |
\end{equation*} |
|
304 |
\caption{The vertices of the $k$-dimensional associahedron are indexed by binary trees on $k+2$ leaves.} |
|
305 |
\label{fig:A4-vertices} |
|
306 |
\end{figure} |
|
307 |
||
308 |
\begin{figure}[!ht] |
|
309 |
\begin{equation*} |
|
310 |
\mathfig{0.7}{associahedron/A4-faces} |
|
311 |
\end{equation*} |
|
312 |
\caption{The faces of the $k$-dimensional associahedron are indexed by trees with $2$ vertices on $k+2$ leaves.} |
|
313 |
\label{fig:A4-vertices} |
|
314 |
\end{figure} |
|
315 |
||
316 |
\newcommand{\tm}{\widetilde{m}} |
|
317 |
||
318 |
Let $\tm_1(a) = a$. |
|
319 |
||
320 |
We now define $\bdy(\tm_k(a_1 \tensor \cdots \tensor a_k))$, first giving an opaque formula, then explaining the combinatorics behind it. |
|
321 |
\begin{align} |
|
322 |
\notag \bdy(\tm_k(a_1 & \tensor \cdots \tensor a_k)) = \\ |
|
323 |
\label{eq:bdy-tm-k-1} & \phantom{+} \sum_{\ell'=0}^{k-1} (-1)^{\abs{\tm_k}+\sum_{j=1}^{\ell'} \abs{a_j}} \tm_k(a_1 \tensor \cdots \tensor \bdy a_{\ell'+1} \tensor \cdots \tensor a_k) + \\ |
|
324 |
\label{eq:bdy-tm-k-2} & + \sum_{\ell=1}^{k-1} \tm_{\ell}(a_1 \tensor \cdots \tensor a_{\ell}) \tensor \tm_{k-\ell}(a_{\ell+1} \tensor \cdots \tensor a_k) + \\ |
|
325 |
\label{eq:bdy-tm-k-3} & + \sum_{\ell=1}^{k-1} \sum_{\ell'=0}^{l-1} (-1)^{\abs{\tm_k}+\sum_{j=1}^{\ell'} \abs{a_j}} \tm_{\ell}(a_1 \tensor \cdots \tensor m_{k-\ell + 1}(a_{\ell' + 1} \tensor \cdots \tensor a_{\ell' + k - \ell + 1}) \tensor \cdots \tensor a_k) |
|
326 |
\end{align} |
|
327 |
The first set of terms in $\bdy(\tm_k(a_1 \tensor \cdots \tensor a_k))$ just have $\bdy$ acting on each argument $a_i$. |
|
328 |
The terms appearing in \eqref{eq:bdy-tm-k-2} and \eqref{eq:bdy-tm-k-3} are indexed by trees with $2$ vertices on $k+1$ leaves. |
|
329 |
Note here that we have one more leaf than there arguments of $\tm_k$. |
|
330 |
(See Figure \ref{fig:A4-vertices}, in which the rightmost branches are helpfully drawn in red.) |
|
331 |
We will treat the vertices which involve a rightmost (red) branch differently from the vertices which only involve the first $k$ leaves. |
|
332 |
The terms in \eqref{eq:bdy-tm-k-2} arise in the cases in which both |
|
333 |
vertices are rightmost, and the corresponding term in $\bdy(\tm_k(a_1 \tensor \cdots \tensor a_k))$ is a tensor product of the form |
|
334 |
$$\tm_{\ell}(a_1 \tensor \cdots \tensor a_{\ell}) \tensor \tm_{k-\ell}(a_{\ell+1} \tensor \cdots \tensor a_k)$$ |
|
335 |
where $\ell + 1$ and $k - \ell + 1$ are the number of branches entering the vertices. |
|
336 |
If only one vertex is rightmost, we get the term $$\tm_{\ell}(a_1 \tensor \cdots \tensor m_{k-\ell+1}(a_{\ell' + 1} \tensor \cdots \tensor a_{\ell' + k - \ell}) \tensor \cdots \tensor a_k)$$ |
|
337 |
in \eqref{eq:bdy-tm-k-3}, |
|
338 |
where again $\ell + 1$ is the number of branches entering the rightmost vertex, $k-\ell+1$ is the number of branches entering the other vertex, and $\ell'$ is the number of edges meeting the rightmost vertex which start to the left of the other vertex. |
|
339 |
For example, we have |
|
340 |
\begin{align*} |
|
341 |
\bdy(\tm_2(a \tensor b)) & = \left(\tm_2(\bdy a \tensor b) + (-1)^{\abs{a}} \tm_2(a \tensor \bdy b)\right) + \\ |
|
342 |
& \qquad - a \tensor b + m_2(a \tensor b) \\ |
|
343 |
\bdy(\tm_3(a \tensor b \tensor c)) & = \left(- \tm_3(\bdy a \tensor b \tensor c) + (-1)^{\abs{a} + 1} \tm_3(a \tensor \bdy b \tensor c) + (-1)^{\abs{a} + \abs{b} + 1} \tm_3(a \tensor b \tensor \bdy c)\right) + \\ |
|
344 |
& \qquad + \left(- \tm_2(a \tensor b) \tensor c + a \tensor \tm_2(b \tensor c)\right) + \\ |
|
345 |
& \qquad + \left(- \tm_2(m_2(a \tensor b) \tensor c) + \tm_2(a, m_2(b \tensor c)) + m_3(a \tensor b \tensor c)\right) |
|
346 |
\end{align*} |
|
347 |
\begin{align*} |
|
348 |
\bdy(& \tm_4(a \tensor b \tensor c \tensor d)) = \left(\tm_4(\bdy a \tensor b \tensor c \tensor d) + \cdots + \tm_4(a \tensor b \tensor c \tensor \bdy d)\right) + \\ |
|
349 |
& + \left(\tm_3(a \tensor b \tensor c) \tensor d + \tm_2(a \tensor b) \tensor \tm_2(c \tensor d) + a \tensor \tm_3(b \tensor c \tensor d)\right) + \\ |
|
350 |
& + \left(\tm_3(m_2(a \tensor b) \tensor c \tensor d) + \tm_3(a \tensor m_2(b \tensor c) \tensor d) + \tm_3(a \tensor b \tensor m_2(c \tensor d))\right. + \\ |
|
351 |
& + \left.\tm_2(m_3(a \tensor b \tensor c) \tensor d) + \tm_2(a \tensor m_3(b \tensor c \tensor d)) + m_4(a \tensor b \tensor c \tensor d)\right) \\ |
|
352 |
\end{align*} |
|
353 |
See Figure \ref{fig:A4-terms}, comparing it against Figure \ref{fig:A4-faces}, to see this illustrated in the case $k=4$. There the $3$ faces closest |
|
354 |
to the top of the diagram have two rightmost vertices, while the other $6$ faces have only one. |
|
355 |
||
356 |
\begin{figure}[!ht] |
|
357 |
\begin{equation*} |
|
358 |
\mathfig{1.0}{associahedron/A4-terms} |
|
359 |
\end{equation*} |
|
360 |
\caption{The terms of $\bdy(\tm_k(a_1 \tensor \cdots \tensor a_k))$ correspond to the faces of the $k-1$ dimensional associahedron.} |
|
361 |
\label{fig:A4-terms} |
|
362 |
\end{figure} |
|
363 |
||
364 |
\begin{lem} |
|
365 |
This definition actually results in a chain complex, that is $\bdy^2 = 0$. |
|
366 |
\end{lem} |
|
367 |
\begin{proof} |
|
368 |
\newcommand{\T}{\text{---}} |
|
369 |
\newcommand{\ssum}[1]{{\sum}^{(#1)}} |
|
370 |
For the duration of this proof, inside a summation over variables $l_1, \ldots, l_m$, an expression with $m$ dashes will be interpreted |
|
371 |
by replacing each dash with contiguous factors from $a_1 \tensor \cdots \tensor a_k$, so the first dash takes the first $l_1$ factors, the second |
|
372 |
takes the next $l_2$ factors, and so on. Further, we'll write $\ssum{m}$ for $\sum_{\sum_{i=1}^m l_i = k}$. |
|
373 |
In this notation, the formula for the differential becomes |
|
374 |
\begin{align} |
|
375 |
\notag |
|
376 |
\bdy \tm(\T) & = \ssum{2} \tm(\T) \tensor \tm(\T) \times \sigma_{0;l_1,l_2} + \ssum{3} \tm(\T \tensor m(\T) \tensor \T) \times \tau_{0;l_1,l_2,l_3} \\ |
|
377 |
\intertext{and we calculate} |
|
378 |
\notag |
|
379 |
\bdy^2 \tm(\T) & = \ssum{2} \bdy \tm(\T) \tensor \tm(\T) \times \sigma_{0;l_1,l_2} \\ |
|
380 |
\notag & \qquad + \ssum{2} \tm(\T) \tensor \bdy \tm(\T) \times \sigma_{0;l_1,l_2} \\ |
|
381 |
\notag & \qquad + \ssum{3} \bdy \tm(\T \tensor m(\T) \tensor \T) \times \tau_{0;l_1,l_2,l_3} \\ |
|
382 |
\label{eq:d21} & = \ssum{3} \tm(\T) \tensor \tm(\T) \tensor \tm(\T) \times \sigma_{0;l_1+l_2,l_3} \sigma_{0;l_1,l_2} \\ |
|
383 |
\label{eq:d22} & \qquad + \ssum{4} \tm(\T \tensor m(\T) \tensor \T) \tensor \tm(\T) \times \sigma_{0;l_1+l_2+l_3,l_4} \tau_{0;l_1,l_2,l_3} \\ |
|
384 |
\label{eq:d23} & \qquad + \ssum{3} \tm(\T) \tensor \tm(\T) \tensor \tm(\T) \times \sigma_{0;l_1,l_2+l_3} \sigma_{l_1;l_2,l_3} \\ |
|
385 |
\label{eq:d24} & \qquad + \ssum{4} \tm(\T) \tensor \tm(\T \tensor m(\T) \tensor \T) \times \sigma_{0;l_1,l_2+l_3+l_4} \tau_{l_1;l_2,l_3,l_4} \\ |
|
386 |
\label{eq:d25} & \qquad + \ssum{4} \tm(\T \tensor m(\T) \tensor \T) \tensor \tm(\T) \times \tau_{0;l_1,l_2,l_3+l_4} ??? \\ |
|
387 |
\label{eq:d26} & \qquad + \ssum{4} \tm(\T) \tensor \tm(\T \tensor m(\T) \tensor \T) \times \tau_{0;l_1+l_2,l_3,l_4} \sigma_{0;l_1,l_2} \\ |
|
388 |
\label{eq:d27} & \qquad + \ssum{5} \tm(\T \tensor m(\T) \tensor \T \tensor m(\T) \tensor \T) \times \tau_{0;l_1+l_2+l_3,l_4,l_5} \tau_{0;l_1,l_2,l_3} \\ |
|
389 |
\label{eq:d28} & \qquad + \ssum{5} \tm(\T \tensor m(\T \tensor m(\T) \tensor \T) \tensor \T) \times \tau_{0;l_1,l_2+l_3+l_4,l_5} ??? \\ |
|
390 |
\label{eq:d29} & \qquad + \ssum{5} \tm(\T \tensor m(\T) \tensor \T \tensor m(\T) \tensor \T) \times \tau_{0;l_1,l_2,l_3+l_4+l_5} ??? |
|
391 |
\end{align} |
|
392 |
Now, we see the the expressions on the right hand side of line \eqref{eq:d21} and those on \eqref{eq:d23} cancel. Similarly, line \eqref{eq:d22} cancels |
|
393 |
with \eqref{eq:d25}, \eqref{eq:d24} with \eqref{eq:d26}, and \eqref{eq:d27} with \eqref{eq:d29}. Finally, we need to see that \eqref{eq:d28} gives $0$, |
|
394 |
by the usual relations between the $m_k$ in an $A_\infty$ algebra. |
|
395 |
\end{proof} |
|
396 |
||
397 |
\nn{Need to let the input $n$-category $C$ be a graded thing (e.g. DG |
|
398 |
$n$-category or $A_\infty$ $n$-category). DG $n$-category case is pretty |
|
399 |
easy, I think, so maybe it should be done earlier??} |
|
400 |
||
401 |
\bigskip |
|
402 |
||
403 |
Outline: |
|
404 |
\begin{itemize} |
|
405 |
\item recall defs of $A_\infty$ category (1-category only), modules, (self-) tensor product. |
|
406 |
use graphical/tree point of view, rather than following Keller exactly |
|
407 |
\item define blob complex in $A_\infty$ case; fat mapping cones? tree decoration? |
|
408 |
\item topological $A_\infty$ cat def (maybe this should go first); also modules gluing |
|
65
15a79fb469e1
edits for "basic properties" section
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
55
diff
changeset
|
409 |
\item motivating example: $C_*(\Maps(X, M))$ |
55 | 410 |
\item maybe incorporate dual point of view (for $n=1$), where points get |
411 |
object labels and intervals get 1-morphism labels |
|
412 |
\end{itemize} |
|
413 |
||
414 |
||
415 |
\subsection{$A_\infty$ action on the boundary} |
|
416 |
\label{sec:boundary-action}% |
|
417 |
Let $Y$ be an $n{-}1$-manifold. |
|
418 |
The collection of complexes $\{\bc_*(Y\times I; a, b)\}$, where $a, b \in \cC(Y)$ are boundary |
|
419 |
conditions on $\bd(Y\times I) = Y\times \{0\} \cup Y\times\{1\}$, has the structure |
|
420 |
of an $A_\infty$ category. |
|
421 |
||
422 |
Composition of morphisms (multiplication) depends of a choice of homeomorphism |
|
423 |
$I\cup I \cong I$. Given this choice, gluing gives a map |
|
424 |
\eq{ |
|
425 |
\bc_*(Y\times I; a, b) \otimes \bc_*(Y\times I; b, c) \to \bc_*(Y\times (I\cup I); a, c) |
|
426 |
\cong \bc_*(Y\times I; a, c) |
|
427 |
} |
|
428 |
Using (\ref{CDprop}) and the inclusion $\Diff(I) \sub \Diff(Y\times I)$ gives the various |
|
429 |
higher associators of the $A_\infty$ structure, more or less canonically. |
|
430 |
||
431 |
\nn{is this obvious? does more need to be said?} |
|
432 |
||
433 |
Let $\cA(Y)$ denote the $A_\infty$ category $\bc_*(Y\times I; \cdot, \cdot)$. |
|
434 |
||
435 |
Similarly, if $Y \sub \bd X$, a choice of collaring homeomorphism |
|
436 |
$(Y\times I) \cup_Y X \cong X$ gives the collection of complexes $\bc_*(X; r, a)$ |
|
437 |
(variable $a \in \cC(Y)$; fixed $r \in \cC(\bd X \setmin Y)$) the structure of a representation of the |
|
438 |
$A_\infty$ category $\{\bc_*(Y\times I; \cdot, \cdot)\}$. |
|
439 |
Again the higher associators come from the action of $\Diff(I)$ on a collar neighborhood |
|
440 |
of $Y$ in $X$. |
|
441 |
||
442 |
In the next section we use the above $A_\infty$ actions to state and prove |
|
443 |
a gluing theorem for the blob complexes of $n$-manifolds. |
|
444 |
||
445 |
||
446 |
\subsection{The gluing formula} |
|
447 |
\label{sec:gluing-formula}% |
|
448 |
Let $Y$ be an $n{-}1$-manifold and let $X$ be an $n$-manifold with a copy |
|
449 |
of $Y \du -Y$ contained in its boundary. |
|
450 |
Gluing the two copies of $Y$ together we obtain a new $n$-manifold $X\sgl$. |
|
451 |
We wish to describe the blob complex of $X\sgl$ in terms of the blob complex |
|
452 |
of $X$. |
|
453 |
More precisely, we want to describe $\bc_*(X\sgl; c\sgl)$, |
|
454 |
where $c\sgl \in \cC(\bd X\sgl)$, |
|
455 |
in terms of the collection $\{\bc_*(X; c, \cdot, \cdot)\}$, thought of as a representation |
|
456 |
of the $A_\infty$ category $\cA(Y\du-Y) \cong \cA(Y)\times \cA(Y)\op$. |
|
457 |
||
458 |
\begin{thm} |
|
459 |
$\bc_*(X\sgl; c\sgl)$ is quasi-isomorphic to the the self tensor product |
|
460 |
of $\{\bc_*(X; c, \cdot, \cdot)\}$ over $\cA(Y)$. |
|
461 |
\end{thm} |
|
462 |
||
463 |
The proof will occupy the remainder of this section. |
|
464 |
||
465 |
\nn{...} |
|
466 |
||
467 |
\bigskip |
|
468 |
||
469 |
\nn{need to define/recall def of (self) tensor product over an $A_\infty$ category} |
|
470 |