author | Kevin Walker <kevin@canyon23.net> |
Tue, 31 Aug 2010 11:18:26 -0700 | |
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%!TEX root = ../blob1.tex |
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\section{The blob complex for \texorpdfstring{$A_\infty$}{A-infinity} \texorpdfstring{$n$}{n}-categories} |
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\label{sec:ainfblob} |
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Given an $A_\infty$ $n$-category $\cC$ and an $n$-manifold $M$, we make the anticlimactically tautological definition of the blob |
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complex $\bc_*(M;\cC)$ to be the homotopy colimit $\cl{\cC}(M)$ of \S\ref{ss:ncat_fields}. |
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We will show below |
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in Corollary \ref{cor:new-old} |
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that when $\cC$ is obtained from a system of fields $\cD$ |
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as the blob complex of an $n$-ball (see Example \ref{ex:blob-complexes-of-balls}), |
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$\cl{\cC}(M)$ is homotopy equivalent to |
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our original definition of the blob complex $\bc_*^\cD(M)$. |
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%\medskip |
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%An important technical tool in the proofs of this section is provided by the idea of ``small blobs". |
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%Fix $\cU$, an open cover of $M$. |
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%Define the ``small blob complex" $\bc^{\cU}_*(M)$ to be the subcomplex of $\bc_*(M)$ |
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%of all blob diagrams in which every blob is contained in some open set of $\cU$, |
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%and moreover each field labeling a region cut out by the blobs is splittable |
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%into fields on smaller regions, each of which is contained in some open set of $\cU$. |
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% |
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%\begin{thm}[Small blobs] \label{thm:small-blobs} |
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%The inclusion $i: \bc^{\cU}_*(M) \into \bc_*(M)$ is a homotopy equivalence. |
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%\end{thm} |
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%The proof appears in \S \ref{appendix:small-blobs}. |
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\subsection{A product formula} |
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\label{ss:product-formula} |
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Given a system of fields $\cE$ and a $n{-}k$-manifold $F$, recall from |
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Example \ref{ex:blob-complexes-of-balls} that there is an $A_\infty$ $k$-category $\cC_F$ |
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defined by $\cC_F(X) = \cE(X\times F)$ if $\dim(X) < k$ and |
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$\cC_F(X) = \bc_*^\cE(X\times F)$ if $\dim(X) = k$. |
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\begin{thm} \label{thm:product} |
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Let $Y$ be a $k$-manifold. |
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Then there is a homotopy equivalence between ``old-fashioned" (blob diagrams) |
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and ``new-fangled" (hocolimit) blob complexes |
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\[ |
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\cB_*(Y \times F) \htpy \cl{\cC_F}(Y) . |
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\]\end{thm} |
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\begin{proof} |
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We will use the concrete description of the homotopy colimit from \S\ref{ss:ncat_fields}. |
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First we define a map |
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\[ |
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\psi: \cl{\cC_F}(Y) \to \bc_*(Y\times F;C) . |
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\] |
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On 0-simplices of the hocolimit |
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we just glue together the various blob diagrams on $X_i\times F$ |
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(where $X_i$ is a component of a permissible decomposition of $Y$) to get a blob diagram on |
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$Y\times F$. |
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For simplices of dimension 1 and higher we define the map to be zero. |
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It is easy to check that this is a chain map. |
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In the other direction, we will define a subcomplex $G_*\sub \bc_*(Y\times F;C)$ |
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and a map |
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\[ |
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\phi: G_* \to \cl{\cC_F}(Y) . |
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\] |
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Given a decomposition $K$ of $Y$ into $k$-balls $X_i$, let $K\times F$ denote the corresponding |
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decomposition of $Y\times F$ into the pieces $X_i\times F$. |
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Let $G_*\sub \bc_*(Y\times F;C)$ be the subcomplex generated by blob diagrams $a$ such that there |
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exists a decomposition $K$ of $Y$ such that $a$ splits along $K\times F$. |
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It follows from Lemma \ref{thm:small-blobs} that $\bc_*(Y\times F; C)$ |
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is homotopic to a subcomplex of $G_*$. |
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(If the blobs of $a$ are small with respect to a sufficiently fine cover then their |
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projections to $Y$ are contained in some disjoint union of balls.) |
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Note that the image of $\psi$ is equal to $G_*$. |
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We will define $\phi: G_* \to \cl{\cC_F}(Y)$ using the method of acyclic models. |
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Let $a$ be a generator of $G_*$. |
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Let $D(a)$ denote the subcomplex of $\cl{\cC_F}(Y)$ generated by all $(b, \ol{K})$ |
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such that $a$ splits along $K_0\times F$ and $b$ is a generator appearing |
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in an iterated boundary of $a$ (this includes $a$ itself). |
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(Recall that $\ol{K} = (K_0,\ldots,K_l)$ denotes a chain of decompositions; |
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see \S\ref{ss:ncat_fields}.) |
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By $(b, \ol{K})$ we really mean $(b^\sharp, \ol{K})$, where $b^\sharp$ is |
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$b$ split according to $K_0\times F$. |
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To simplify notation we will just write plain $b$ instead of $b^\sharp$. |
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Roughly speaking, $D(a)$ consists of 0-simplices which glue up to give |
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$a$ (or one of its iterated boundaries), 1-simplices which connect all the 0-simplices, |
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2-simplices which kill the homology created by the |
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1-simplices, and so on. |
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More formally, |
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\begin{lemma} \label{lem:d-a-acyclic} |
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$D(a)$ is acyclic. |
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\end{lemma} |
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\begin{proof} |
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We will prove acyclicity in the first couple of degrees, and |
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%\nn{in this draft, at least} |
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leave the general case to the reader. |
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Let $K$ and $K'$ be two decompositions (0-simplices) of $Y$ compatible with $a$. |
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We want to find 1-simplices which connect $K$ and $K'$. |
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We might hope that $K$ and $K'$ have a common refinement, but this is not necessarily |
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the case. |
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(Consider the $x$-axis and the graph of $y = x^2\sin(1/x)$ in $\r^2$.) |
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However, we {\it can} find another decomposition $L$ such that $L$ shares common |
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refinements with both $K$ and $K'$. |
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Let $KL$ and $K'L$ denote these two refinements. |
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Then 1-simplices associated to the four anti-refinements |
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$KL\to K$, $KL\to L$, $K'L\to L$ and $K'L\to K'$ |
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give the desired chain connecting $(a, K)$ and $(a, K')$ |
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(see Figure \ref{zzz4}). |
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\begin{figure}[!ht] |
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\begin{equation*} |
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\begin{tikzpicture} |
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\foreach \x/\label in {-3/K, 0/L, 3/K'} { |
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\node(\label) at (\x,0) {$\label$}; |
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} |
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\foreach \x/\la/\lb in {-1.5/K/L, 1.5/K'/L} { |
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\node(\la \lb) at (\x,-1.5) {$\la \lb$}; |
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\draw[->] (\la \lb) -- (\la); |
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\draw[->] (\la \lb) -- (\lb); |
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} |
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\end{tikzpicture} |
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\end{equation*} |
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\caption{Connecting $K$ and $K'$ via $L$} |
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\label{zzz4} |
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\end{figure} |
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Consider a different choice of decomposition $L'$ in place of $L$ above. |
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This leads to a cycle of 1-simplices. |
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We want to find 2-simplices which fill in this cycle. |
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Choose a decomposition $M$ which has common refinements with each of |
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$K$, $KL$, $L$, $K'L$, $K'$, $K'L'$, $L'$ and $KL'$. |
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(We also also require that $KLM$ antirefines to $KM$, etc.) |
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Then we have 2-simplices, as shown in Figure \ref{zzz5}, which do the trick. |
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(Each small triangle in Figure \ref{zzz5} can be filled with a 2-simplex.) |
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\begin{figure}[!ht] |
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%\begin{equation*} |
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%\mathfig{1.0}{tempkw/zz5} |
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%\end{equation*} |
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\begin{equation*} |
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\begin{tikzpicture} |
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\node(M) at (0,0) {$M$}; |
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\foreach \angle/\label in {0/K', 45/K'L, 90/L, 135/KL, 180/K, 225/KL', 270/L', 315/K'L'} { |
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\node(\label) at (\angle:4) {$\label$}; |
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} |
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\foreach \label in {K', L, K, L'} { |
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\node(\label M) at ($(M)!0.6!(\label)$) {$\label M$}; |
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\draw[->] (\label M)--(M); |
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\draw[->] (\label M)--(\label); |
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} |
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\foreach \k in {K, K'} { |
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\foreach \l in {L, L'} { |
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\node(\k \l M) at (intersection cs: first line={(\k M)--(\l)}, second line={(\l M)--(\k)}) {$\k \l M$}; |
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\draw[->] (\k \l M)--(M); |
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\draw[->] (\k \l M)--(\k \l ); |
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\draw[->] (\k \l M)--(\k M); |
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\draw[->] (\k \l M)--(\l); |
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\draw[->] (\k \l M)--(\l M); |
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\draw[->] (\k \l M)--(\k); |
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} |
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} |
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\draw[->] (K'L') to[bend right=10] (K'); |
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\draw[->] (K'L') to[bend left=10] (L'); |
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\draw[->] (KL') to[bend left=10] (K); |
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\draw[->] (KL') to[bend right=10] (L'); |
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\draw[->] (K'L) to[bend left=10] (K'); |
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\draw[->] (K'L) to[bend right=10] (L); |
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\draw[->] (KL) to[bend right=10] (K); |
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\draw[->] (KL) to[bend left=10] (L); |
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\end{tikzpicture} |
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119 | 177 |
\end{equation*} |
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\caption{Filling in $K$-$KL$-$L$-$K'L$-$K'$-$K'L'$-$L'$-$KL'$-$K$} |
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\label{zzz5} |
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\end{figure} |
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116 | 181 |
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Continuing in this way we see that $D(a)$ is acyclic. |
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\end{proof} |
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We are now in a position to apply the method of acyclic models to get a map |
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$\phi:G_* \to \cl{\cC_F}(Y)$. |
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We may assume that $\phi(a)$ has the form $(a, K) + r$, where $(a, K)$ is a 0-simplex |
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and $r$ is a sum of simplices of dimension 1 or higher. |
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We now show that $\phi\circ\psi$ and $\psi\circ\phi$ are homotopic to the identity. |
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First, $\psi\circ\phi$ is the identity on the nose: |
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\[ |
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\psi(\phi(a)) = \psi((a,K)) + \psi(r) = a + 0. |
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\] |
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Roughly speaking, $(a, K)$ is just $a$ chopped up into little pieces, and |
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$\psi$ glues those pieces back together, yielding $a$. |
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We have $\psi(r) = 0$ since $\psi$ is zero on $(\ge 1)$-simplices. |
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Second, $\phi\circ\psi$ is the identity up to homotopy by another argument based on the method of acyclic models. |
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To each generator $(b, \ol{K})$ of $G_*$ we associate the acyclic subcomplex $D(b)$ defined above. |
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Both the identity map and $\phi\circ\psi$ are compatible with this |
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collection of acyclic subcomplexes, so by the usual method of acyclic models argument these two maps |
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are homotopic. |
123 | 205 |
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This concludes the proof of Theorem \ref{thm:product}. |
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\end{proof} |
208 |
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\nn{need to prove a version where $E$ above has dimension $m<n$; result is an $n{-}m$-category} |
113 | 210 |
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123 | 211 |
\medskip |
113 | 212 |
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Taking $F$ above to be a point, we obtain the following corollary. |
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123 | 215 |
\begin{cor} |
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\label{cor:new-old} |
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Let $\cE$ be a system of fields (with local relations) and let $\cC_\cE$ be the $A_\infty$ |
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$n$-category obtained from $\cE$ by taking the blob complex of balls. |
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Then for all $n$-manifolds $Y$ the old-fashioned and new-fangled blob complexes are |
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homotopy equivalent: |
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\[ |
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\bc^\cE_*(Y) \htpy \cl{\cC_\cE}(Y) . |
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\] |
123 | 224 |
\end{cor} |
113 | 225 |
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\medskip |
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133 | 227 |
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Theorem \ref{thm:product} extends to the case of general fiber bundles |
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\[ |
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F \to E \to Y , |
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\] |
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an indeed even to the case of general maps |
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\[ |
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M\to Y . |
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\] |
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We outline two approaches to these generalizations. |
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The first is somewhat tautological, while the second is more amenable to |
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calculation. |
312 | 239 |
|
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We can generalize the definition of a $k$-category by replacing the categories |
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of $j$-balls ($j\le k$) with categories of $j$-balls $D$ equipped with a map $p:D\to Y$ |
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(c.f. \cite{MR2079378}). |
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Call this a $k$-category over $Y$. |
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A fiber bundle $F\to E\to Y$ gives an example of a $k$-category over $Y$: |
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assign to $p:D\to Y$ the blob complex $\bc_*(p^*(E))$, if $\dim(D) = k$, |
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or the fields $\cE(p^*(E))$, if $\dim(D) < k$. |
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($p^*(E)$ denotes the pull-back bundle over $D$.) |
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Let $\cF_E$ denote this $k$-category over $Y$. |
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We can adapt the homotopy colimit construction (based decompositions of $Y$ into balls) to |
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get a chain complex $\cl{\cF_E}(Y)$. |
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The proof of Theorem \ref{thm:product} goes through essentially unchanged |
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to show that |
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\begin{thm} |
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Let $F \to E \to Y$ be a fiber bundle and let $\cF_E$ be the $k$-category over $Y$ defined above. |
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Then |
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\[ |
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\bc_*(E) \simeq \cl{\cF_E}(Y) . |
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\] |
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\qed |
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\end{thm} |
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|
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We can generalize this result still further by noting that it is not really necessary |
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for the definition of $\cF_E$ that $E\to Y$ be a fiber bundle. |
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Let $M\to Y$ be a map, with $\dim(M) = n$ and $\dim(Y) = k$. |
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Call a map $D^j\to Y$ ``good" with respect to $M$ if the fibered product |
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$D\widetilde{\times} M$ is a manifold of dimension $n-k+j$ with a collar structure along the boundary of $D$. |
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(If $D\to Y$ is an embedding then $D\widetilde{\times} M$ is just the part of $M$ |
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lying above $D$.) |
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We can define a $k$-category $\cF_M$ based on maps of balls into $Y$ which a re good with respect to $M$. |
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We can again adapt the homotopy colimit construction to |
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get a chain complex $\cl{\cF_M}(Y)$. |
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The proof of Theorem \ref{thm:product} again goes through essentially unchanged |
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to show that |
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\begin{thm} |
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Let $M \to Y$ be a map of manifolds and let $\cF_M$ be the $k$-category over $Y$ defined above. |
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Then |
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\[ |
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\bc_*(M) \simeq \cl{\cF_M}(Y) . |
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\] |
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\qed |
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\end{thm} |
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|
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|
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\medskip |
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|
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In the second approach we use a decorated colimit (as in \S \ref{ssec:spherecat}) |
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and various sphere modules based on $F \to E \to Y$ |
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or $M\to Y$, instead of an undecorated colimit with fancier $k$-categories over $Y$. |
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Information about the specific map to $Y$ has been taken out of the categories |
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and put into sphere modules and decorations. |
526 | 291 |
\nn{...} |
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|
526 | 293 |
%Let $F \to E \to Y$ be a fiber bundle as above. |
294 |
%Choose a decomposition $Y = \cup X_i$ |
|
295 |
%such that the restriction of $E$ to $X_i$ is a product $F\times X_i$, |
|
296 |
%and choose trivializations of these products as well. |
|
297 |
% |
|
298 |
%\nn{edit marker} |
|
299 |
%To each codim-1 face $X_i\cap X_j$ we have a bimodule ($S^0$-module). |
|
300 |
%And more generally to each codim-$j$ face we have an $S^{j-1}$-module. |
|
301 |
%Decorate the decomposition with these modules and do the colimit. |
|
302 |
% |
|
303 |
% |
|
304 |
%\nn{There is a version of this last construction for arbitrary maps $E \to Y$ |
|
305 |
%(not necessarily a fibration).} |
|
306 |
% |
|
307 |
% |
|
308 |
% |
|
309 |
%Note that Theorem \ref{thm:gluing} can be viewed as a special case of this one. |
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310 |
%Let $X_1$ and $X_2$ be $n$-manifolds |
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311 |
% |
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|
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|
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\subsection{A gluing theorem} |
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\label{sec:gluing} |
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|
133 | 317 |
Next we prove a gluing theorem. |
318 |
Let $X$ be a closed $k$-manifold with a splitting $X = X'_1\cup_Y X'_2$. |
|
319 |
We will need an explicit collar on $Y$, so rewrite this as |
|
320 |
$X = X_1\cup (Y\times J) \cup X_2$. |
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Given this data we have: |
133 | 322 |
\begin{itemize} |
323 |
\item An $A_\infty$ $n{-}k$-category $\bc(X)$, which assigns to an $m$-ball |
|
324 |
$D$ fields on $D\times X$ (for $m+k < n$) or the blob complex $\bc_*(D\times X; c)$ |
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(for $m+k = n$). |
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(See Example \ref{ex:blob-complexes-of-balls}.) |
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%\nn{need to explain $c$}. |
133 | 328 |
\item An $A_\infty$ $n{-}k{+}1$-category $\bc(Y)$, defined similarly. |
329 |
\item Two $\bc(Y)$ modules $\bc(X_1)$ and $\bc(X_2)$, which assign to a marked |
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330 |
$m$-ball $(D, H)$ either fields on $(D\times Y) \cup (H\times X_i)$ (if $m+k < n$) |
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331 |
or the blob complex $\bc_*((D\times Y) \cup (H\times X_i))$ (if $m+k = n$). |
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(See Example \ref{bc-module-example}.) |
133 | 333 |
\end{itemize} |
334 |
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\nn{statement (and proof) is only for case $k=n$; need to revise either above or below; maybe |
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just say that until we define functors we can't do more} |
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|
133 | 338 |
\begin{thm} |
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\label{thm:gluing} |
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$\bc(X) \simeq \bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$. |
133 | 341 |
\end{thm} |
342 |
||
343 |
\begin{proof} |
|
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We will assume $k=n$; the other cases are similar. |
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The proof is similar to that of Theorem \ref{thm:product}. |
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We give a short sketch with emphasis on the differences from |
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the proof of Theorem \ref{thm:product}. |
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348 |
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Let $\cT$ denote the chain complex $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$. |
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Recall that this is a homotopy colimit based on decompositions of the interval $J$. |
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342 | 352 |
We define a map $\psi:\cT\to \bc_*(X)$. |
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On 0-simplices it is given |
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by gluing the pieces together to get a blob diagram on $X$. |
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On simplices of dimension 1 and greater $\psi$ is zero. |
133 | 356 |
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The image of $\psi$ is the subcomplex $G_*\sub \bc(X)$ generated by blob diagrams which split |
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over some decomposition of $J$. |
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It follows from Lemma \ref{thm:small-blobs} that $\bc_*(X)$ is homotopic to |
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a subcomplex of $G_*$. |
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361 |
|
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Next we define a map $\phi:G_*\to \cT$ using the method of acyclic models. |
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As in the proof of Theorem \ref{thm:product}, we assign to a generator $a$ of $G_*$ |
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an acyclic subcomplex which is (roughly) $\psi\inv(a)$. |
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The proof of acyclicity is easier in this case since any pair of decompositions of $J$ have |
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a common refinement. |
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367 |
|
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The proof that these two maps are inverse to each other is the same as in |
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369 |
Theorem \ref{thm:product}. |
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370 |
\end{proof} |
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|
133 | 372 |
\medskip |
211 | 373 |
|
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374 |
\subsection{Reconstructing mapping spaces} |
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375 |
\label{sec:map-recon} |
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376 |
|
211 | 377 |
The next theorem shows how to reconstruct a mapping space from local data. |
378 |
Let $T$ be a topological space, let $M$ be an $n$-manifold, |
|
379 |
and recall the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$ |
|
380 |
of Example \ref{ex:chains-of-maps-to-a-space}. |
|
381 |
Think of $\pi^\infty_{\leq n}(T)$ as encoding everything you would ever |
|
382 |
want to know about spaces of maps of $k$-balls into $T$ ($k\le n$). |
|
383 |
To simplify notation, let $\cT = \pi^\infty_{\leq n}(T)$. |
|
384 |
||
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385 |
\begin{thm} |
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386 |
\label{thm:map-recon} |
342 | 387 |
The blob complex for $M$ with coefficients in the fundamental $A_\infty$ $n$-category for $T$ |
388 |
is quasi-isomorphic to singular chains on maps from $M$ to $T$. |
|
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389 |
$$\cB^\cT(M) \simeq C_*(\Maps(M\to T)).$$ |
211 | 390 |
\end{thm} |
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391 |
\begin{rem} |
342 | 392 |
Lurie has shown in \cite[Theorem 3.8.6]{0911.0018} that the topological chiral homology |
393 |
of an $n$-manifold $M$ with coefficients in a certain $E_n$ algebra constructed from $T$ recovers |
|
394 |
the same space of singular chains on maps from $M$ to $T$, with the additional hypothesis that $T$ is $n-1$-connected. |
|
395 |
This extra hypothesis is not surprising, in view of the idea described in Example \ref{ex:e-n-alg} |
|
396 |
that an $E_n$ algebra is roughly equivalent data to an $A_\infty$ $n$-category which |
|
397 |
is trivial at all but the topmost level. |
|
398 |
Ricardo Andrade also told us about a similar result. |
|
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399 |
\end{rem} |
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400 |
|
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401 |
\begin{proof} |
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402 |
The proof is again similar to that of Theorem \ref{thm:product}. |
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403 |
|
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404 |
We begin by constructing chain map $\psi: \cB^\cT(M) \to C_*(\Maps(M\to T))$. |
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405 |
|
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406 |
Recall that |
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407 |
the 0-simplices of the homotopy colimit $\cB^\cT(M)$ |
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408 |
are a direct sum of chain complexes with the summands indexed by |
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409 |
decompositions of $M$ which have their $n{-}1$-skeletons labeled by $n{-}1$-morphisms |
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410 |
of $\cT$. |
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411 |
Since $\cT = \pi^\infty_{\leq n}(T)$, this means that the summands are indexed by pairs |
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412 |
$(K, \vphi)$, where $K$ is a decomposition of $M$ and $\vphi$ is a continuous |
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413 |
map from the $n{-}1$-skeleton of $K$ to $T$. |
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414 |
The summand indexed by $(K, \vphi)$ is |
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415 |
\[ |
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416 |
\bigotimes_b D_*(b, \vphi), |
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417 |
\] |
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418 |
where $b$ runs through the $n$-cells of $K$ and $D_*(b, \vphi)$ denotes |
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419 |
chains of maps from $b$ to $T$ compatible with $\vphi$. |
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420 |
We can take the product of these chains of maps to get chains of maps from |
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421 |
all of $M$ to $K$. |
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422 |
This defines $\psi$ on 0-simplices. |
325 | 423 |
|
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424 |
We define $\psi$ to be zero on $(\ge1)$-simplices. |
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425 |
It is not hard to see that this defines a chain map from |
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426 |
$\cB^\cT(M)$ to $C_*(\Maps(M\to T))$. |
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|
427 |
|
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|
428 |
The image of $\psi$ is the subcomplex $G_*\sub C_*(\Maps(M\to T))$ generated by |
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|
429 |
families of maps whose support is contained in a disjoint union of balls. |
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430 |
It follows from Lemma \ref{extension_lemma_c} |
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431 |
that $C_*(\Maps(M\to T))$ is homotopic to a subcomplex of $G_*$. |
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432 |
|
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|
433 |
We will define a map $\phi:G_*\to \cB^\cT(M)$ via acyclic models. |
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434 |
Let $a$ be a generator of $G_*$. |
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435 |
Define $D(a)$ to be the subcomplex of $\cB^\cT(M)$ generated by all |
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436 |
pairs $(b, \ol{K})$, where $b$ is a generator appearing in an iterated boundary of $a$ |
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437 |
and $\ol{K}$ is an index of the homotopy colimit $\cB^\cT(M)$. |
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438 |
(See the proof of Theorem \ref{thm:product} for more details.) |
355
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439 |
The same proof as of Lemma \ref{lem:d-a-acyclic} shows that $D(a)$ is acyclic. |
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440 |
By the usual acyclic models nonsense, there is a (unique up to homotopy) |
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441 |
map $\phi:G_*\to \cB^\cT(M)$ such that $\phi(a)\in D(a)$. |
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442 |
Furthermore, we may choose $\phi$ such that for all $a$ |
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|
443 |
\[ |
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|
444 |
\phi(a) = (a, K) + r |
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445 |
\] |
448
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446 |
where $(a, K)$ is a 0-simplex and $r$ is a sum of simplices of dimension 1 and greater. |
355
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|
447 |
|
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|
448 |
It is now easy to see that $\psi\circ\phi$ is the identity on the nose. |
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|
449 |
Another acyclic models argument shows that $\phi\circ\psi$ is homotopic to the identity. |
400
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450 |
(See the proof of Theorem \ref{thm:product} for more details.) |
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|
451 |
\end{proof} |
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452 |
|
212 | 453 |
\nn{maybe should also mention version where we enrich over |
325 | 454 |
spaces rather than chain complexes;} |
211 | 455 |
|
113 | 456 |