author | Scott Morrison <scott@tqft.net> |
Tue, 01 Jun 2010 23:07:42 -0700 | |
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%!TEX root = ../blob1.tex |
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\section{The blob complex for $A_\infty$ $n$-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 define the blob |
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complex $\bc_*(M)$ to the be the homotopy colimit $\cC(M)$ of Section \ref{sec:ncats}. |
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\nn{say something about this being anticlimatically tautological?} |
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We will show below |
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in Corollary \ref{cor:new-old} |
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that this agrees (up to homotopy) with our original definition of the blob complex |
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in the case of plain $n$-categories. |
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When we need to distinguish between the new and old definitions, we will refer to the |
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new-fangled and old-fashioned blob complex. |
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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 $M$. Define the `small blob complex' $\bc^{\cU}_*(M)$ to be the subcomplex of $\bc_*(M)$ of all blob diagrams in which every blob is contained in some open set of $\cU$. |
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\nn{KW: We need something a little stronger: Every blob diagram (even a 0-blob diagram) is splittable into pieces which are small w.r.t.\ $\cU$. |
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If field have potentially large coupons/boxes, then this is a non-trivial constraint. |
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On the other hand, we could probably get away with ignoring this point. |
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Maybe the exposition will be better if we sweep this technical detail under the rug?} |
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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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\noop{ |
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Let $Y$ be a $k$-manifold, $F$ be an $n{-}k$-manifold, and |
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\[ |
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E = Y\times F . |
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\] |
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Let $\cC$ be an $n$-category. |
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Let $\cF$ be the $k$-category of Example \ref{ex:blob-complexes-of-balls}, |
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\[ |
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\cF(X) = \cC(X\times F) |
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\] |
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for $X$ an $m$-ball with $m\le k$. |
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} |
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\nn{need to settle on notation; proof and statement are inconsistent} |
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\begin{thm} \label{product_thm} |
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Given a topological $n$-category $C$ and a $n{-}k$-manifold $F$, recall from Example \ref{ex:blob-complexes-of-balls} that there is an $A_\infty$ $k$-category $C^{\times F}$ defined by |
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\begin{equation*} |
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C^{\times F}(B) = \cB_*(B \times F, C). |
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\end{equation*} |
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Now, given a $k$-manifold $Y$, there is a homotopy equivalence between the `old-fashioned' blob complex for $Y \times F$ with coefficients in $C$ and the `new-fangled' (i.e.\ homotopy colimit) blob complex for $Y$ with coefficients in $C^{\times F}$: |
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\begin{align*} |
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\cB_*(Y \times F, C) & \htpy \cB_*(Y, C^{\times F}) |
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\end{align*} |
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\end{thm} |
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\begin{proof}%[Proof of Theorem \ref{product_thm}] |
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We will use the concrete description of the colimit from Subsection \ref{ss:ncat_fields}. |
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First we define a map |
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\[ |
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\psi: \bc_*^\cF(Y) \to \bc_*^C(Y\times F) . |
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\] |
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In filtration degree 0 we just glue together the various blob diagrams on $X\times F$ |
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(where $X$ 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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In filtration degrees 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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Next we define a map |
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\[ |
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\phi: \bc_*^C(Y\times F) \to \bc_*^\cF(Y) . |
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\] |
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Actually, we will define it on the homotopy equivalent subcomplex |
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$\cS_* \sub \bc_*^C(Y\times F)$ generated by blob diagrams which are small with |
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respect to some open cover |
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of $Y\times F$ |
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(Proposition \ref{thm:small-blobs}). |
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We will have to show eventually that this is independent (up to homotopy) of the choice of cover. |
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Also, for a fixed choice of cover we will only be able to define the map for blob degree less than |
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some bound, but this bound goes to infinity as the cover become finer. |
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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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%We will define $\phi$ inductively, starting at blob degree 0. |
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%Given a 0-blob diagram $x$ on $Y\times F$, we can choose a decomposition $K$ of $Y$ |
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%such that $x$ is splittable with respect to $K\times F$. |
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%This defines a filtration degree 0 element of $\bc_*^\cF(Y)$ |
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We will define $\phi$ using a variant of the method of acyclic models. |
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Let $a\in \cS_m$ be a blob diagram on $Y\times F$. |
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For $m$ sufficiently small there exists a decomposition $K$ of $Y$ into $k$-balls such that the |
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codimension 1 cells of $K\times F$ miss the blobs of $a$, and more generally such that $a$ is splittable along (the codimension-1 part of) $K\times F$. |
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Let $D(a)$ denote the subcomplex of $\bc_*^\cF(Y)$ generated by all $(a, \ol{K})$ |
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such that each $K_i$ has the aforementioned splittable property. |
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(Recall that $\ol{K} = (K_0,\ldots,K_l)$ denotes a chain of decompositions; |
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see Subsection \ref{ss:ncat_fields}.) |
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\nn{need to define $D(a)$ more clearly; also includes $(b_j, \ol{K})$ where |
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$\bd(a) = \sum b_j$.} |
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(By $(a, \ol{K})$ we really mean $(a^\sharp, \ol{K})$, where $a^\sharp$ is |
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$a$ split according to $K_0\times F$. |
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To simplify notation we will just write plain $a$ instead of $a^\sharp$.) |
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Roughly speaking, $D(a)$ consists of filtration degree 0 stuff which glues up to give |
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$a$, filtration degree 1 stuff which makes all of the filtration degree 0 stuff homologous, |
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filtration degree 2 stuff which kills the homology created by the |
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filtration degree 1 stuff, and so on. |
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More formally, |
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\begin{lemma} |
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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 \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 of $Y$ compatible with $a$. |
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We want to show that $(a, K)$ and $(a, K')$ are homologous via filtration degree 1 stuff. |
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\nn{need to say this better; these two chains don't have the same boundary.} |
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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 filtration degree 1 chains associated to the four anti-refinemnts |
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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 consisting of filtration degree 1 stuff. |
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We want to show that this cycle bounds a chain of filtration degree 2 stuff. |
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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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\nn{need to also require that $KLM$ antirefines to $KM$, etc.} |
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Then we have a filtration degree 2 chain, as shown in Figure \ref{zzz5}, which does the trick. |
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(Each small triangle in Figure \ref{zzz5} can be filled with a filtration degree 2 chain.) |
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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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\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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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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$\phi:\cS_* \to \bc_*^\cF(Y)$. |
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This map is defined in sufficiently low degrees, sends a blob diagram $a$ to $D(a)$, |
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and is well-defined up to (iterated) homotopy. |
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The subcomplex $\cS_* \subset \bc_*^C(Y\times F)$ depends on choice of cover of $Y\times F$. |
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If we refine that cover, we get a complex $\cS'_* \subset \cS_*$ |
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and a map $\phi':\cS'_* \to \bc_*^\cF(Y)$. |
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$\phi'$ is defined only on homological degrees below some bound, but this bound is higher than |
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the corresponding bound for $\phi$. |
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We must show that $\phi$ and $\phi'$ agree, up to homotopy, |
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on the intersection of the subcomplexes on which they are defined. |
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This is clear, since the acyclic subcomplexes $D(a)$ above used in the definition of |
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$\phi$ and $\phi'$ do not depend on the choice of cover. |
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%\nn{need to say (and justify) that we now have a map $\phi$ indep of choice of cover} |
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We now show that $\phi\circ\psi$ and $\psi\circ\phi$ are homotopic to the identity. |
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$\psi\circ\phi$ is the identity on the nose. |
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$\phi$ takes a blob diagram $a$ and chops it into pieces |
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according to some decomposition $K$ of $Y$. |
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$\psi$ glues those pieces back together, yielding the same $a$ we started with. |
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$\phi\circ\psi$ is the identity up to homotopy by another MoAM argument.... |
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This concludes the proof of Theorem \ref{product_thm}. |
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\end{proof} |
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\nn{need to say something about dim $< n$ above} |
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\medskip |
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\begin{cor} |
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\label{cor:new-old} |
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The new-fangled and old-fashioned blob complexes are homotopic. |
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\end{cor} |
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\begin{proof} |
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Apply Theorem \ref{product_thm} with the fiber $F$ equal to a point. |
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\end{proof} |
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\medskip |
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Theorem \ref{product_thm} 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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We outline two approaches. |
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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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\nn{need citation to other work that does this; Stolz and Teichner?} |
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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 $\bc_*(p^*(E))$. |
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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 $\cF_E(Y)$. |
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The proof of Theorem \ref{product_thm} goes through essentially unchanged |
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to show that |
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\[ |
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\bc_*(E) \simeq \cF_E(Y) . |
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\] |
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\nn{The second approach: Choose a decomposition $Y = \cup X_i$ |
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such that the restriction of $E$ to $X_i$ is a product $F\times X_i$. |
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Choose the product structure as well. |
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To each codim-1 face $D_i\cap D_j$ we have a bimodule ($S^0$-module). |
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And more generally to each codim-$j$ face we have an $S^{j-1}$-module. |
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Decorate the decomposition with these modules and do the colimit. |
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} |
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\nn{There is a version of this last construction for arbitrary maps $E \to Y$ |
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(not necessarily a fibration).} |
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\subsection{A gluing theorem} |
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\label{sec:gluing} |
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133 | 288 |
Next we prove a gluing theorem. |
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Let $X$ be a closed $k$-manifold with a splitting $X = X'_1\cup_Y X'_2$. |
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We will need an explicit collar on $Y$, so rewrite this as |
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$X = X_1\cup (Y\times J) \cup X_2$. |
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\nn{need figure} |
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Given this data we have: \nn{need refs to above for these} |
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\begin{itemize} |
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\item An $A_\infty$ $n{-}k$-category $\bc(X)$, which assigns to an $m$-ball |
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$D$ fields on $D\times X$ (for $m+k < n$) or the blob complex $\bc_*(D\times X; c)$ |
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(for $m+k = n$). \nn{need to explain $c$}. |
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\item An $A_\infty$ $n{-}k{+}1$-category $\bc(Y)$, defined similarly. |
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\item Two $\bc(Y)$ modules $\bc(X_1)$ and $\bc(X_2)$, which assign to a marked |
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$m$-ball $(D, H)$ either fields on $(D\times Y) \cup (H\times X_i)$ (if $m+k < n$) |
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or the blob complex $\bc_*((D\times Y) \cup (H\times X_i))$ (if $m+k = n$). |
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\end{itemize} |
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\begin{thm} |
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\label{thm:gluing} |
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$\bc(X) \cong \bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$. |
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\end{thm} |
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\begin{proof} |
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The proof is similar to that of Theorem \ref{product_thm}. |
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\nn{need to say something about dimensions less than $n$, |
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but for now concentrate on top dimension.} |
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Let $\cT$ denote the $n{-}k$-category $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$. |
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Let $D$ be an $n{-}k$-ball. |
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There is an obvious map from $\cT(D)$ to $\bc_*(D\times X)$. |
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To get a map in the other direction, we replace $\bc_*(D\times X)$ with a subcomplex |
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$\cS_*$ which is adapted to a fine open cover of $D\times X$. |
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For sufficiently small $j$ (depending on the cover), we can find, for each $j$-blob diagram $b$ |
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on $D\times X$, a decomposition of $J$ such that $b$ splits on the corresponding |
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decomposition of $D\times X$. |
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The proof that these two maps are inverse to each other is the same as in |
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Theorem \ref{product_thm}. |
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\end{proof} |
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This establishes Property \ref{property:gluing}. |
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\medskip |
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211 | 329 |
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\subsection{Reconstructing mapping spaces} |
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The next theorem shows how to reconstruct a mapping space from local data. |
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Let $T$ be a topological space, let $M$ be an $n$-manifold, |
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and recall the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$ |
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of Example \ref{ex:chains-of-maps-to-a-space}. |
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Think of $\pi^\infty_{\leq n}(T)$ as encoding everything you would ever |
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want to know about spaces of maps of $k$-balls into $T$ ($k\le n$). |
|
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To simplify notation, let $\cT = \pi^\infty_{\leq n}(T)$. |
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\begin{thm} \label{thm:map-recon} |
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The blob complex for $M$ with coefficients in the fundamental $A_\infty$ $n$-category for $T$ is quasi-isomorphic to singular chains on maps from $M$ to $T$. |
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$$\cB^\cT(M) \simeq C_*(\Maps(M\to T)).$$ |
211 | 343 |
\end{thm} |
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\begin{rem} |
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\nn{This just isn't true, Lurie doesn't do this! I just heard this from Ricardo...} |
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Lurie has shown in \cite{0911.0018} that the topological chiral homology of an $n$-manifold $M$ with coefficients in \nn{a certain $E_n$ algebra constructed from $T$} recovers the same space of singular chains on maps from $M$ to $T$, with the additional hypothesis that $T$ is $n-1$-connected. This extra hypothesis is not surprising, in view of the idea that an $E_n$ algebra is roughly equivalent data as an $A_\infty$ $n$-category which is trivial at all but the topmost level. |
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\end{rem} |
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211 | 349 |
\begin{proof} |
212 | 350 |
We begin by constructing chain map $g: \cB^\cT(M) \to C_*(\Maps(M\to T))$. |
351 |
We then use \ref{extension_lemma_b} to show that $g$ induces isomorphisms on homology. |
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352 |
||
353 |
Recall that the homotopy colimit $\cB^\cT(M)$ is constructed out of a series of |
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$j$-fold mapping cylinders, $j \ge 0$. |
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355 |
So, as an abelian group (but not as a chain complex), |
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\[ |
|
357 |
\cB^\cT(M) = \bigoplus_{j\ge 0} C^j, |
|
358 |
\] |
|
359 |
where $C^j$ denotes the new chains introduced by the $j$-fold mapping cylinders. |
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361 |
Recall that $C^0$ is a direct sum of chain complexes with the summands indexed by |
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decompositions of $M$ which have their $n{-}1$-skeletons labeled by $n{-}1$-morphisms |
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of $\cT$. |
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364 |
Since $\cT = \pi^\infty_{\leq n}(T)$, this means that the summands are indexed by pairs |
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$(K, \vphi)$, where $K$ is a decomposition of $M$ and $\vphi$ is a continuous |
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maps from the $n{-}1$-skeleton of $K$ to $T$. |
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The summand indexed by $(K, \vphi)$ is |
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\[ |
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\bigotimes_b D_*(b, \vphi), |
|
370 |
\] |
|
371 |
where $b$ runs through the $n$-cells of $K$ and $D_*(b, \vphi)$ denotes |
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chains of maps from $b$ to $T$ compatible with $\vphi$. |
|
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We can take the product of these chains of maps to get a chains of maps from |
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all of $M$ to $K$. |
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375 |
This defines $g$ on $C^0$. |
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376 |
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377 |
We define $g(C^j) = 0$ for $j > 0$. |
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378 |
It is not hard to see that this defines a chain map from |
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379 |
$\cB^\cT(M)$ to $C_*(\Maps(M\to T))$. |
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380 |
||
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|
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%%%%%%%%%%%%%%%%% |
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383 |
\noop{ |
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213 | 384 |
Next we show that $g$ induces a surjection on homology. |
385 |
Fix $k > 0$ and choose an open cover $\cU$ of $M$ fine enough so that the union |
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of any $k$ open sets of $\cU$ is contained in a disjoint union of balls in $M$. |
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\nn{maybe should refer to elsewhere in this paper where we made a very similar argument} |
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Let $S_*$ be the subcomplex of $C_*(\Maps(M\to T))$ generated by chains adapted to $\cU$. |
|
389 |
It follows from Lemma \ref{extension_lemma_b} that $C_*(\Maps(M\to T))$ |
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390 |
retracts onto $S_*$. |
|
391 |
||
392 |
Let $S_{\le k}$ denote the chains of $S_*$ of degree less than or equal to $k$. |
|
393 |
We claim that $S_{\le k}$ lies in the image of $g$. |
|
394 |
Let $c$ be a generator of $S_{\le k}$ --- that is, a $j$-parameter family of maps $M\to T$, |
|
395 |
$j \le k$. |
|
396 |
We chose $\cU$ fine enough so that the support of $c$ is contained in a disjoint union of balls |
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397 |
in $M$. |
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398 |
It follow that we can choose a decomposition $K$ of $M$ so that the support of $c$ is |
|
399 |
disjoint from the $n{-}1$-skeleton of $K$. |
|
400 |
It is now easy to see that $c$ is in the image of $g$. |
|
401 |
||
402 |
Next we show that $g$ is injective on homology. |
|
214 | 403 |
} |
213 | 404 |
|
405 |
||
406 |
||
212 | 407 |
\nn{...} |
408 |
||
409 |
||
410 |
||
211 | 411 |
\end{proof} |
412 |
||
212 | 413 |
\nn{maybe should also mention version where we enrich over |
291 | 414 |
spaces rather than chain complexes; should comment on Lurie's \cite{0911.0018} (and others') similar result |
211 | 415 |
for the $E_\infty$ case, and mention that our version does not require |
416 |
any connectivity assumptions} |
|
417 |
||
418 |
\medskip |
|
113 | 419 |
\hrule |
420 |
\medskip |
|
421 |
||
422 |
\nn{to be continued...} |
|
423 |
\medskip |
|
133 | 424 |
\nn{still to do: fiber bundles, general maps} |
113 | 425 |
|
134 | 426 |
\todo{} |
427 |
Various citations we might want to make: |
|
428 |
\begin{itemize} |
|
429 |
\item \cite{MR2061854} McClure and Smith's review article |
|
430 |
\item \cite{MR0420610} May, (inter alia, definition of $E_\infty$ operad) |
|
431 |
\item \cite{MR0236922,MR0420609} Boardman and Vogt |
|
432 |
\item \cite{MR1256989} definition of framed little-discs operad |
|
433 |
\end{itemize} |
|
434 |
||
435 |
We now turn to establishing the gluing formula for blob homology, restated from Property \ref{property:gluing} in the Introduction |
|
436 |
\begin{itemize} |
|
437 |
%\mbox{}% <-- gets the indenting right |
|
438 |
\item For any $(n-1)$-manifold $Y$, the blob homology of $Y \times I$ is |
|
439 |
naturally an $A_\infty$ category. % We'll write $\bc_*(Y)$ for $\bc_*(Y \times I)$ below. |
|
440 |
||
441 |
\item For any $n$-manifold $X$, with $Y$ a codimension $0$-submanifold of its boundary, the blob homology of $X$ is naturally an |
|
442 |
$A_\infty$ module for $\bc_*(Y \times I)$. |
|
443 |
||
444 |
\item For any $n$-manifold $X$, with $Y \cup Y^{\text{op}}$ a codimension |
|
445 |
$0$-submanifold of its boundary, the blob homology of $X'$, obtained from |
|
446 |
$X$ by gluing along $Y$, is the $A_\infty$ self-tensor product of |
|
447 |
$\bc_*(X)$ as an $\bc_*(Y \times I)$-bimodule. |
|
448 |
\begin{equation*} |
|
449 |
\bc_*(X') \iso \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y \times I)}} \!\!\!\!\!\!\xymatrix{ \ar@(ru,rd)@<-1ex>[]} |
|
450 |
\end{equation*} |
|
451 |
\end{itemize} |
|
452 |