blob: comparison text/a_inf

equal deleted inserted replaced

-:f5af4f863a8f
+:a80cc9f9a65b
 Given an $A_\infty$ $n$-category $\cC$ and an $n$-manifold $M$, we make the following
 anticlimactically tautological definition of the blob
 complex.
 \begin{defn}
 The blob complex $\bc_*(M;\cC)$ of an $n$-manifold $M$ with coefficients in
-an $A_\infty$ $n$-category $\cC$ is the homotopy colimit $\cl{\cC}(M)$ of \S\ref{ss:ncat_fields}.
+an $A_\infty$ $n$-category $\cC$ is the homotopy colimit $\colimit{\cC}(M)$ of \S\ref{ss:ncat_fields}.
 \end{defn}
 We will show below
 in Corollary \ref{cor:new-old}
 that when $\cC$ is obtained from a system of fields $\cE$
 as the blob complex of an $n$-ball (see Example \ref{ex:blob-complexes-of-balls}),
-$\cl{\cC}(M)$ is homotopy equivalent to
+$\colimit{\cC}(M)$ is homotopy equivalent to
 our original definition of the blob complex $\bc_*(M;\cE)$.
 %\medskip
 %An important technical tool in the proofs of this section is provided by the idea of ``small blobs".
 Let $Y$ be a $k$-manifold which admits a ball decomposition
 (e.g.\ any triangulable manifold).
 Then there is a homotopy equivalence between ``old-fashioned" (blob diagrams)
 and ``new-fangled" (hocolimit) blob complexes
 \[
-	\cB_*(Y \times F) \htpy \cl{\cC_F}(Y) .
+	\cB_*(Y \times F) \htpy \colimit{\cC_F}(Y) .
 \]\end{thm}
 \begin{proof}
 We will use the concrete description of the homotopy colimit from \S\ref{ss:ncat_fields}.
 First we define a map
 \[
-	\psi: \cl{\cC_F}(Y) \to \bc_*(Y\times F;\cE) .
+	\psi: \colimit{\cC_F}(Y) \to \bc_*(Y\times F;\cE) .
 \]
 On 0-simplices of the hocolimit
 we just glue together the various blob diagrams on $X_i\times F$
 (where $X_i$ is a component of a permissible decomposition of $Y$) to get a blob diagram on
 $Y\times F$.
 It is easy to check that this is a chain map.
 In the other direction, we will define (in the next few paragraphs)
 a subcomplex $G_*\sub \bc_*(Y\times F;\cE)$ and a map
 \[
-	\phi: G_* \to \cl{\cC_F}(Y) .
+	\phi: G_* \to \colimit{\cC_F}(Y) .
 \]
 Given a decomposition $K$ of $Y$ into $k$-balls $X_i$, let $K\times F$ denote the corresponding
 decomposition of $Y\times F$ into the pieces $X_i\times F$.
 is homotopic to a subcomplex of $G_*$.
 (If the blobs of $a$ are small with respect to a sufficiently fine cover then their
 projections to $Y$ are contained in some disjoint union of balls.)
 Note that the image of $\psi$ is equal to $G_*$.
-We will define $\phi: G_* \to \cl{\cC_F}(Y)$ using the method of acyclic models.
+We will define $\phi: G_* \to \colimit{\cC_F}(Y)$ using the method of acyclic models.
 Let $a$ be a generator of $G_*$.
-Let $D(a)$ denote the subcomplex of $\cl{\cC_F}(Y)$ generated by all $(b, \ol{K})$
+Let $D(a)$ denote the subcomplex of $\colimit{\cC_F}(Y)$ generated by all $(b, \ol{K})$
 where $b$ is a generator appearing
 in an iterated boundary of $a$ (this includes $a$ itself)
 and $b$ splits along $K_0\times F$.
 (Recall that $\ol{K} = (K_0,\ldots,K_l)$ denotes a chain of decompositions;
 see \S\ref{ss:ncat_fields}.)
 Continuing in this way we see that $D(a)$ is acyclic.
 By Lemma \ref{lemma:vcones} we can fill in any cycle with a V-Cone.
 \end{proof}
 We are now in a position to apply the method of acyclic models to get a map
-$\phi:G_* \to \cl{\cC_F}(Y)$.
+$\phi:G_* \to \colimit{\cC_F}(Y)$.
 We may assume that $\phi(a)$ has the form $(a, K) + r$, where $(a, K)$ is a 0-simplex
 and $r$ is a sum of simplices of dimension 1 or higher.
 We now show that $\phi\circ\psi$ and $\psi\circ\phi$ are homotopic to the identity.
 Roughly speaking, $(a, K)$ is just $a$ chopped up into little pieces, and
 $\psi$ glues those pieces back together, yielding $a$.
 We have $\psi(r) = 0$ since $\psi$ is zero on $(\ge 1)$-simplices.
 Second, $\phi\circ\psi$ is the identity up to homotopy by another argument based on the method of acyclic models.
-To each generator $(b, \ol{K})$ of $\cl{\cC_F}(Y)$ we associate the acyclic subcomplex $D(b)$ defined above.
+To each generator $(b, \ol{K})$ of $\colimit{\cC_F}(Y)$ we associate the acyclic subcomplex $D(b)$ defined above.
 Both the identity map and $\phi\circ\psi$ are compatible with this
 collection of acyclic subcomplexes, so by the usual method of acyclic models argument these two maps
 are homotopic.
 This concludes the proof of Theorem \ref{thm:product}.
 If $Y$ has dimension $k-m$, then we have an $m$-category $\cC_{Y\times F}$ whose value at
 a $j$-ball $X$ is either $\cE(X\times Y\times F)$ (if $j<m$) or $\bc_*(X\times Y\times F)$
 (if $j=m$).
 (See Example \ref{ex:blob-complexes-of-balls}.)
-Similarly we have an $m$-category whose value at $X$ is $\cl{\cC_F}(X\times Y)$.
+Similarly we have an $m$-category whose value at $X$ is $\colimit{\cC_F}(X\times Y)$.
 These two categories are equivalent, but since we do not define functors between
 disk-like $n$-categories in this paper we are unable to say precisely
 what ``equivalent" means in this context.
 We hope to include this stronger result in a future paper.
 Let $\cE$ be a system of fields (with local relations) and let $\cC_\cE$ be the $A_\infty$
 $n$-category obtained from $\cE$ by taking the blob complex of balls.
 Then for all $n$-manifolds $Y$ the old-fashioned and new-fangled blob complexes are
 homotopy equivalent:
 \[
-	\bc^\cE_*(Y) \htpy \cl{\cC_\cE}(Y) .
+	\bc^\cE_*(Y) \htpy \colimit{\cC_\cE}(Y) .
 \]
 \end{cor}
 \medskip
 assign to $p:D\to Y$ the blob complex $\bc_*(p^*(E))$, when $\dim(D) = k$,
 or the fields $\cE(p^*(E))$, when $\dim(D) < k$.
 (Here $p^*(E)$ denotes the pull-back bundle over $D$.)
 Let $\cF_E$ denote this $k$-category over $Y$.
 We can adapt the homotopy colimit construction (based on decompositions of $Y$ into balls) to
-get a chain complex $\cl{\cF_E}(Y)$.
+get a chain complex $\colimit{\cF_E}(Y)$.
 \begin{thm}
 Let $F \to E \to Y$ be a fiber bundle and let $\cF_E$ be the $k$-category over $Y$ defined above.
 Then
 \[
-	\bc_*(E) \simeq \cl{\cF_E}(Y) .
+	\bc_*(E) \simeq \colimit{\cF_E}(Y) .
 \]
 \qed
 \end{thm}
 \begin{proof}
 The proof is nearly identical to the proof of Theorem \ref{thm:product}, so we will only give a sketch which
 emphasizes the few minor changes that need to be made.
 As before, we define a map
 \[
-	\psi: \cl{\cF_E}(Y) \to \bc_*(E) .
+	\psi: \colimit{\cF_E}(Y) \to \bc_*(E) .
 \]
-The 0-simplices of the homotopy colimit $\cl{\cF_E}(Y)$ are glued up to give an element of $\bc_*(E)$.
+The 0-simplices of the homotopy colimit $\colimit{\cF_E}(Y)$ are glued up to give an element of $\bc_*(E)$.
 Simplices of positive degree are sent to zero.
 Let $G_* \sub \bc_*(E)$ be the image of $\psi$.
 By Lemma \ref{thm:small-blobs}, $\bc_*(Y\times F; \cE)$
 is homotopic to a subcomplex of $G_*$.
 We will define a homotopy inverse of $\psi$ on $G_*$, using acyclic models.
-To each generator $a$ of $G_*$ we assign an acyclic subcomplex $D(a) \sub \cl{\cF_E}(Y)$ which consists of
+To each generator $a$ of $G_*$ we assign an acyclic subcomplex $D(a) \sub \colimit{\cF_E}(Y)$ which consists of
 0-simplices which map via $\psi$ to $a$, plus higher simplices (as described in the proof of Theorem \ref{thm:product})
 which insure that $D(a)$ is acyclic.
 \end{proof}
 We can generalize this result still further by noting that it is not really necessary
 $D\widetilde{\times} M$ is a manifold of dimension $n-k+j$ with a collar structure along the boundary of $D$.
 (If $D\to Y$ is an embedding then $D\widetilde{\times} M$ is just the part of $M$
 lying above $D$.)
 We can define a $k$-category $\cF_M$ based on maps of balls into $Y$ which are good with respect to $M$.
 We can again adapt the homotopy colimit construction to
-get a chain complex $\cl{\cF_M}(Y)$.
+get a chain complex $\colimit{\cF_M}(Y)$.
 The proof of Theorem \ref{thm:product} again goes through essentially unchanged
 to show that
 %\begin{thm}
 %Let $M \to Y$ be a map of manifolds and let $\cF_M$ be the $k$-category over $Y$ defined above.
 %Then
 \[
-	\bc_*(M) \simeq \cl{\cF_M}(Y) .
+	\bc_*(M) \simeq \colimit{\cF_M}(Y) .
 \]
 %\qed
 %\end{thm}

changeset 978	a80cc9f9a65b
parent 953	ec1c5ccef482