
diff -r f5af4f863a8f -r a80cc9f9a65b text/a_inf_blob.tex
--- a/text/a_inf_blob.tex	Thu Apr 26 06:57:24 2012 -0600
+++ b/text/a_inf_blob.tex	Fri Apr 27 22:37:14 2012 -0700
@@ -7,14 +7,14 @@
 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
@@ -47,7 +47,7 @@
 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}
@@ -55,7 +55,7 @@
 
 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 $X_i\times F$
@@ -67,7 +67,7 @@
 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
@@ -81,9 +81,9 @@
 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 $K_0\times F$.
@@ -198,7 +198,7 @@
 \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.
 
@@ -213,7 +213,7 @@
 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.
@@ -227,7 +227,7 @@
 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.
@@ -244,7 +244,7 @@
 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}
 
@@ -272,13 +272,13 @@
 (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}
@@ -289,16 +289,16 @@
 
 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}
@@ -312,14 +312,14 @@
 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}




