# HG changeset patch
# User Scott Morrison <scott@tqft.net>
# Date 1274994246 25200
# Node ID cb16992373be2b9bcf240d4e57c866ee28f8b603
# Parent  79f7b1bd7b1ac68101c0091e42bc0fa9365b9d83
\mapsfrom

diff -r 79f7b1bd7b1a -r cb16992373be preamble.tex
--- a/preamble.tex	Tue May 25 16:50:55 2010 -0700
+++ b/preamble.tex	Thu May 27 14:04:06 2010 -0700
@@ -13,6 +13,7 @@
 
 \usepackage[section]{placeins}
 \usepackage{leftidx}
+\usepackage{stmaryrd} % additional math symbols, e.g. \mapsfrom
 
 \SelectTips{cm}{}
 % This may speed up compilation of complex documents with many xymatrices.
diff -r 79f7b1bd7b1a -r cb16992373be text/ncat.tex
--- a/text/ncat.tex	Tue May 25 16:50:55 2010 -0700
+++ b/text/ncat.tex	Thu May 27 14:04:06 2010 -0700
@@ -581,7 +581,16 @@
 
 See also Theorem \ref{thm:map-recon} below, recovering $C_*(\Maps(M \to T))$ up to homotopy the blob complex of $M$ with coefficients in $\pi^\infty_{\le n}(T)$.
 
-\begin{example}[Blob complexes of balls (with a fiber)]
\rm
\label{ex:blob-complexes-of-balls}
Fix an $m$-dimensional manifold $F$ and system of fields $\cE$.
We will define an $A_\infty$ $(n-m)$-category $\cC$.
When $X$ is a $k$-ball or $k$-sphere, with $k<n-m$, define $\cC(X) = \cE(X\times F)$.
When $X$ is an $(n-m)$-ball,
define $\cC(X; c) = \bc^\cE_*(X\times F; c)$
where $\bc^\cE_*$ denotes the blob complex based on $\cE$.
\end{example}
+\begin{example}[Blob complexes of balls (with a fiber)]
+\rm
+\label{ex:blob-complexes-of-balls}
+Fix an $m$-dimensional manifold $F$ and system of fields $\cE$.
+We will define an $A_\infty$ $(n-m)$-category $\cC$.
+When $X$ is a $k$-ball or $k$-sphere, with $k<n-m$, define $\cC(X) = \cE(X\times F)$.
+When $X$ is an $(n-m)$-ball,
+define $\cC(X; c) = \bc^\cE_*(X\times F; c)$
+where $\bc^\cE_*$ denotes the blob complex based on $\cE$.
+\end{example}
 
 This example will be essential for Theorem \ref{product_thm} below, which allows us to compute the blob complex of a product. Notice that with $F$ a point, the above example is a construction turning a topological $n$-category into an $A_\infty$ $n$-category. We think of this as providing a `free resolution' of the topological $n$-category. \todo{Say more here!} In fact, there is also a trivial way to do this: we can think of each vector space associated to an $n$-ball as a chain complex concentrated in degree $0$, and take $\CD{B}$ to act trivially. 
 
@@ -1104,9 +1113,8 @@
 \begin{eqnarray*}
 	\hom_A(X_B\ot {_BY_A} \to Z_A) &\cong& \hom_B(X_B \to \hom_A( {_BY_A} \to Z_A)) \\
 	f &\mapsto& [x \mapsto f(x\ot -)] \\
-	{}[x\ot y \mapsto g(x)(y)] & \leftarrowtail & g .
+	{}[x\ot y \mapsto g(x)(y)] & \mapsfrom & g .
 \end{eqnarray*}
-\nn{how to do a left-pointing ``$\mapsto$"?}
 If $A$ and $Z_A$ are both the ground field $\k$, this simplifies to
 \[
 	(X_B\ot {_BY})^* \cong  \hom_B(X_B \to (_BY)^*) .