
diff -r f4e13802a181 -r ae29491942e6 text/ncat.tex
--- a/text/ncat.tex	Sat May 15 10:46:37 2010 -0500
+++ b/text/ncat.tex	Sat May 15 16:00:31 2010 -0700
@@ -581,17 +581,7 @@
 
 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 $\pi^\infty_{\le n}(T)$.
 
-\todo{Yikes! We can't do this yet, because we haven't explained how to produce a system of fields and local relations from a topological $n$-category.}
-
-\begin{example}[Blob complexes of balls (with a fiber)]
-\rm
-\label{ex:blob-complexes-of-balls}
-Fix an $m$-dimensional manifold $F$. We will define an $A_\infty$ $(n-m)$-category $\cC$.
-Given a plain $n$-category $C$, 
-when $X$ is a $k$-ball or $k$-sphere, with $k<n-m$, define $\cC(X) = C(X)$. When $X$ is an $(n-m)$-ball,
-define $\cC(X; c) = \bc^C_*(X\times F; c)$
-where $\bc^C_*$ denotes the blob complex based on $C$.
-\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. 
 




