# HG changeset patch
# User Kevin Walker <kevin@canyon23.net>
# Date 1285213367 25200
# Node ID 11532ce39ec00569965f550d7dcb7e9c98468cbe
# Parent  62d6db8c2ead3270dacadea897acffb9989af4b1
making "no functors" excuses; other minor stuff

diff -r 62d6db8c2ead -r 11532ce39ec0 text/a_inf_blob.tex
--- a/text/a_inf_blob.tex	Wed Sep 22 19:01:41 2010 -0700
+++ b/text/a_inf_blob.tex	Wed Sep 22 20:42:47 2010 -0700
@@ -31,7 +31,7 @@
 \label{ss:product-formula}
 
 
-Given a system of fields $\cE$ and a $n{-}k$-manifold $F$, recall from 
+Given an $n$-dimensional system of fields $\cE$ and a $n{-}k$-manifold $F$, recall from 
 Example \ref{ex:blob-complexes-of-balls} that there is an  $A_\infty$ $k$-category $\cC_F$ 
 defined by $\cC_F(X) = \cE(X\times F)$ if $\dim(X) < k$ and
 $\cC_F(X) = \bc_*(X\times F;\cE)$ if $\dim(X) = k$.
@@ -200,11 +200,21 @@
 This concludes the proof of Theorem \ref{thm:product}.
 \end{proof}
 
-\nn{need to prove a version where $E$ above has dimension $m<n$; result is an $n{-}m$-category}
+%\nn{need to prove a version where $E$ above has dimension $m<n$; result is an $n{-}m$-category}
+
+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 $\cl{\cC_F}(X\times Y)$.
+These two categories are equivalent, but since we do not define functors between
+topological $n$-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.
 
 \medskip
 
-Taking $F$ above to be a point, we obtain the following corollary.
+Taking $F$ in Theorem \ref{thm:product} to be a point, we obtain the following corollary.
 
 \begin{cor}
 \label{cor:new-old}
@@ -324,18 +334,24 @@
 $m$-ball $(D, H)$ either fields on $(D\times Y) \cup (H\times X_i)$ (if $m+k < n$)
 or the blob complex $\bc_*((D\times Y) \cup (H\times X_i))$ (if $m+k = n$).
 (See Example \ref{bc-module-example}.)
+\item The tensor product $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$, which is
+an $A_\infty$ $n{-}k$-category.
+(See \S \ref{moddecss}.)
 \end{itemize}
 
-\nn{statement (and proof) is only for case $k=n$; need to revise either above or below; maybe
-just say that until we define functors we can't do more}
+It is the case that the $n{-}k$-categories $\bc(X)$ and $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$
+are equivalent for all $k$, but since we do not develop a definition of functor between $n$-categories
+in this paper, we cannot state this precisely.
+(It will appear in a future paper.)
+So we content ourselves with
 
 \begin{thm}
 \label{thm:gluing}
-$\bc(X) \simeq \bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$.
+When $k=n$ above, $\bc(X)$ is homotopy equivalent to $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$.
 \end{thm}
 
 \begin{proof}
-We will assume $k=n$; the other cases are similar.
+%We will assume $k=n$; the other cases are similar.
 The proof is similar to that of Theorem \ref{thm:product}.
 We give a short sketch with emphasis on the differences from 
 the proof of Theorem \ref{thm:product}.
diff -r 62d6db8c2ead -r 11532ce39ec0 text/evmap.tex
--- a/text/evmap.tex	Wed Sep 22 19:01:41 2010 -0700
+++ b/text/evmap.tex	Wed Sep 22 20:42:47 2010 -0700
@@ -191,7 +191,6 @@
 and with $\supp(x_k) = U$.
 We can now take $d_j \deq \sum x_k$.
 It is clear that $\bd d_j = \sum (g_{j-1}(e_k) - g_j(e_k)) = g_{j-1}(s(\bd b)) - g_{j}(s(\bd b))$, as desired.
-\nn{should have figure}
 
 We now define 
 \[
diff -r 62d6db8c2ead -r 11532ce39ec0 text/ncat.tex
--- a/text/ncat.tex	Wed Sep 22 19:01:41 2010 -0700
+++ b/text/ncat.tex	Wed Sep 22 20:42:47 2010 -0700
@@ -1354,7 +1354,7 @@
 plain ball case.
 Note that a marked pinched product can be decomposed into either
 two marked pinched products or a plain pinched product and a marked pinched product.
-\nn{should maybe give figure}
+%\nn{should maybe give figure}
 
 \begin{module-axiom}[Product (identity) morphisms]
 For each pinched product $\pi:E\to M$, with $M$ a marked $k$-ball and $E$ a marked