# HG changeset patch
# User Kevin Walker <kevin@canyon23.net>
# Date 1285270456 25200
# Node ID 5fdf1488ce2008edc680eab1c06afbd5c5bb12b8
# Parent  4f008d0a29d4ec54c682360905ee55d3ec51ad64
resolving two more nns

diff -r 4f008d0a29d4 -r 5fdf1488ce20 text/a_inf_blob.tex
--- a/text/a_inf_blob.tex	Thu Sep 23 10:03:26 2010 -0700
+++ b/text/a_inf_blob.tex	Thu Sep 23 12:34:16 2010 -0700
@@ -292,27 +292,27 @@
 or $M\to Y$, instead of an undecorated colimit with fancier $k$-categories over $Y$.
 Information about the specific map to $Y$ has been taken out of the categories
 and put into sphere modules and decorations.
-\nn{just say that one could do something along these lines}
+
+Let $F \to E \to Y$ be a fiber bundle as above.
+Choose a decomposition $Y = \cup X_i$
+such that the restriction of $E$ to $X_i$ is homeomorphic to a product $F\times X_i$,
+and choose trivializations of these products as well.
 
-%Let $F \to E \to Y$ be a fiber bundle as above.
-%Choose a decomposition $Y = \cup X_i$
-%such that the restriction of $E$ to $X_i$ is a product $F\times X_i$,
-%and choose trivializations of these products as well.
-%
-%\nn{edit marker}
-%To each codim-1 face $X_i\cap X_j$ we have a bimodule ($S^0$-module).
-%And more generally to each codim-$j$ face we have an $S^{j-1}$-module.
-%Decorate the decomposition with these modules and do the colimit.
-%
-%
-%\nn{There is a version of this last construction for arbitrary maps $E \to Y$
-%(not necessarily a fibration).}
-%
-%
-%
+Let $\cF$ be the $k$-category associated to $F$.
+To each codimension-1 face $X_i\cap X_j$ we have a bimodule ($S^0$-module) for $\cF$.
+More generally, to each codimension-$m$ face we have an $S^{m-1}$-module for a $(k{-}m{+}1)$-category
+associated to the (decorated) link of that face.
+We can decorate the strata of the decomposition of $Y$ with these sphere modules and form a 
+colimit as in \S \ref{ssec:spherecat}.
+This colimit computes $\bc_*(E)$.
+
+There is a similar construction for general maps $M\to Y$.
+
 %Note that Theorem \ref{thm:gluing} can be viewed as a special case of this one.
 %Let $X_1$ and $X_2$ be $n$-manifolds
-%
+%\nn{...}
+
+
 
 
 \subsection{A gluing theorem}
diff -r 4f008d0a29d4 -r 5fdf1488ce20 text/ncat.tex
--- a/text/ncat.tex	Thu Sep 23 10:03:26 2010 -0700
+++ b/text/ncat.tex	Thu Sep 23 12:34:16 2010 -0700
@@ -1845,7 +1845,7 @@
 	\draw[line width=1pt, green!50!brown, ->] (M\n.\qm+135) to[out=\qm+135,in=\qm+90] (\qm+5:1.3);
 }
 \end{tikzpicture}
-\caption{Cone on a marked circle}
+\caption{Cone on a marked circle, the prototypical 1-marked ball}
 \label{feb21d}
 \end{figure}
 
@@ -2237,9 +2237,11 @@
 
 To define (binary) composition of $n{+}1$-morphisms, choose the obvious common equator
 then compose the module maps.
+Associativity of this composition rules follows from repeated application of the adjoint identity between
+the maps of Figures \ref{jun23b} and \ref{jun23c}.
 
 
-\nn{still to do: associativity}
+%\nn{still to do: associativity}
 
 \medskip