# HG changeset patch
# User Scott Morrison <scott@tqft.net>
# Date 1278519441 21600
# Node ID 257066702f600669f57e678a2fde3784ed599418
# Parent  a96f3d2ef852821fa1dee7c212e5c01a0af393ec
minor

diff -r a96f3d2ef852 -r 257066702f60 text/a_inf_blob.tex
--- a/text/a_inf_blob.tex	Mon Jul 05 07:47:23 2010 -0600
+++ b/text/a_inf_blob.tex	Wed Jul 07 10:17:21 2010 -0600
@@ -41,9 +41,9 @@
 Given a topological $n$-category $C$ and a $n{-}k$-manifold $F$, recall from 
 Example \ref{ex:blob-complexes-of-balls} that there is an  $A_\infty$ $k$-category $\bc_*(F; C)$ defined by
 \begin{equation*}
-\bc_*(F; C) = \cB_*(B \times F, C).
+\bc_*(F; C)(B) = \cB_*(F \times B; C).
 \end{equation*}
-Now, given a $k$-manifold $Y$, there is a homotopy equivalence between the `old-fashioned' 
+Now, given a $k$-manifold $Y$, there is a homotopy equivalence between the ``old-fashioned'' 
 blob complex for $Y \times F$ with coefficients in $C$ and the ``new-fangled" 
 (i.e.\ homotopy colimit) blob complex for $Y$ with coefficients in $\bc_*(F; C)$:
 \begin{align*}
diff -r a96f3d2ef852 -r 257066702f60 text/ncat.tex
--- a/text/ncat.tex	Mon Jul 05 07:47:23 2010 -0600
+++ b/text/ncat.tex	Wed Jul 07 10:17:21 2010 -0600
@@ -378,7 +378,6 @@
 \[
 	d: \Delta^{k+m}\to\Delta^k .
 \]
-In other words, \nn{each point has a neighborhood blah blah...}
 (We thank Kevin Costello for suggesting this approach.)
 
 Note that for each interior point $x\in X$, $\pi\inv(x)$ is an $m$-ball,
@@ -518,7 +517,7 @@
 
 We start with the plain $n$-category case.
 
-\begin{axiom}[Isotopy invariance in dimension $n$]{\textup{\textbf{[preliminary]}}}
+\begin{axiom}[\textup{\textbf{[preliminary]}} Isotopy invariance in dimension $n$]
 Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts
 to the identity on $\bd X$ and is isotopic (rel boundary) to the identity.
 Then $f$ acts trivially on $\cC(X)$; $f(a) = a$ for all $a\in \cC(X)$.
@@ -592,7 +591,7 @@
 The revised axiom is
 
 \addtocounter{axiom}{-1}
-\begin{axiom}{\textup{\textbf{[topological  version]}} Extended isotopy invariance in dimension $n$.}
+\begin{axiom}[\textup{\textbf{[topological  version]}} Extended isotopy invariance in dimension $n$.]
 \label{axiom:extended-isotopies}
 Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts
 to the identity on $\bd X$ and isotopic (rel boundary) to the identity.
@@ -610,7 +609,7 @@
 
 
 \addtocounter{axiom}{-1}
-\begin{axiom}{\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$.}
+\begin{axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$.]
 For each $n$-ball $X$ and each $c\in \cl{\cC}(\bd X)$ we have a map of chain complexes
 \[
 	C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) .
@@ -628,7 +627,7 @@
 and we can replace the class of all intervals $J$ with intervals contained in $\r$.
 Having chains on the space of collar maps act gives rise to coherence maps involving
 weak identities.
-We will not pursue this in this draft of the paper.
+We will not pursue this in detail here.
 
 Note that if we take homology of chain complexes, we turn an $A_\infty$ $n$-category
 into a plain $n$-category (enriched over graded groups).
@@ -916,7 +915,7 @@
 and we  will define $\cC(W)$ as a suitable colimit 
 (or homotopy colimit in the $A_\infty$ case) of this functor. 
 We'll later give a more explicit description of this colimit.
-In the case that the $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain complexes to $n$-manifolds with boundary data), 
+In the case that the $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain complexes to $n$-balls with boundary data), 
 then the resulting colimit is also enriched, that is, the set associated to $W$ splits into subsets according to boundary data, and each of these subsets has the appropriate structure (e.g. a vector space or chain complex).
 
 \begin{defn}
@@ -971,7 +970,7 @@
 fix a field on $\bd W$
 (i.e. fix an element of the colimit associated to $\bd W$).
 
-Finally, we construct $\cC(W)$ as the appropriate colimit of $\psi_{\cC;W}$.
+Finally, we construct $\cl{\cC}(W)$ as the appropriate colimit of $\psi_{\cC;W}$.
 
 \begin{defn}[System of fields functor]
 \label{def:colim-fields}
@@ -1036,7 +1035,7 @@
 
 $\cC(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. Restricting the $k$-spheres, we have now proved Lemma \ref{lem:spheres}.
 
-\todo{This next sentence is circular: these maps are an axiom, not a consequence of anything. -S} It is easy to see that
+It is easy to see that
 there are well-defined maps $\cC(W)\to\cC(\bd W)$, and that these maps
 comprise a natural transformation of functors.
 
@@ -1338,10 +1337,10 @@
 $(B,N) \to (S,T)$ and on each marked $n$-ball $(B,N)$ it consists of all 
 such maps modulo homotopies fixed on $\bdy B \setminus N$.
 This is a module over the fundamental $n$-category $\pi_{\leq n}(S)$ of $S$, from Example \ref{ex:maps-to-a-space}.
+\end{example}
 Modifications corresponding to Examples \ref{ex:maps-to-a-space-with-a-fiber} and 
 \ref{ex:linearized-maps-to-a-space} are also possible, and there is an $A_\infty$ version analogous to 
 Example \ref{ex:chains-of-maps-to-a-space} given by taking singular chains.
-\end{example}
 
 \subsection{Modules as boundary labels (colimits for decorated manifolds)}
 \label{moddecss}