--- a/text/appendixes/comparing_defs.tex Wed Jul 14 11:06:11 2010 -0600
+++ b/text/appendixes/comparing_defs.tex Wed Jul 14 11:06:20 2010 -0600
@@ -200,11 +200,10 @@
\subsection{$A_\infty$ $1$-categories}
\label{sec:comparing-A-infty}
-In this section, we make contact between the usual definition of an $A_\infty$ algebra
-and our definition of a topological $A_\infty$ algebra, from Definition \ref{defn:topological-Ainfty-category}.
+In this section, we make contact between the usual definition of an $A_\infty$ category
+and our definition of a topological $A_\infty$ $1$-category, from \S \ref{???}.
-We begin be restricting the data of a topological $A_\infty$ algebra to the standard interval $[0,1]$,
-which we can alternatively characterise as:
+That definition associates a chain complex to every interval, and we begin by giving an alternative definition that is entirely in terms of the chain complex associated to the standard interval $[0,1]$.
\begin{defn}
A \emph{topological $A_\infty$ category on $[0,1]$} $\cC$ has a set of objects $\Obj(\cC)$,
and for each $a,b \in \Obj(\cC)$, a chain complex $\cC_{a,b}$, along with
@@ -222,7 +221,7 @@
In the $X$-labeled case, we insist that the appropriate labels match up.
Saying we have an action of this operad means that for each labeled cell decomposition
$0 < x_1< \cdots < x_k < 1$, $a_0, \ldots, a_{k+1} \subset \Obj(\cC)$, there is a chain
-map $$\cC_{a_0,a_1} \tensor \cdots \tensor \cC_{a_k,a_{k+1}} \to \cC(a_0,a_{k+1})$$ and these
+map $$\cC_{a_0,a_1} \tensor \cdots \tensor \cC_{a_k,a_{k+1}} \to \cC_{a_0,a_{k+1}}$$ and these
chain maps compose exactly as the cell decompositions.
An action of $\CD{[0,1]}$ is compatible with an action of the cell decomposition operad
if given a decomposition $\pi$, and a family of diffeomorphisms $f \in \CD{[0,1]}$ which
--- a/text/comm_alg.tex Wed Jul 14 11:06:11 2010 -0600
+++ b/text/comm_alg.tex Wed Jul 14 11:06:20 2010 -0600
@@ -193,5 +193,6 @@
\item compare the topological computation for truncated polynomial algebra with \cite{MR1600246}
\item multivariable truncated polynomial algebras (at least mention them)
\item ideally, say something more about higher hochschild homology (maybe sketch idea for proof of equivalence)
+\item say something about SMCs as $n$-categories, e.g. Vect and K-theory.
\end{itemize}
--- a/text/evmap.tex Wed Jul 14 11:06:11 2010 -0600
+++ b/text/evmap.tex Wed Jul 14 11:06:20 2010 -0600
@@ -122,7 +122,7 @@
Now for a little more detail.
(But we're still just motivating the full, gory details, which will follow.)
-Choose a metric on $X$, and let $\cU_\gamma$ be the open cover of balls of radius $\gamma$.
+Choose a metric on $X$, and let $\cU_\gamma$ be the open cover of $X$ by balls of radius $\gamma$.
By Lemma \ref{extension_lemma} we can restrict our attention to $k$-parameter families
$p$ of homeomorphisms such that $\supp(p)$ is contained in the union of $k$ $\gamma$-balls.
For fixed blob diagram $b$ and fixed $k$, it's not hard to show that for $\gamma$ small enough
@@ -151,7 +151,7 @@
We'll use the notation $|b| = \supp(b)$ and $|p| = \supp(p)$.
Choose a metric on $X$.
-Choose a monotone decreasing sequence of positive real numbers $\ep_i$ converging monotonically to zero
+Choose a monotone decreasing sequence of positive real numbers $\ep_i$ converging to zero
(e.g.\ $\ep_i = 2^{-i}$).
Choose another sequence of positive real numbers $\delta_i$ such that $\delta_i/\ep_i$
converges monotonically to zero (e.g.\ $\delta_i = \ep_i^2$).
@@ -175,7 +175,7 @@
is homeomorphic to a disjoint union of balls and
\[
N_{i,k}(p\ot b) \subeq V_0 \subeq N_{i,k+1}(p\ot b)
- \subeq V_1 \subeq \cdots \subeq V_m \subeq N_{i,k+m+1}(p\ot b) .
+ \subeq V_1 \subeq \cdots \subeq V_m \subeq N_{i,k+m+1}(p\ot b) ,
\]
and further $\bd(p\ot b) \in G_*^{i,m}$.
We also require that $b$ is splitable (transverse) along the boundary of each $V_l$.
@@ -343,7 +343,8 @@
\begin{proof}
There exists $\lambda > 0$ such that for every subset $c$ of the blobs of $b$ $\Nbd_u(c)$ is homeomorphic to $|c|$ for all $u < \lambda$ .
-(Here we are using the fact that the blobs are piecewise-linear and thatthat $\bd c$ is collared.)
+(Here we are using the fact that the blobs are
+piecewise smooth or piecewise-linear and that $\bd c$ is collared.)
We need to consider all such $c$ because all generators appearing in
iterated boundaries of $p\ot b$ must be in $G_*^{i,m}$.)
--- a/text/ncat.tex Wed Jul 14 11:06:11 2010 -0600
+++ b/text/ncat.tex Wed Jul 14 11:06:20 2010 -0600
@@ -1434,11 +1434,6 @@
let $\{Y_i\}$ be a collection of disjoint codimension 0 submanifolds of $\bd W$,
and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to $Y_i$.
-%Let $\cC$ be an [$A_\infty$] $n$-category, let $W$ be a $k$-manifold ($k\le n$),
-%and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to each boundary
-%component $\bd_i W$ of $W$.
-%(More generally, each $\cN_i$ could label some codimension zero submanifold of $\bd W$.)
-
We will define a set $\cC(W, \cN)$ using a colimit construction similar to
the one appearing in \S \ref{ss:ncat_fields} above.
(If $k = n$ and our $n$-categories are enriched, then
@@ -1448,15 +1443,18 @@
\[
W = \left(\bigcup_a X_a\right) \cup \left(\bigcup_{i,b} M_{ib}\right) ,
\]
-where each $X_a$ is a plain $k$-ball (disjoint from $\bd W$) and
-each $M_{ib}$ is a marked $k$-ball intersecting $\bd_i W$,
+where each $X_a$ is a plain $k$-ball (disjoint from $\cup Y_i$) and
+each $M_{ib}$ is a marked $k$-ball intersecting $Y_i$,
with $M_{ib}\cap Y_i$ being the marking.
(See Figure \ref{mblabel}.)
-\begin{figure}[!ht]\begin{equation*}
+\begin{figure}[t]
+\begin{equation*}
\mathfig{.4}{ncat/mblabel}
-\end{equation*}\caption{A permissible decomposition of a manifold
+\end{equation*}
+\caption{A permissible decomposition of a manifold
whose boundary components are labeled by $\cC$ modules $\{\cN_i\}$.
-Marked balls are shown shaded, plain balls are unshaded.}\label{mblabel}\end{figure}
+Marked balls are shown shaded, plain balls are unshaded.}\label{mblabel}
+\end{figure}
Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement
of $y$, or write $x \le y$, if each ball of $y$ is a union of balls of $x$.
This defines a partial ordering $\cell(W)$, which we will think of as a category.
@@ -1472,23 +1470,25 @@
\]
such that the restrictions to the various pieces of shared boundaries amongst the
$X_a$ and $M_{ib}$ all agree.
-(That is, the fibered product over the boundary maps.)
+(That is, the fibered product over the boundary restriction maps.)
If $x$ is a refinement of $y$, define a map $\psi_\cN(x)\to\psi_\cN(y)$
via the gluing (composition or action) maps from $\cC$ and the $\cN_i$.
We now define the set $\cC(W, \cN)$ to be the colimit of the functor $\psi_\cN$.
-(As usual, if $k=n$ and we are in the $A_\infty$ case, then ``colimit" means
-homotopy colimit.)
+(As in \S\ref{ss:ncat-coend}, if $k=n$ we take a colimit in whatever
+category we are enriching over, and if additionally we are in the $A_\infty$ case,
+then we use a homotopy colimit.)
+
+\medskip
If $D$ is an $m$-ball, $0\le m \le n-k$, then we can similarly define
$\cC(D\times W, \cN)$, where in this case $\cN_i$ labels the submanifold
$D\times Y_i \sub \bd(D\times W)$.
It is not hard to see that the assignment $D \mapsto \cC(D\times W, \cN)$
-has the structure of an $n{-}k$-category, which we call $\cT(W, \cN)(D)$.
+has the structure of an $n{-}k$-category.
\medskip
-
We will use a simple special case of the above
construction to define tensor products
of modules.
@@ -1497,7 +1497,7 @@
a left module and the other a right module.)
Choose a 1-ball $J$, and label the two boundary points of $J$ by $\cM_1$ and $\cM_2$.
Define the tensor product $\cM_1 \tensor \cM_2$ to be the
-$n{-}1$-category $\cT(J, \{\cM_1, \cM_2\})$.
+$n{-}1$-category associated as above to $J$ with its boundary labeled by $\cM_1$ and $\cM_2$.
This of course depends (functorially)
on the choice of 1-ball $J$.