--- a/blob1.tex Sun Jul 04 23:32:48 2010 -0600
+++ b/blob1.tex Mon Jul 05 07:47:23 2010 -0600
@@ -16,7 +16,7 @@
\maketitle
-[revision $\ge$ 417; $\ge$ 4 July 2010]
+[revision $\ge$ 418; $\ge$ 5 July 2010]
{\color[rgb]{.9,.5,.2} \large \textbf{Draft version, read with caution.}}
We're in the midst of revising this, and hope to have a version on the arXiv soon.
--- a/text/a_inf_blob.tex Sun Jul 04 23:32:48 2010 -0600
+++ b/text/a_inf_blob.tex Mon Jul 05 07:47:23 2010 -0600
@@ -248,7 +248,7 @@
\bc_*(E) \simeq \cF_E(Y) .
\]
-
+\nn{remark further that this still works when the map is not even a fibration?}
\nn{put this later}
@@ -261,7 +261,8 @@
}
\nn{There is a version of this last construction for arbitrary maps $E \to Y$
-(not necessarily a fibration).}
+(not necessarily a fibration).
+In fact, there is also a version of the first construction for non-fibrations.}
--- a/text/ncat.tex Sun Jul 04 23:32:48 2010 -0600
+++ b/text/ncat.tex Mon Jul 05 07:47:23 2010 -0600
@@ -663,11 +663,13 @@
We now describe several classes of examples of $n$-categories satisfying our axioms.
+We typically specify only the morphisms; the rest of the data for the category
+(restriction maps, gluing, product morphisms, action of homeomorphisms) is usually obvious.
\begin{example}[Maps to a space]
\rm
\label{ex:maps-to-a-space}%
-Fix a ``target space" $T$, any topological space.
+Let $T$be a topological space.
We define $\pi_{\leq n}(T)$, the fundamental $n$-category of $T$, as follows.
For $X$ a $k$-ball with $k < n$, define $\pi_{\leq n}(T)(X)$ to be the set of
all continuous maps from $X$ to $T$.
@@ -676,10 +678,14 @@
(Note that homotopy invariance implies isotopy invariance.)
For $a\in \cC(X)$ define the product morphism $a\times D \in \cC(X\times D)$ to
be $a\circ\pi_X$, where $\pi_X : X\times D \to X$ is the projection.
+\end{example}
+\noop{
Recall we described a system of fields and local relations based on maps to $T$ in Example \ref{ex:maps-to-a-space(fields)} above.
Constructing a system of fields from $\pi_{\leq n}(T)$ recovers that example.
-\end{example}
+\nn{shouldn't this go elsewhere? we haven't yet discussed constructing a system of fields from
+an n-cat}
+}
\begin{example}[Maps to a space, with a fiber]
\rm
@@ -701,7 +707,8 @@
the $R$-module of finite linear combinations of continuous maps from $X\times F$ to $T$,
modulo the relation that if $a$ is homotopic to $b$ (rel boundary) via a homotopy
$h: X\times F\times I \to T$, then $a = \alpha(h)b$.
-\nn{need to say something about fundamental classes, or choose $\alpha$ carefully}
+(In order for this to be well-defined we must choose $\alpha$ to be zero on degenerate simplices.
+Alternatively, we could equip the balls with fundamental classes.)
\end{example}
The next example is only intended to be illustrative, as we don't specify which definition of a ``traditional $n$-category" we intend.
@@ -723,8 +730,12 @@
to be the set of all $C$-labeled embedded cell complexes of $X\times F$.
Define $\cC(X; c)$, for $X$ an $n$-ball,
to be the dual Hilbert space $A(X\times F; c)$.
-\nn{refer elsewhere for details?}
+(See Subsection \ref{sec:constructing-a-tqft}.)
+\end{example}
+\noop{
+\nn{shouldn't this go elsewhere? we haven't yet discussed constructing a system of fields from
+an n-cat}
Recall we described a system of fields and local relations based on a ``traditional $n$-category"
$C$ in Example \ref{ex:traditional-n-categories(fields)} above.
\nn{KW: We already refer to \S \ref{sec:fields} above}
@@ -734,11 +745,8 @@
where the quotient is built in.
but (string diagrams)/(relations) is isomorphic to
(pasting diagrams composed of smaller string diagrams)/(relations)}
-\end{example}
+}
-Finally, we describe a version of the bordism $n$-category suitable to our definitions.
-
-\nn{should also include example of ncats coming from TQFTs, or refer ahead to where we discuss that example}
\newcommand{\Bord}{\operatorname{Bord}}
\begin{example}[The bordism $n$-category, plain version]
@@ -766,15 +774,19 @@
%We have two main examples of $A_\infty$ $n$-categories, coming from maps to a target space and from the blob complex.
-\begin{example}[Chains of maps to a space]
+\begin{example}[Chains (or space) of maps to a space]
\rm
\label{ex:chains-of-maps-to-a-space}
We can modify Example \ref{ex:maps-to-a-space} above to define the fundamental $A_\infty$ $n$-category $\pi^\infty_{\le n}(T)$ of a topological space $T$.
For a $k$-ball $X$, with $k < n$, the set $\pi^\infty_{\leq n}(T)(X)$ is just $\Maps(X \to T)$.
Define $\pi^\infty_{\leq n}(T)(X; c)$ for an $n$-ball $X$ and $c \in \pi^\infty_{\leq n}(T)(\bdy X)$ to be the chain complex
-$$C_*(\Maps_c(X\times F \to T)),$$ where $\Maps_c$ denotes continuous maps restricting to $c$ on the boundary,
+\[
+ C_*(\Maps_c(X\times F \to T)),
+\]
+where $\Maps_c$ denotes continuous maps restricting to $c$ on the boundary,
and $C_*$ denotes singular chains.
-\nn{maybe should also mention version where we enrich over spaces rather than chain complexes}
+Alternatively, if we take the $n$-morphisms to be simply $\Maps_c(X\times F \to T)$,
+we get an $A_\infty$ $n$-category enriched over spaces.
\end{example}
See also Theorem \ref{thm:map-recon} below, recovering $C_*(\Maps(M \to T))$ up to
@@ -783,7 +795,7 @@
\begin{example}[Blob complexes of balls (with a fiber)]
\rm
\label{ex:blob-complexes-of-balls}
-Fix an $n-k$-dimensional manifold $F$ and an $n$-dimensional system of fields $\cE$.
+Fix an $n{-}k$-dimensional manifold $F$ and an $n$-dimensional system of fields $\cE$.
We will define an $A_\infty$ $k$-category $\cC$.
When $X$ is a $m$-ball, with $m<k$, define $\cC(X) = \cE(X\times F)$.
When $X$ is an $k$-ball,
@@ -795,9 +807,8 @@
Notice that with $F$ a point, the above example is a construction turning a topological
$n$-category $\cC$ into an $A_\infty$ $n$-category which we'll denote by $\bc_*(\cC)$.
We think of this as providing a ``free resolution"
-\nn{``cofibrant replacement"?}
of the topological $n$-category.
-\todo{Say more here!}
+\nn{say something about cofibrant replacements?}
In fact, there is also a trivial, but mostly uninteresting, 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.