--- a/text/intro.tex Mon May 31 17:27:17 2010 -0700
+++ b/text/intro.tex Mon May 31 23:42:37 2010 -0700
@@ -26,15 +26,18 @@
Nevertheless, when we attempt to establish all of the observed properties of the blob complex, we find this situation unsatisfactory. Thus, in the second part of the paper (section \S \ref{sec:ncats}) we pause and give yet another definition of an $n$-category, or rather a definition of an $n$-category with strong duality. (It appears that removing the duality conditions from our definition would make it more complicated rather than less.) We call these ``topological $n$-categories'', to differentiate them from previous versions. Moreover, we find that we need analogous $A_\infty$ $n$-categories, and we define these as well following very similar axioms.
-The basic idea is that each potential definition of an $n$-category makes a choice about the `shape' of morphisms. We try to be as lax as possible: a topological $n$-category associates a vector space to every $B$ homeomorphic to the $n$-ball. These vector spaces glue together associatively, and we require that there is an action of the homeomorphism group.
+The basic idea is that each potential definition of an $n$-category makes a choice about the `shape' of morphisms. We try to be as lax as possible: a topological $n$-category associates a vector space to every $B$ homeomorphic to the $n$-ball. These vector spaces glue together associatively, and we require that there is an action of the homeomorphism groupoid.
For an $A_\infty$ $n$-category, we associate a chain complex instead of a vector space to each such $B$ and ask that the action of homeomorphisms extends to a suitably defined action of the complex of singular chains of homeomorphisms. The axioms for an $A_\infty$ $n$-category are designed to capture two main examples: the blob complexes of $n$-balls labelled by a topological $n$-category, and the complex $\CM{-}{T}$ of maps to a fixed target space $T$.
-In \S \ref{sec:ainfblob} we explain how to construct a system of fields from a topological $n$-category, and give an alternative definition of the blob complex for an $A_\infty$ $n$-category on an $n$-manifold. Using these definitions, we show how to use the blob complex to `resolve' any topological $n$-category as an $A_\infty$ $n$-category, and relate the first and second definitions of the blob complex. We use the blob complex for $A_\infty$ $n$-categories to establish important properties of the blob complex (in both variants), in particular the `gluing formula' of Property \ref{property:gluing} below.
+In \S \ref{sec:ainfblob} we explain how to construct a system of fields from a topological $n$-category (using a colimit along cellulations of a manifold), and give an alternative definition of the blob complex for an $A_\infty$ $n$-category on an $n$-manifold (analogously, using a homotopy colimit). Using these definitions, we show how to use the blob complex to `resolve' any topological $n$-category as an $A_\infty$ $n$-category, and relate the first and second definitions of the blob complex. We use the blob complex for $A_\infty$ $n$-categories to establish important properties of the blob complex (in both variants), in particular the `gluing formula' of Property \ref{property:gluing} below.
+
+The relationship between all these ideas is sketched in Figure \ref{fig:outline}.
\nn{KW: the previous two paragraphs seem a little awkward to me, but I don't presently have a good idea for fixing them.}
\tikzstyle{box} = [rectangle, rounded corners, draw,outer sep = 5pt, inner sep = 5pt, line width=0.5pt]
+\begin{figure}[!ht]
{\center
\begin{tikzpicture}[align=center,line width = 1.5pt]
@@ -69,6 +72,9 @@
\end{tikzpicture}
}
+\caption{The main gadgets and constructions of the paper.}
+\label{fig:outline}
+\end{figure}
Finally, later sections address other topics. Section \S \ref{sec:comm_alg} describes the blob complex when $\cC$ is a commutative algebra, thought of as a topological $n$-category, in terms of the topology of $M$. Section \S \ref{sec:deligne} states (and in a later edition of this paper, hopefully proves) a higher dimensional generalization of the Deligne conjecture (that the little discs operad acts on Hochschild cohomology) in terms of the blob complex. The appendixes prove technical results about $\CH{M}$, and make connections between our definitions of $n$-categories and familiar definitions for $n=1$ and $n=2$, as well as relating the $n=1$ case of our $A_\infty$ $n$-categories with usual $A_\infty$ algebras.
@@ -167,7 +173,7 @@
\end{property}
As a consequence, there is an action of $\Homeo(X)$ on the chain complex $\bc_*^\cC(X)$; this action is extended to all of $C_*(\Homeo(X))$ in Property \ref{property:evaluation} below.
-The blob complex is also functorial with respect to $\cC$, although we will not address this in detail here. \todo{exact w.r.t $\cC$?}
+The blob complex is also functorial (indeed, exact) with respect to $\cC$, although we will not address this in detail here.
\begin{property}[Disjoint union]
\label{property:disjoint-union}
@@ -220,19 +226,20 @@
\end{property}
In the following $\CH{X}$ is the singular chain complex of the space of homeomorphisms of $X$, fixed on $\bdy X$.
-\begin{property}[$C_*(\Homeo(-))$ action]
+\begin{property}[$C_*(\Homeo(-))$ action]\mbox{}\\
+\vspace{-0.5cm}
\label{property:evaluation}%
-There is a chain map
+\begin{enumerate}
+\item There is a chain map
\begin{equation*}
\ev_X: \CH{X} \tensor \bc_*(X) \to \bc_*(X).
\end{equation*}
-Restricted to $C_0(\Homeo(X))$ this is just the action of homeomorphisms described in Property \ref{property:functoriality}.
-\nn{should probably say something about associativity here (or not?)}
+\item Restricted to $C_0(\Homeo(X))$ this is the action of homeomorphisms described in Property \ref{property:functoriality}.
-For
+\item For
any codimension $0$-submanifold $Y \sqcup Y^\text{op} \subset \bdy X$ the following diagram
-(using the gluing maps described in Property \ref{property:gluing-map}) commutes.
+(using the gluing maps described in Property \ref{property:gluing-map}) commutes (up to homotopy).
\begin{equation*}
\xymatrix@C+2cm{
\CH{X \bigcup_Y \selfarrow} \otimes \bc_*(X \bigcup_Y \selfarrow) \ar[r]^<<<<<<<<<<<<{\ev_{(X \bigcup_Y \scalebox{0.5}{\selfarrow})}} & \bc_*(X \bigcup_Y \selfarrow) \\
@@ -241,15 +248,23 @@
\bc_*(X) \ar[u]_{\gl_Y}
}
\end{equation*}
-
-\nn{unique up to homotopy?}
+\item Any such chain map satisfying points 2. and 3. above is unique, up to an iterated homotopy. (That is, any pair of homotopies have a homotopy between them, and so on.)
+\item This map is associative, in the sense that the following diagram commutes (up to homotopy).
+\begin{equation*}
+\xymatrix{
+\CH{X} \tensor \CH{X} \tensor \bc_*(X) \ar[r]^<<<<<{\id \tensor \ev_X} \ar[d]^{\compose \tensor \id} & \CH{X} \tensor \bc_*(X) \ar[d]^{\ev_X} \\
+\CH{X} \tensor \bc_*(X) \ar[r]^{\ev_X} & \bc_*(X)
+}
+\end{equation*}
+\end{enumerate}
\end{property}
-Since the blob complex is functorial in the manifold $X$, we can use this to build a chain map
+Since the blob complex is functorial in the manifold $X$, this is equivalent to having chain maps
$$ev_{X \to Y} : \CH{X \to Y} \tensor \bc_*(X) \to \bc_*(Y)$$
+for any homeomorphic pair $X$ and $Y$,
satisfying corresponding conditions.
-In \S \ref{sec:ncats} we introduce the notion of topological $n$-categories, from which we can construct systems of fields. Below, we talk about the blob complex associated to a topological $n$-category, implicitly passing to the system of fields. Further, in \S \ref{sec:ncats} we also have the notion of an $A_\infty$ $n$-category.
+In \S \ref{sec:ncats} we introduce the notion of topological $n$-categories, from which we can construct systems of fields. Below, we talk about the blob complex associated to a topological $n$-category, implicitly passing first to the system of fields. Further, in \S \ref{sec:ncats} we also have the notion of an $A_\infty$ $n$-category.
\begin{property}[Blob complexes of (products with) balls form an $A_\infty$ $n$-category]
\label{property:blobs-ainfty}