--- a/text/intro.tex Mon Jul 26 22:57:43 2010 -0400
+++ b/text/intro.tex Tue Jul 27 09:25:38 2010 -0400
@@ -111,7 +111,7 @@
\draw[->] (C) -- node[left=10pt] {
Example \ref{ex:traditional-n-categories(fields)} \\ and \S \ref{ss:ncat_fields}
- %$\displaystyle \cF(M) = \DirectSum_{c \in\cell(M)} \cC(c)$ \\ $\displaystyle \cU(B) = \DirectSum_{c \in \cell(B)} \ker \ev: \cC(c) \to \cC(B)$
+ %$\displaystyle \cF(M) = \DirectSum_{c \in\cell(M)} \cC(c)$ \\ $\displaystyle \cU(B) = \DirectSum_{c \in \cell(B)} \ker e: \cC(c) \to \cC(B)$
} (FU);
\draw[->] (BC) -- node[left] {$H_0$} node[right] {c.f. Theorem \ref{thm:skein-modules}} (A);
@@ -134,12 +134,9 @@
Appendix \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 $M$.
-Throughout the paper we typically prefer concrete categories (vector spaces, chain complexes)
-\nn{...}
-
-
-%\item related: we are being unsophisticated from a homotopy theory point of
-%view and using chain complexes in many places where we could get by with spaces
+%%%% this is said later in the intro
+%Throughout the paper we typically prefer concrete categories (vector spaces, chain complexes)
+%even when we could work in greater generality (symmetric monoidal categories, model categories, etc.).
%\item ? one of the points we make (far) below is that there is not really much
%difference between (a) systems of fields and local relations and (b) $n$-cats;
@@ -151,8 +148,6 @@
\label{sec:motivations}
We will briefly sketch our original motivation for defining the blob complex.
-\nn{this is adapted from an old draft of the intro; it needs further modification
-in order to better integrate it into the current intro.}
As a starting point, consider TQFTs constructed via fields and local relations.
(See \S\ref{sec:tqftsviafields} or \cite{kw:tqft}.)
@@ -166,7 +161,7 @@
It associates a bigraded vector space $A_{Kh}(W^4, L)$ to a 4-manifold $W$ together
with a link $L \subset \bd W$.
The original Khovanov homology of a link in $S^3$ is recovered as $A_{Kh}(B^4, L)$.
-\todo{I'm tempted to replace $A_{Kh}$ with $\cl{Kh}$ throughout this page -S}
+%\todo{I'm tempted to replace $A_{Kh}$ with $\cl{Kh}$ throughout this page -S}
How would we go about computing $A_{Kh}(W^4, L)$?
For the Khovanov homology of a link in $S^3$ the main tool is the exact triangle (long exact sequence)
@@ -178,8 +173,8 @@
associated to $B^3$ (with appropriate boundary conditions).
The coend is not an exact functor, so the exactness of the triangle breaks.
-
-The obvious solution to this problem is to replace the coend with its derived counterpart.
+The obvious solution to this problem is to replace the coend with its derived counterpart,
+Hochschild homology.
This presumably works fine for $S^1\times B^3$ (the answer being the Hochschild homology
of an appropriate bimodule), but for more complicated 4-manifolds this leaves much to be desired.
If we build our manifold up via a handle decomposition, the computation
@@ -187,7 +182,9 @@
A different handle decomposition of the same manifold would yield a different
sequence of derived coends.
To show that our definition in terms of derived coends is well-defined, we
-would need to show that the above two sequences of derived coends yield the same answer.
+would need to show that the above two sequences of derived coends yield
+isomorphic answers, and that the isomorphism does not depend on any
+choices we made along the way.
This is probably not easy to do.
Instead, we would prefer a definition for a derived version of $A_{Kh}(W^4, L)$
@@ -201,7 +198,7 @@
\[
\text{linear combinations of fields} \;\big/\; \text{local relations} ,
\]
-with an appropriately free resolution (the ``blob complex")
+with an appropriately free resolution (the blob complex)
\[
\cdots\to \bc_2(W, L) \to \bc_1(W, L) \to \bc_0(W, L) .
\]
@@ -210,11 +207,6 @@
$\bc_2$ is linear combinations of relations amongst relations on $W$,
and so on.
-None of the above ideas depend on the details of the Khovanov homology example,
-so we develop the general theory in this paper and postpone specific applications
-to later papers.
-
-
\subsection{Formal properties}
\label{sec:properties}
@@ -236,10 +228,12 @@
The blob complex is also functorial (indeed, exact) with respect to $\cF$,
although we will not address this in detail here.
+\nn{KW: what exactly does ``exact in $\cF$" mean?
+Do we mean a similar statement for module labels?}
\begin{property}[Disjoint union]
\label{property:disjoint-union}
-The blob complex of a disjoint union is naturally the tensor product of the blob complexes.
+The blob complex of a disjoint union is naturally isomorphic to the tensor product of the blob complexes.
\begin{equation*}
\bc_*(X_1 \du X_2) \iso \bc_*(X_1) \tensor \bc_*(X_2)
\end{equation*}
@@ -264,16 +258,19 @@
\begin{property}[Contractibility]
\label{property:contractibility}%
-With field coefficients, the blob complex on an $n$-ball is contractible in the sense that it is homotopic to its $0$-th homology.
-Moreover, the $0$-th homology of balls can be canonically identified with the vector spaces associated by the system of fields $\cF$ to balls.
+With field coefficients, the blob complex on an $n$-ball is contractible in the sense
+that it is homotopic to its $0$-th homology.
+Moreover, the $0$-th homology of balls can be canonically identified with the vector spaces
+associated by the system of fields $\cF$ to balls.
\begin{equation*}
-\xymatrix{\bc_*(B^n;\cF) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & H_0(\bc_*(B^n;\cF)) \ar[r]^(0.6)\iso & \cF(B^n)}
+\xymatrix{\bc_*(B^n;\cF) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & H_0(\bc_*(B^n;\cF)) \ar[r]^(0.6)\iso & A_\cF(B^n)}
\end{equation*}
\end{property}
Properties \ref{property:functoriality} will be immediate from the definition given in
\S \ref{sec:blob-definition}, and we'll recall it at the appropriate point there.
-Properties \ref{property:disjoint-union}, \ref{property:gluing-map} and \ref{property:contractibility} are established in \S \ref{sec:basic-properties}.
+Properties \ref{property:disjoint-union}, \ref{property:gluing-map} and
+\ref{property:contractibility} are established in \S \ref{sec:basic-properties}.
\subsection{Specializations}
\label{sec:specializations}
@@ -298,13 +295,14 @@
The blob complex for a $1$-category $\cC$ on the circle is
quasi-isomorphic to the Hochschild complex.
\begin{equation*}
-\xymatrix{\bc_*(S^1;\cC) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & \HC_*(\cC).}
+\xymatrix{\bc_*(S^1;\cC) \ar[r]^(0.47){\iso}_(0.47){\text{qi}} & \HC_*(\cC).}
\end{equation*}
\end{thm:hochschild}
Theorem \ref{thm:skein-modules} is immediate from the definition, and
Theorem \ref{thm:hochschild} is established in \S \ref{sec:hochschild}.
-We also note \S \ref{sec:comm_alg} which describes the blob complex when $\cC$ is a one of certain commutative algebras thought of as an $n$-category.
+We also note \S \ref{sec:comm_alg} which describes the blob complex when $\cC$ is a one of
+certain commutative algebras thought of as $n$-categories.
\subsection{Structure of the blob complex}
@@ -318,7 +316,7 @@
\label{thm:evaluation}%
There is a chain map
\begin{equation*}
-\ev_X: \CH{X} \tensor \bc_*(X) \to \bc_*(X).
+e_X: \CH{X} \tensor \bc_*(X) \to \bc_*(X).
\end{equation*}
such that
\begin{enumerate}
@@ -330,14 +328,15 @@
\begin{equation*}
\xymatrix@C+2cm{
\CH{X} \otimes \bc_*(X)
- \ar[r]_{\ev_{X}} \ar[d]^{\gl^{\Homeo}_Y \otimes \gl_Y} &
+ \ar[r]_{e_{X}} \ar[d]^{\gl^{\Homeo}_Y \otimes \gl_Y} &
\bc_*(X) \ar[d]_{\gl_Y} \\
- \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)
+ \CH{X \bigcup_Y \selfarrow} \otimes \bc_*(X \bigcup_Y \selfarrow) \ar[r]^<<<<<<<<<<<<{e_{(X \bigcup_Y \scalebox{0.5}{\selfarrow})}} & \bc_*(X \bigcup_Y \selfarrow)
}
\end{equation*}
\end{enumerate}
Moreover any such chain map is unique, up to an iterated homotopy.
(That is, any pair of homotopies have a homotopy between them, and so on.)
+\nn{revisit this after proof below has stabilized}
\end{thm:CH}
\newtheorem*{thm:CH-associativity}{Theorem \ref{thm:CH-associativity}}
@@ -348,8 +347,8 @@
The chain map of Theorem \ref{thm:CH} 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)
+\CH{X} \tensor \CH{X} \tensor \bc_*(X) \ar[r]^<<<<<{\id \tensor e_X} \ar[d]^{\compose \tensor \id} & \CH{X} \tensor \bc_*(X) \ar[d]^{e_X} \\
+\CH{X} \tensor \bc_*(X) \ar[r]^{e_X} & \bc_*(X)
}
\end{equation*}
\end{thm:CH-associativity}