--- a/text/intro.tex Sun Nov 01 20:29:33 2009 +0000
+++ b/text/intro.tex Sun Nov 01 20:29:41 2009 +0000
@@ -11,13 +11,11 @@
on the configurations space of unlabeled points in $M$.
%$$H_*(\bc_*(M; k[t])) = H^{\text{sing}}_*(\Delta^\infty(M), k).$$
\end{itemize}
-The blob complex has good formal properties, summarized in \S \ref{sec:properties}. These include an action of $\CD{M}$,
-\nn{maybe replace Diff with Homeo?}
-extending the usual $\Diff(M)$ action on the TQFT space $H_0$ (see Property \ref{property:evaluation}) and a gluing formula allowing calculations by cutting manifolds into smaller parts (see Property \ref{property:gluing}).
+The blob complex definition is motivated by the desire for a derived analogue of the usual TQFT Hilbert space (replacing quotient of fields by local relations with some sort of resolution),
+and for a generalization of Hochschild homology to higher $n$-categories. We would also like to be able to talk about $\CM{M}{T}$ when $T$ is an $n$-category rather than a manifold. The blob complex allows us to do all of these! More detailed motivations are described in \S \ref{sec:motivations}.
-The blob complex definition is motivated by the desire for a `derived' analogue of the usual TQFT Hilbert space (replacing quotient of fields by local relations with some sort of `resolution'),
-\nn{are the quotes around `derived' and `resolution' necessary?}
-and for a generalization of Hochschild homology to higher $n$-categories. We would also like to be able to talk about $\CM{M}{T}$ when $T$ is an $n$-category rather than a manifold. The blob complex allows us to do all of these! More detailed motivations are described in \S \ref{sec:motivations}.
+The blob complex has good formal properties, summarized in \S \ref{sec:properties}. These include an action of $\CH{M}$,
+extending the usual $\Homeo(M)$ action on the TQFT space $H_0$ (see Property \ref{property:evaluation}) and a gluing formula allowing calculations by cutting manifolds into smaller parts (see Property \ref{property:gluing}).
We expect applications of the blob complex to contact topology and Khovanov homology but do not address these in this paper. See \S \ref{sec:future} for slightly more detail.
@@ -29,14 +27,14 @@
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's not clear that we could remove the duality conditions from our definition, even if we wanted to.) 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.
\nn{Not sure that the next para is appropriate here}
-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 $B$ diffeomorphic to the $n$-ball. This vector spaces glue together associatively. For an $A_\infty$ $n$-category, we instead associate a chain complex to each such $B$. We require that diffeomorphisms (or the complex of singular chains of diffeomorphisms in the $A_\infty$ case) act. The axioms for an $A_\infty$ $n$-category are designed to capture two main examples: the blob complexes of $n$-balls, using a topological $n$-category, and the complex $\CM{-}{X}$ of maps to a fixed target space $X$.
+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. This vector spaces glue together associatively. For an $A_\infty$ $n$-category, we instead associate a chain complex to each such $B$. We require that homeomorphisms (or the complex of singular chains of homeomorphisms in the $A_\infty$ case) act. The axioms for an $A_\infty$ $n$-category are designed to capture two main examples: the blob complexes of $n$-balls, using a topological $n$-category, and the complex $\CM{-}{X}$ of maps to a fixed target space $X$.
\nn{we might want to make our choice of notation here ($B$, $X$) consistent with later sections ($X$, $T$), or vice versa}
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 $n$-manifold and an $A_\infty$ $n$-category. 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.
\nn{KW: the previous two paragraphs seem a little awkward to me, but I don't presently have a good idea for fixing them.}
-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 generalisation 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 $\CD{M}$, and make connections between our definitions of $n$-categories and familar 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.
+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 generalisation 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 familar 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.
\nn{some more things to cover in the intro}
@@ -58,12 +56,12 @@
in order to better integrate it into the current intro.}
As a starting point, consider TQFTs constructed via fields and local relations.
-(See Section \ref{sec:tqftsviafields} or \cite{kwtqft}.)
+(See Section \ref{sec:tqftsviafields} or \cite{kw:tqft}.)
This gives a satisfactory treatment for semisimple TQFTs
(i.e.\ TQFTs for which the cylinder 1-category associated to an
$n{-}1$-manifold $Y$ is semisimple for all $Y$).
-For non-semiemple TQFTs, this approach is less satisfactory.
+For non-semi-simple TQFTs, this approach is less satisfactory.
Our main motivating example (though we will not develop it in this paper)
is the (decapitated) $4{+}1$-dimensional TQFT associated to Khovanov homology.
It associates a bigraded vector space $A_{Kh}(W^4, L)$ to a 4-manifold $W$ together
@@ -72,7 +70,7 @@
How would we go about computing $A_{Kh}(W^4, L)$?
For $A_{Kh}(B^4, L)$, the main tool is the exact triangle (long exact sequence)
-\nn{... $L_1, L_2, L_3$}.
+relating resolutions of a crossing.
Unfortunately, the exactness breaks if we glue $B^4$ to itself and attempt
to compute $A_{Kh}(S^1\times B^3, L)$.
According to the gluing theorem for TQFTs-via-fields, gluing along $B^3 \subset \bd B^4$
@@ -113,7 +111,7 @@
and so on.
None of the above ideas depend on the details of the Khovanov homology example,
-so we develop the general theory in the paper and postpone specific applications
+so we develop the general theory in this paper and postpone specific applications
to later papers.
@@ -186,22 +184,22 @@
\end{equation*}
\end{property}
-Here $\CD{X}$ is the singular chain complex of the space of diffeomorphisms of $X$, fixed on $\bdy X$.
-\begin{property}[$C_*(\Diff(-))$ action]
+Here $\CH{X}$ is the singular chain complex of the space of homeomorphisms of $X$, fixed on $\bdy X$.
+\begin{property}[$C_*(\Homeo(-))$ action]
\label{property:evaluation}%
There is a chain map
\begin{equation*}
-\ev_X: \CD{X} \tensor \bc_*(X) \to \bc_*(X).
+\ev_X: \CH{X} \tensor \bc_*(X) \to \bc_*(X).
\end{equation*}
-Restricted to $C_0(\Diff(X))$ this is just the action of diffeomorphisms described in Property \ref{property:functoriality}. Further, for
+Restricted to $C_0(\Homeo(X))$ this is just the action of homeomorphisms described in Property \ref{property:functoriality}. Further, for
any codimension $1$-submanifold $Y \subset X$ dividing $X$ into $X_1 \cup_Y X_2$, the following diagram
(using the gluing maps described in Property \ref{property:gluing-map}) commutes.
\begin{equation*}
\xymatrix{
- \CD{X} \otimes \bc_*(X) \ar[r]^{\ev_X} & \bc_*(X) \\
- \CD{X_1} \otimes \CD{X_2} \otimes \bc_*(X_1) \otimes \bc_*(X_2)
- \ar@/_4ex/[r]_{\ev_{X_1} \otimes \ev_{X_2}} \ar[u]^{\gl^{\Diff}_Y \otimes \gl_Y} &
+ \CH{X} \otimes \bc_*(X) \ar[r]^{\ev_X} & \bc_*(X) \\
+ \CH{X_1} \otimes \CH{X_2} \otimes \bc_*(X_1) \otimes \bc_*(X_2)
+ \ar@/_4ex/[r]_{\ev_{X_1} \otimes \ev_{X_2}} \ar[u]^{\gl^{\Homeo}_Y \otimes \gl_Y} &
\bc_*(X_1) \otimes \bc_*(X_2) \ar[u]_{\gl_Y}
}
\end{equation*}
@@ -212,9 +210,9 @@
(using the gluing maps described in Property \ref{property:gluing-map}) commutes.
\begin{equation*}
\xymatrix@C+2cm{
- \CD{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) \\
- \CD{X} \otimes \bc_*(X)
- \ar[r]_{\ev_{X}} \ar[u]^{\gl^{\Diff}_Y \otimes \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} \otimes \bc_*(X)
+ \ar[r]_{\ev_{X}} \ar[u]^{\gl^{\Homeo}_Y \otimes \gl_Y} &
\bc_*(X) \ar[u]_{\gl_Y}
}
\end{equation*}
@@ -240,9 +238,9 @@
Let $A_*(Y)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology (see Property \ref{property:blobs-ainfty}).
Then
\[
- \bc_*(Y^{n-k}\times W^k, \cC) \simeq \bc_*(W, A_*(Y)) .
+ \bc_*(Y\times W, \cC) \simeq \bc_*(W, A_*(Y)) .
\]
-Note on the right here we have the version of the blob complex for $A_\infty$ $n$-categories.
+Note on the right hand side we have the version of the blob complex for $A_\infty$ $n$-categories.
\end{property}
It seems reasonable to expect a generalization describing an arbitrary fibre bundle. See in particular \S \ref{moddecss} for the framework as such a statement.
@@ -293,7 +291,7 @@
Throughout, we have resisted the temptation to work in the greatest generality possible (don't worry, it wasn't that hard).
In most of the places where we say ``set" or ``vector space", any symmetric monoidal category would do.
\nn{maybe make similar remark about chain complexes and $(\infty, 0)$-categories}
-More could be said about finite characteristic (there appears in be $2$-torsion in $\bc_1(S^2, \cC)$ for any spherical $2$-category $\cC$). Much more could be said about other types of manifolds, in particular oriented, $\operatorname{Spin}$ and $\operatorname{Pin}^{\pm}$ manifolds, where boundary issues become more complicated. (We'd recommend thinking about boundaries as germs, rather than just codimension $1$ manifolds.) We've also take the path of least resistance by considering $\operatorname{PL}$ manifolds; there may be some differences for topological manifolds and smooth manifolds.
+More could be said about finite characteristic (there appears in be $2$-torsion in $\bc_1(S^2, \cC)$ for any spherical $2$-category $\cC$, for example). Much more could be said about other types of manifolds, in particular oriented, $\operatorname{Spin}$ and $\operatorname{Pin}^{\pm}$ manifolds, where boundary issues become more complicated. (We'd recommend thinking about boundaries as germs, rather than just codimension $1$ manifolds.) We've also take the path of least resistance by considering $\operatorname{PL}$ manifolds; there may be some differences for topological manifolds and smooth manifolds.
Many results in Hochschild homology can be understood `topologically' via the blob complex. For example, we expect that the shuffle product on the Hochschild homology of a commutative algebra $A$ simply corresponds to the gluing operation on $\bc_*(S^1 \times [0,1], A)$, but haven't investigated the details.