--- a/pnas/pnas.tex Fri Oct 29 11:42:35 2010 +0900
+++ b/pnas/pnas.tex Thu Nov 04 17:01:56 2010 +0900
@@ -139,7 +139,7 @@
%% When adding keywords, separate each term with a straight line: |
-\keywords{term | term | term}
+\keywords{n-categories | topological quantum field theory | hochschild homology}
%% Optional for entering abbreviations, separate the abbreviation from
%% its definition with a comma, separate each pair with a semicolon:
@@ -159,8 +159,6 @@
%% \subsection{}
%% \subsubsection{}
-\todo{Check font size in an actual PNAS article: this looks a little big}
-
\nn{
background: TQFTs are important, historically, semisimple categories well-understood.
Many new examples arising recently which do not fit this framework, e.g. SW and OS theory.
@@ -178,7 +176,45 @@
\section{Definitions}
\subsection{$n$-categories} \mbox{}
-\todo{This is just a copy and paste of the statements of the axioms. We need to rewrite this into something that's both compact and comprehensible! The first few at least aren't that terrifying, but we definitely don't want to derail the reader with the actual product axiom, for example.}
+\nn{rough draft of n-cat stuff...}
+
+\nn{maybe say something about goals: well-suited to TQFTs; avoid proliferation of coherency axioms;
+non-recursive (n-cats not defined n terms of (n-1)-cats; easy to show that the motivating
+examples satisfy the axioms; strong duality; both plain and infty case;
+(?) easy to see that axioms are correct, in the sense of nothing missing (need
+to say this better if we keep it)}
+
+\nn{maybe: the typical n-cat definition tries to do two things at once: (1) give a list of basic properties
+which are weak enough to include the basic examples and strong enough to support the proofs
+of the main theorems; and (2) specify a minimal set of generators and/or axioms.
+We separate these two tasks, and address only the first, which becomes much easier when not burdened by the second.
+More specifically, life is easier when working with maximal, rather than minimal, collections of axioms.}
+
+\nn{say something about defining plain and infty cases simultaneously}
+
+There are five basic ingredients of an $n$-category definition:
+$k$-morphisms (for $0\le k \le n$), domain and range, composition,
+identity morphisms, and special behavior in dimension $n$ (e.g. enrichment
+in some auxiliary category, or strict associativity instead of weak associativity).
+We will treat each of these in turn.
+
+To motivate our morphism axiom, consider the venerable notion of the Moore loop space
+\nn{need citation -- \S 2.2 of Adams' ``Infinite Loop Spaces''?}.
+In the standard definition of a loop space, loops are always parameterized by the unit interval $I = [0,1]$,
+so composition of loops requires a reparameterization $I\cup I \cong I$, and this leads to a proliferation
+of higher associativity relations.
+While this proliferation is manageable for 1-categories (and indeed leads to an elegant theory
+of Stasheff polyhedra and $A_\infty$ categories), it becomes undesirably complex for higher categories.
+In a Moore loop space, we have a separate space $\Omega_r$ for each interval $[0,r]$, and a
+{\it strictly associative} composition $\Omega_r\times \Omega_s\to \Omega_{r+s}$.
+Thus we can have the simplicity of strict associativity in exchange for more morphisms.
+We wish to imitate this strategy in higher categories.
+Because we are mainly interested in the case of strong duality, we replace the intervals $[0,r]$ not with
+a product of $k$ intervals \nn{cf xxxx} but rather with any $k$-ball, that is, any $k$-manifold which is homeomorphic
+to the standard $k$-ball $B^k$.
+\nn{maybe add that in addition we want functoriality}
+
+\nn{say something about different flavors of balls; say it here? later?}
\begin{axiom}[Morphisms]
\label{axiom:morphisms}
@@ -186,16 +222,30 @@
the category of $k$-balls and
homeomorphisms to the category of sets and bijections.
\end{axiom}
-\begin{lem}
-\label{lem:spheres}
-For each $1 \le k \le n$, we have a functor $\cl{\cC}_{k-1}$ from
-the category of $k{-}1$-spheres and
-homeomorphisms to the category of sets and bijections.
-\end{lem}
+
+Note that the functoriality in the above axiom allows us to operate via \nn{fragment?}
+
+Next we consider domains and ranges of $k$-morphisms.
+Because we assume strong duality, it doesn't make much sense to subdivide the boundary of a morphism
+into domain and range --- the duality operations can convert domain to range and vice-versa.
+Instead, we will use a unified domain/range, which we will call a ``boundary".
+
+In order to state the axiom for boundaries, we need to extend the functors $\cC_k$
+of $k$-balls to functors $\cl{\cC}_{k-1}$ of $k$-spheres.
+This extension is described in xxxx below.
+
+%\begin{lem}
+%\label{lem:spheres}
+%For each $1 \le k \le n$, we have a functor $\cl{\cC}_{k-1}$ from
+%the category of $k{-}1$-spheres and
+%homeomorphisms to the category of sets and bijections.
+%\end{lem}
+
\begin{axiom}[Boundaries]\label{nca-boundary}
For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cl{\cC}_{k-1}(\bd X)$.
These maps, for various $X$, comprise a natural transformation of functors.
\end{axiom}
+
\begin{lem}[Boundary from domain and range]
\label{lem:domain-and-range}
Let $S = B_1 \cup_E B_2$, where $S$ is a $k{-}1$-sphere $(1\le k\le n)$,
@@ -210,6 +260,7 @@
(When $k=1$ we stipulate that $\cl{\cC}(E)$ is a point, so that the above fibered product
becomes a normal product.)
\end{lem}
+
\begin{axiom}[Composition]
\label{axiom:composition}
Let $B = B_1 \cup_Y B_2$, where $B$, $B_1$ and $B_2$ are $k$-balls ($0\le k\le n$)
@@ -229,11 +280,12 @@
we require that $\gl_Y$ is injective.
(For $k=n$ in the plain (non-$A_\infty$) case, see below.)
\end{axiom}
+
\begin{axiom}[Strict associativity] \label{nca-assoc}
The composition (gluing) maps above are strictly associative.
-Given any splitting of a ball $B$ into smaller balls
+Given any decomposition of a ball $B$ into smaller balls
$$\bigsqcup B_i \to B,$$
-any sequence of gluings (in the sense of Definition \ref{defn:gluing-decomposition}, where all the intermediate steps are also disjoint unions of balls) yields the same result.
+any sequence of gluings (where all the intermediate steps are also disjoint unions of balls) yields the same result.
\end{axiom}
\begin{axiom}[Product (identity) morphisms]
\label{axiom:product}
@@ -319,6 +371,11 @@
\subsection{The blob complex}
\subsubsection{Decompositions of manifolds}
+\nn{KW: I'm inclined to suppress all discussion of the subtleties of decompositions.
+Maybe just a single remark that we are omitting some details which appear in our
+longer paper.}
+\nn{SM: for now I disagree: the space expense is pretty minor, and it always us to be "in principle" complete. Let's see how we go for length.}
+
A \emph{ball decomposition} of $W$ is a
sequence of gluings $M_0\to M_1\to\cdots\to M_m = W$ such that $M_0$ is a disjoint union of balls
$\du_a X_a$ and each $M_i$ is a manifold.
@@ -383,21 +440,24 @@
The $k$-blob group $\bc_k(W; \cC)$ is generated by the $k$-blob diagrams. A $k$-blob diagram consists of
\begin{itemize}
-\item a permissible collection of $k$ embedded balls (called `blobs') in $W$,
+\item a permissible collection of $k$ embedded balls,
\item an ordering of the balls, and
\item for each resulting piece of $W$, a field,
\end{itemize}
-such that for any innermost blob $B$, the field on $B$ goes to zero under the composition map from $\cC$.
+such that for any innermost blob $B$, the field on $B$ goes to zero under the composition map from $\cC$. We call such a field a `null field on $B$'.
The differential acts on a $k$-blob diagram by summing over ways to forget one of the $k$ blobs, with signs given by the ordering.
\todo{Say why this really is the homotopy colimit}
-\todo{Spell out $k=0, 1, 2$}
+
+We now spell this out for some small values of $k$. For $k=0$, the $0$-blob group is simply fields on $W$. For $k=1$, a generator consists of a field on $W$ and a ball, such that the restriction of the field that that ball is a null field. The differential simply forgets the ball. Thus we see that $H_0$ of the blob complex is the quotient of fields by null fields.
+
+For $k=2$, we have a two types of generators; they each consists of a field $f$ on $W$, and two balls $B_1$ and $B_2$. In the first case, the balls are disjoint, and $f$ restricted to either of the $B_i$ is a null field. In the second case, the balls are properly nested, say $B_1 \subset B_2$, and $f$ restricted to $B_1$ is null. Note that this implies that $f$ restricted to $B_2$ is also null, by the associativity of the gluing operation. This ensures that the differential is well-defined.
\section{Properties of the blob complex}
\subsection{Formal properties}
\label{sec:properties}
-The blob complex enjoys the following list of formal properties.
+The blob complex enjoys the following list of formal properties. The first three properties are immediate from the definitions.
\begin{property}[Functoriality]
\label{property:functoriality}%
@@ -447,7 +507,12 @@
\end{property}
\nn{maybe should say something about the $A_\infty$ case}
-Properties \ref{property:functoriality}, \ref{property:disjoint-union} and \ref{property:gluing-map} are immediate from the definition. Property \ref{property:contractibility} \todo{}
+\begin{proof}(Sketch)
+For $k\ge 1$, the contracting homotopy sends a $k$-blob diagram to the $(k{+}1)$-blob diagram
+obtained by adding an outer $(k{+}1)$-st blob consisting of all $B^n$.
+For $k=0$ we choose a splitting $s: H_0(\bc_*(B^n)) \to \bc_0(B^n)$ and send
+$x\in \bc_0(B^n)$ to $x - s([x])$, where $[x]$ denotes the image of $x$ in $H_0(\bc_*(B^n))$.
+\end{proof}
\subsection{Specializations}
\label{sec:specializations}
@@ -590,7 +655,7 @@
This says that we can recover (up to homotopy) the space of maps to $T$ via blob homology from local data.
Note that there is no restriction on the connectivity of $T$ as there is for the corresponding result in topological chiral homology \cite[Theorem 3.8.6]{0911.0018}.
-
+\todo{sketch proof}
\begin{thm}[Higher dimensional Deligne conjecture]
\label{thm:deligne}
@@ -599,12 +664,13 @@
this implies that the little $n{+}1$-balls operad acts on blob cochains of the $n$-ball.
\end{thm}
-An $n$-dimensional surgery cylinder is an alternating sequence of mapping cylinders and surgeries, modulo changing the order of distant surgeries, and conjugating the submanifold not modified in a surgery by a homeomorphism. See Figure \ref{delfig2}. Surgery cylinders form an operad, by gluing the outer boundary of one cylinder into an inner boundary of another.
+An $n$-dimensional surgery cylinder is a sequence of mapping cylinders and surgeries (Figure \ref{delfig2}), modulo changing the order of distant surgeries, and conjugating a submanifold not modified in a surgery by a homeomorphism. Surgery cylinders form an operad, by gluing the outer boundary of one cylinder into an inner boundary of another.
-\todo{Explain blob cochains}
+By the `blob cochains' of a manifold $X$, we mean the $A_\infty$ maps of $\bc_*(X)$ as a $\bc_*(\bdy X)$ $A_\infty$-module.
+
\todo{Sketch proof}
-The little disks operad $LD$ is homotopy equivalent to the $n=1$ case of the $n$-SC operad. The blob complex $\bc_*(I, \cC)$ is a bimodule over itself, and the $A_\infty$-bimodule intertwiners are homotopy equivalent to the Hochschild cohains $Hoch^*(C, C)$. The usual Deligne conjecture (proved variously in \cite{hep-th/9403055, MR1805894, MR2064592, MR1805923} gives a map
+The little disks operad $LD$ is homotopy equivalent to the $n=1$ case of the $n$-SC operad. The blob complex $\bc_*(I, \cC)$ is a bimodule over itself, and the $A_\infty$-bimodule intertwiners are homotopy equivalent to the Hochschild cohains $Hoch^*(C, C)$. The usual Deligne conjecture (proved variously in \cite{hep-th/9403055, MR1805894, MR2064592, MR1805923}) gives a map
\[
C_*(LD_k)\otimes \overbrace{Hoch^*(C, C)\otimes\cdots\otimes Hoch^*(C, C)}^{\text{$k$ copies}}
\to Hoch^*(C, C),