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 \begin{article}
 \begin{abstract}
+\nn{needs revision}
 We explain the need for new axioms for topological quantum field theories that include ideas from derived
 categories and homotopy theory. We summarize our axioms for higher categories, and describe the ``blob complex".
 Fixing an $n$-category $\cC$, the blob complex associates a chain complex $\bc_*(W;\cC)$ to any $n$-manifold $W$.
 The $0$-th homology of this chain complex recovers the usual TQFT invariants of $W$.
 The higher homology groups should be viewed as generalizations of Hochschild homology.
 in a concrete version of the homotopy colimit.)
 We then review some basic properties of the blob complex, and finish by showing how it
 yields a higher categorical and higher dimensional generalization of Deligne's
 conjecture on Hochschild cochains and the little 2-disks operad.
+\nn{needs revision}
 Of course, there are currently many interesting alternative notions of $n$-category and of TQFT.
 We note that our $n$-categories are both more and less general
 than the ``fully dualizable" ones which play a prominent role in \cite{0905.0465}.
 They are more general in that we make no duality assumptions in the top dimension $n{+}1$.
 They are less general in that we impose stronger duality requirements in dimensions 0 through $n$.
 %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.}
 We will define two variations simultaneously,  as all but one of the axioms are identical in the two cases.
-These variations are ``isotopy $n$-categories", where homeomorphisms fixing the boundary
+These variations are ``plain $n$-categories", where homeomorphisms fixing the boundary
 act trivially on the sets associated to $n$-balls
 (and these sets are usually vector spaces or more generally modules over a commutative ring)
 and ``$A_\infty$ $n$-categories",  where there is a homotopy action of
 $k$-parameter families of homeomorphisms on these sets
 (which are usually chain complexes or topological spaces).
 which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions
 to the intersection of the boundaries of $B$ and $B_i$.
 If $k < n$,
 or if $k=n$ and we are in the $A_\infty$ case,
 we require that $\gl_Y$ is injective.
-(For $k=n$ in the isotopy $n$-category case, see Axiom \ref{axiom:extended-isotopies}.)
+(For $k=n$ in the plain $n$-category case, see Axiom \ref{axiom:extended-isotopies}.)
 \end{axiom}
 \begin{axiom}[Strict associativity] \label{nca-assoc}\label{axiom:associativity}
 The gluing maps above are strictly associative.
 Given any decomposition of a ball $B$ into smaller balls
 where the first arrow is gluing with a product morphism on $Y\times_p J$ and
 the second is induced by a homeomorphism from $X\cup_Y (Y\times_p J)$ to $X$ which restricts
 to the identity on the boundary.
-\begin{axiom}[\textup{\textbf{[for isotopy  $n$-categories]}} Extended isotopy invariance in dimension $n$.]
+\begin{axiom}[\textup{\textbf{[for plain  $n$-categories]}} Extended isotopy invariance in dimension $n$.]
 \label{axiom:extended-isotopies}
 Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts
 to the identity on $\bd X$ and isotopic (rel boundary) to the identity.
 Then $f$ acts trivially on $\cC(X)$.
 In addition, collar maps act trivially on $\cC(X)$.
 \begin{equation*}
 %\label{eq:psi-C}
 	\psi_{\cC;W}(x) \subset \prod_a \cC(X_a)\spl
 \end{equation*}
 where the restrictions to the various pieces of shared boundaries amongst the balls
-$X_a$ all agree (this is a fibered product of all the labels of $k$-balls over the labels of $k-1$-manifolds).
+$X_a$ all agree (similar to a fibered product).
 When $k=n$, the ``subset" and ``product" in the above formula should be
 interpreted in the appropriate enriching category.
 If $x$ is a refinement of $y$, the map $\psi_{\cC;W}(x) \to \psi_{\cC;W}(y)$ is given by the composition maps of $\cC$.
 \end{defn}
-We will use the term ``field on $W$" to refer to a point of this functor,
+%We will use the term ``field on $W$" to refer to a point of this functor,
-that is, a permissible decomposition $x$ of $W$ together with an element of $\psi_{\cC;W}(x)$.
+%that is, a permissible decomposition $x$ of $W$ together with an element of $\psi_{\cC;W}(x)$.
 \subsubsection{Colimits}
 Recall that our definition of an $n$-category is essentially a collection of functors
 defined on the categories of homeomorphisms of $k$-balls
 It is natural to hope to extend such functors to the
 larger categories of all $k$-manifolds (again, with homeomorphisms).
 In fact, the axioms stated above already require such an extension to $k$-spheres for $k<n$.
 The natural construction achieving this is a colimit along the poset of permissible decompositions.
-Given an isotopy $n$-category $\cC$,
+Given a plain $n$-category $\cC$,
 we will denote its extension to all manifolds by $\cl{\cC}$. On a $k$-manifold $W$, with $k \leq n$,
 this is defined to be the colimit along $\cell(W)$ of the functor $\psi_{\cC;W}$.
 Note that Axioms \ref{axiom:composition} and \ref{axiom:associativity}
 imply that $\cl{\cC}(X)  \iso \cC(X)$ when $X$ is a $k$-ball with $k<n$.
 Suppose that $\cC$ is enriched in vector spaces: this means that given boundary conditions $c \in \cl{\cC}(\bdy X)$, for $X$ an $n$-ball,
 %When $\cC$ is a topological $n$-category,
 %the flexibility available in the construction of a homotopy colimit allows
 %us to give a much more explicit description of the blob complex which we'll write as $\bc_*(W; \cC)$.
 %\todo{either need to explain why this is the same, or significantly rewrite this section}
-When $\cC$ is the isotopy $n$-category based on string diagrams for a traditional
+When $\cC$ is the plain $n$-category based on string diagrams for a traditional
 $n$-category $C$,
 one can show \cite{1009.5025} that the above two constructions of the homotopy colimit
 are equivalent to the more concrete construction which we describe next, and which we denote $\bc_*(W; \cC)$.
 Roughly speaking, the generators of $\bc_k(W; \cC)$ are string diagrams on $W$ together with
 a configuration of $k$ balls (or ``blobs") in $W$ whose interiors are pairwise disjoint or nested.
 each $B_i$ appears as a connected component of one of the $M_j$.
 Note that this forces the balls to be pairwise either disjoint or nested.
 Such a collection of balls cuts $W$ into pieces, the connected components of $W \setminus \bigcup \bdy B_i$.
 These pieces need not be manifolds, but they do automatically have permissible decompositions.
-The $k$-blob group $\bc_k(W; \cC)$ is generated by the $k$-blob diagrams. A $k$-blob diagram consists of
+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, 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 gluing 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 alternating signs.
-We now spell this out for some small values of $k$. For $k=0$, the $0$-blob group is simply fields on $W$.
+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 to 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 fields which are null on some ball.
 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 $B_2$.
 The blob complex has several important special cases.
 \begin{thm}[Skein modules]
 \label{thm:skein-modules}
-Suppose $\cC$ is an isotopy $n$-category.
+Suppose $\cC$ is a plain $n$-category.
 The $0$-th blob homology of $X$ is the usual
 (dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$
 by $\cC$.
 \begin{equation*}
 H_0(\bc_*(X;\cC)) \iso \cl{\cC}(X)
 blob complexes for the $A_\infty$ $n$-categories constructed as above.
 \begin{thm}[Product formula]
 \label{thm:product}
 Let $W$ be a $k$-manifold and $Y$ be an $n{-}k$ manifold.
-Let $\cC$ be an isotopy $n$-category.
+Let $\cC$ be a plain $n$-category.
 Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ as above.
 Then
 \[
 	\bc_*(Y\times W; \cC) \simeq \clh{\bc_*(Y;\cC)}(W).
 \]
 A standard acyclic models argument now constructs the homotopy inverse.
 \end{proof}
 %\nn{Theorem \ref{thm:product} is proved in \S \ref{ss:product-formula}, and Theorem \ref{thm:gluing} in \S \ref{sec:gluing}.}
-\section{Deligne's conjecture for $n$-categories}
+\section{Extending Deligne's conjecture to $n$-categories}
 \label{sec:applications}
 Let $M$ and $N$ be $n$-manifolds with common boundary $E$.
 Recall (Theorem \ref{thm:gluing}) that the $A_\infty$ category $A = \bc_*(E)$
 acts on $\bc_*(M)$ and $\bc_*(N)$.
 which satisfy the operad compatibility conditions.
 \begin{proof} (Sketch.)
 We have already defined the action of mapping cylinders, in Theorem \ref{thm:evaluation},
 and the action of surgeries is just composition of maps of $A_\infty$-modules.
-We only need to check that the relations of the surgery cylinded operad are satisfied.
+We only need to check that the relations of the surgery cylinder operad are satisfied.
 This follows from the locality of the action of $\CH{-}$ (i.e., that it is compatible with gluing) and associativity.
 \end{proof}
 Consider the special case where $n=1$ and all of the $M_i$'s and $N_i$'s are intervals.
 We have that $SC^1_{\overline{M}, \overline{N}}$ is homotopy equivalent to the little

changeset 662	57bd9fab3827
parent 661	6345c3679795
child 663	001fc6183d19