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%% \title{Almost sharp fronts for the surface quasi-geostrophic equation}

\title{Higher categories, colimits, and the blob complex}

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%% Molecular, Murcia, Spain}, \and Franklin Sonnery\affil{2}{}}

\author{Scott Morrison\affil{1}{Miller Institute for Basic Research, UC Berkeley, CA 94704, USA} 

\and Kevin Walker\affil{2}{Microsoft Station Q, 2243 CNSI Building, UC Santa Barbara, CA 93106, USA}}

\contributor{Submitted to Proceedings of the National Academy of Sciences

of the United States of America}

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\begin{article}

\begin{abstract}

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 topological quantum field theory invariants of $W$. 

The higher homology groups should be viewed as generalizations of Hochschild homology (indeed, when $W=S^1$ they coincide). 

The blob complex has a very natural definition in terms of homotopy colimits along decompositions of the manifold $W$. 

We outline the important properties of the blob complex, and sketch the proof of a generalization of 

Deligne's conjecture on Hochschild cohomology and the little discs operad to higher dimensions.

\end{abstract}

%% When adding keywords, separate each term with a straight line: |

\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:

%% for example:

%% \abbreviations{SAM, self-assembled monolayer; OTS,

%% octadecyltrichlorosilane}

% \abbreviations{TQFT, topological quantum field theory}

%% The first letter of the article should be drop cap: \dropcap{}

%\dropcap{I}n this article we study the evolution of ''almost-sharp'' fronts

%% Enter the text of your article beginning here and ending before

%% \begin{acknowledgements}

%% Section head commands for your reference:

%% \section{}

%% \subsection{}

%% \subsubsection{}

\dropcap{T}he aim of this paper is to describe a derived category analogue of topological quantum field theories.

For our purposes, an $n{+}1$-dimensional TQFT is a locally defined system of

invariants of manifolds of dimensions 0 through $n{+}1$. In particular,

the TQFT invariant $A(Y)$ of a closed $k$-manifold $Y$ is a linear $(n{-}k)$-category.

If $Y$ has boundary then $A(Y)$ is a collection of $(n{-}k)$-categories which afford

a representation of the $(n{-}k{+}1)$-category $A(\bd Y)$.

(See \cite{1009.5025} and \cite{kw:tqft};

for a more homotopy-theoretic point of view see \cite{0905.0465}.)

We now comment on some particular values of $k$ above.

A linear 0-category is a vector space, and a representation

of a vector space is an element of the dual space.

Thus a TQFT assigns to each closed $n$-manifold $Y$ a vector space $A(Y)$,

and to each $(n{+}1)$-manifold $W$ an element of $A(\bd W)^*$.

For the remainder of this paper we will in fact be interested in so-called $(n{+}\epsilon)$-dimensional

TQFTs, which are slightly weaker structures in that they assign 

invariants to mapping cylinders of homeomorphisms between $n$-manifolds, but not to general $(n{+}1)$-manifolds.

When $k=n{-}1$ we have a linear 1-category $A(S)$ for each $(n{-}1)$-manifold $S$,

and a representation of $A(\bd Y)$ for each $n$-manifold $Y$.

The TQFT gluing rule in dimension $n$ states that

$A(Y_1\cup_S Y_2) \cong A(Y_1) \ot_{A(S)} A(Y_2)$,

where $Y_1$ and $Y_2$ are $n$-manifolds with common boundary $S$.

When $k=0$ we have an $n$-category $A(pt)$.

This can be thought of as the local part of the TQFT, and the full TQFT can be reconstructed from $A(pt)$

via colimits (see below).

We call a TQFT semisimple if $A(S)$ is a semisimple 1-category for all $(n{-}1)$-manifolds $S$

and $A(Y)$ is a finite-dimensional vector space for all $n$-manifolds $Y$.

Examples of semisimple TQFTs include Witten-Reshetikhin-Turaev theories, 

Turaev-Viro theories, and Dijkgraaf-Witten theories.

These can all be given satisfactory accounts in the framework outlined above.

(The WRT invariants need to be reinterpreted as $3{+}1$-dimensional theories with only a weak 

dependence on interiors in order to be

extended all the way down to dimension 0.)

For other non-semisimple TQFT-like invariants, however, the above framework seems to be inadequate.

For example, the gluing rule for 3-manifolds in Ozsv\'ath-Szab\'o/Seiberg-Witten theory

involves a tensor product over an $A_\infty$ 1-category associated to 2-manifolds \cite{1003.0598,1005.1248}.

Long exact sequences are important computational tools in these theories,

and also in Khovanov homology, but the colimit construction breaks exactness.

For these reasons and others, it is desirable to 

extend to above framework to incorporate ideas from derived categories.

One approach to such a generalization might be to simply define a

TQFT via its gluing formulas, replacing tensor products with

derived tensor products (c.f. \cite{1011.1958}).

However, it is probably difficult to prove

the invariance of such a definition, as the object associated to a manifold

will a priori depend on the explicit presentation used to apply the gluing formulas.

We instead give a manifestly invariant construction, and

deduce from it the gluing formulas based on $A_\infty$ tensor products.

This paper is organized as follows.

We first give an account of our version of $n$-categories.

According to our definition, $n$-categories are, among other things,

functorial invariants of $k$-balls, $0\le k \le n$, which behave well with respect to gluing.

We then show how to extend an $n$-category from balls to arbitrary $k$-manifolds,

using colimits and homotopy colimits.

This extension, which we call the blob complex, has as $0$-th homology the usual TQFT invariant.

(The name comes from the ``blobs" which feature prominently

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.

At several points we only sketch an argument briefly; full details can be found in \cite{1009.5025}. 

In this paper we attempt to give a clear view of the big picture without getting 

bogged down in technical details.

\section{Definitions}

\subsection{$n$-categories} \mbox{}

In this section we give a definition of $n$-categories designed to work well with TQFTs.

The main idea is to base the definition on actual balls, rather combinatorial models of them.

This has the advantages of avoiding a proliferation of coherency axioms and building in a strong

version of duality from the start.

%\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.}

Of course, there are currently many interesting alternative notions of $n$-category.

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 $n$.

Thus our $n$-categories correspond to $(n{+}\epsilon)$-dimensional {\it unoriented} or {\it oriented} TQFTs, while

Lurie's (fully dualizable) $n$-categories correspond to $(n{+}1)$-dimensional {\it framed} TQFTs.

We will define two variations simultaneously,  as all but one of the axioms are identical in the two cases.

These variations are ``ordinary $n$-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).

There are five basic ingredients 

\cite{life-of-brian} 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

\cite[\S 2.2]{MR505692}.

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.