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%% For titles, only capitalize the first letter

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

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

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\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}

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\keywords{n-categories | topological quantum field theory | hochschild homology}

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% \abbreviations{TQFT, topological quantum field theory}

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

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\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 and do not assign anything to general $(n{+}1)$-manifolds, but only to mapping cylinders.

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 $Y_2$ and $n$-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\'{a}th-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.

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 $n$.

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

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

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.}

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

in the two cases. These variations are `linear $n$-categories', where the sets associated to $n$-balls with specified boundary conditions are in fact vector spaces, and `$A_\infty$ $n$-categories', where these sets are chain complexes.

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.

Because we are mainly interested in the case of strong duality, we replace the intervals $[0,r]$ not with

a product of $k$ intervals (c.f. \cite{0909.2212}) but rather with any $k$-ball, that is, any $k$-manifold which is homeomorphic

to the standard $k$-ball $B^k$.

By default our balls are unoriented,

but it is useful at times to vary this,

for example by considering oriented or Spin balls.

We can also consider more exotic structures, such as balls with a map to some target space,

or equipped with $m$ independent vector fields.

(The latter structure would model $n$-categories with less duality than we usually assume.)

\begin{axiom}[Morphisms]

\label{axiom:morphisms}

For each $0 \le k \le n$, we have a functor $\cC_k$ from 

the category of $k$-balls and 

homeomorphisms to the category of sets and bijections.

\end{axiom}

Note that the functoriality in the above axiom allows us to operate via

homeomorphisms which are not the identity on the boundary of the $k$-ball.

The action of these homeomorphisms gives the ``strong duality" structure.

As such, we don't subdivide the boundary of a morphism

into domain and range --- the duality operations can convert between domain and range.

Later we inductively define an extension of the functors $\cC_k$ to functors $\cl{\cC}_k$ from arbitrary manifolds to sets. We need the restriction of these functors to $k$-spheres, for $k<n$, for the next axiom.

\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}

For $c\in \cl{\cC}_{k-1}(\bd X)$ we define $\cC_k(X; c) = \bd^{-1}(c)$.

Many of the examples we are interested in are enriched in some auxiliary category $\cS$

(e.g. vector spaces or rings, or, in the $A_\infty$ case, chain complexes or topological spaces).

This means that in the top dimension $k=n$ the sets $\cC_n(X; c)$ have the structure

of an object of $\cS$, and all of the structure maps of the category (above and below) are

compatible with the $\cS$ structure on $\cC_n(X; c)$.

Given two hemispheres (a `domain' and `range') that agree on the equator, we need to be able to assemble them into a boundary value of the entire sphere.

\begin{lem}

\label{lem:domain-and-range}

Let $S = B_1 \cup_E B_2$, where $S$ is a $k{-}1$-sphere $(1\le k\le n)$,

$B_i$ is a $k{-}1$-ball, and $E = B_1\cap B_2$ is a $k{-}2$-sphere (Figure \ref{blah3}).

Let $\cC(B_1) \times_{\cl{\cC}(E)} \cC(B_2)$ denote the fibered product of the 

two maps $\bd: \cC(B_i)\to \cl{\cC}(E)$.

Then we have an injective map

\[

	\gl_E : \cC(B_1) \times_{\cl{\cC}(E)} \cC(B_2) \into \cl{\cC}(S)

\]

which is natural with respect to the actions of homeomorphisms.

%(When $k=1$ we stipulate that $\cl{\cC}(E)$ is a point, so that the above fibered product

%becomes a normal product.)

\end{lem}

If $\bdy B = S$, we denote $\bdy^{-1}(\im(\gl_E))$ by $\cC(B)_E$.

\begin{axiom}[Gluing]

\label{axiom:composition}

Let $B = B_1 \cup_Y B_2$, where $B$, $B_1$ and $B_2$ are $k$-balls ($0\le k\le n$)

and $Y = B_1\cap B_2$ is a $k{-}1$-ball (Figure \ref{blah5}).

Let $E = \bd Y$, which is a $k{-}2$-sphere.

%Note that each of $B$, $B_1$ and $B_2$ has its boundary split into two $k{-}1$-balls by $E$.

We have restriction maps $\cC(B_i)_E \to \cC(Y)$.

Let $\cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E$ denote the fibered product of these two maps. 

We have a map

\[

	\gl_Y : \cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E \to \cC(B)_E

\]

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 linear case, see below.)

\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

$$\bigsqcup B_i \to B,$$ 

any sequence of gluings (where all the intermediate steps are also disjoint unions of balls) yields the same result.

\end{axiom}

For the next axiom, a \emph{pinched product} is a map locally modeled on a degeneracy map between simplices.

\begin{axiom}[Product (identity) morphisms]

\label{axiom:product}

For each pinched product $\pi:E\to X$, with $X$ a $k$-ball and $E$ a $k{+}m$-ball ($m\ge 1$),

there is a map $\pi^*:\cC(X)\to \cC(E)$.

These maps must be

\begin{enumerate}

\item natural with respect to maps of pinched products,

\item functorial with respect to composition of pinched products, 

\item compatible with gluing and restriction of pinched products.

\end{enumerate}

%%% begin noop %%%

% this was the original list of conditions, which I've replaced with the much terser list above -S

\noop{

These maps must satisfy the following conditions.

\begin{enumerate}

\item

If $\pi:E\to X$ and $\pi':E'\to X'$ are pinched products, and

if $f:X\to X'$ and $\tilde{f}:E \to E'$ are maps such that the diagram

\[ \xymatrix{

	E \ar[r]^{\tilde{f}} \ar[d]_{\pi} & E' \ar[d]^{\pi'} \\

	X \ar[r]^{f} & X'

} \]

commutes, then we have 

\[

	\pi'^*\circ f = \tilde{f}\circ \pi^*.

\]

\item

Product morphisms are compatible with gluing.

Let $\pi:E\to X$, $\pi_1:E_1\to X_1$, and $\pi_2:E_2\to X_2$ 

be pinched products with $E = E_1\cup E_2$.

Let $a\in \cC(X)$, and let $a_i$ denote the restriction of $a$ to $X_i\subset X$.

Then 

\[

	\pi^*(a) = \pi_1^*(a_1)\bullet \pi_2^*(a_2) .

\]

\item

Product morphisms are associative.

If $\pi:E\to X$ and $\rho:D\to E$ are pinched products then

\[

	\rho^*\circ\pi^* = (\pi\circ\rho)^* .

\]

\item

Product morphisms are compatible with restriction.

If we have a commutative diagram

\[ \xymatrix{

	D \ar@{^(->}[r] \ar[d]_{\rho} & E \ar[d]^{\pi} \\

	Y \ar@{^(->}[r] & X

} \]

such that $\rho$ and $\pi$ are pinched products, then

\[

	\res_D\circ\pi^* = \rho^*\circ\res_Y .

\]

\end{enumerate}

} %%% end \noop %%%

\end{axiom}

To state the next axiom we need the notion of {\it collar maps} on $k$-morphisms.

Let $X$ be a $k$-ball and $Y\subset\bd X$ be a $(k{-}1)$-ball.

Let $J$ be a 1-ball.

Let $Y\times_p J$ denote $Y\times J$ pinched along $(\bd Y)\times J$.

A collar map is an instance of the composition

\[

	\cC(X) \to \cC(X\cup_Y (Y\times_p J)) \to \cC(X) ,

\]

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{[linear  version]}} 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 $\cC(X)$.

\end{axiom}

\smallskip

For $A_\infty$ $n$-categories, we replace

isotopy invariance with the requirement that families of homeomorphisms act.

For the moment, assume that our $n$-morphisms are enriched over chain complexes.

Let $\Homeo_\bd(X)$ denote homeomorphisms of $X$ which fix $\bd X$ and

$C_*(\Homeo_\bd(X))$ denote the singular chains on this space.

\begin{axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$.]

\label{axiom:families}

For each $n$-ball $X$ and each $c\in \cl{\cC}(\bd X)$ we have a map of chain complexes

\[

	C_*(\Homeo_\bd(X))\tensor \cC(X; c) \to \cC(X; c) .

\]

These action maps are required to be associative up to homotopy,

and also compatible with composition (gluing) in the sense that

a diagram like the one in Theorem \ref{thm:CH} commutes.

\end{axiom}

\subsection{Example (the fundamental $n$-groupoid)}

We will define $\pi_{\le n}(T)$, the fundamental $n$-groupoid of a topological space $T$.

When $X$ is a $k$-ball with $k<n$, define $\pi_{\le n}(T)(X)$

to be the set of continuous maps from $X$ to $T$.

When $X$ is an $n$-ball, define $\pi_{\le n}(T)(X)$ to be homotopy classes (rel boundary) of such maps.

Define boundary restrictions and gluing in the obvious way.

If $\rho:E\to X$ is a pinched product and $f:X\to T$ is a $k$-morphism,

define the product morphism $\rho^*(f)$ to be $f\circ\rho$.

We can also define an $A_\infty$ version $\pi_{\le n}^\infty(T)$ of the fundamental $n$-groupoid.

For $X$ an $n$-ball define $\pi_{\le n}^\infty(T)(X)$ to be the space of all maps from $X$ to $T$

(if we are enriching over spaces) or the singular chains on that space (if we are enriching over chain complexes).

\subsection{Example (string diagrams)}

Fix a `traditional' $n$-category $C$ with strong duality (e.g.\ a pivotal 2-category).

Let $X$ be a $k$-ball and define $\cS_C(X)$ to be the set of $C$ string diagrams drawn on $X$;

that is, certain cell complexes embedded in $X$, with the codimension-$j$ cells labeled by $j$-morphisms of $C$.

If $X$ is an $n$-ball, identify two such string diagrams if they evaluate to the same $n$-morphism of $C$.

Boundary restrictions and gluing are again straightforward to define.

Define product morphisms via product cell decompositions.

\subsection{Example (bordism)}

When $X$ is a $k$-ball with $k<n$, $\Bord^n(X)$ is the set of all $k$-dimensional

submanifolds $W$ in $X\times \bbR^\infty$ which project to $X$ transversely

to $\bd X$.

For an $n$-ball $X$ define $\Bord^n(X)$ to be homeomorphism classes rel boundary of such $n$-dimensional submanifolds.

There is an $A_\infty$ analogue enriched in topological spaces, where at the top level we take all such submanifolds, rather than homeomorphism classes. For each fixed $\bdy W \subset \bdy X \times \bbR^\infty$, we can topologize the set of submanifolds by ambient isotopy rel boundary.