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%% For titles, only capitalize the first letter

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

\title{$n$-categories, colimits and the blob complex}

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%% Javier de Ruiz Garcia\affil{2}{Universidad de Murcia, Bioquimica y Biologia

%% 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} -- enter abstract text here -- \end{abstract}

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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}opological quantum field theories (TQFTs) provide local invariants of manifolds, which are determined by the algebraic data of a higher category.

An $n+1$-dimensional TQFT $\cA$ associates a vector space $\cA(M)$

(or more generally, some object in a specified symmetric monoidal category)

to each $n$-dimensional manifold $M$, and a linear map

$\cA(W): \cA(M_0) \to \cA(M_1)$ to each $n+1$-dimensional manifold $W$

with incoming boundary $M_0$ and outgoing boundary $M_1$.

An $n+\epsilon$-dimensional TQFT provides slightly less;

it only assigns linear maps to mapping cylinders.

There is a standard formalism for constructing an $n+\epsilon$-dimensional

TQFT from any $n$-category with sufficiently strong duality,

and with a further finiteness condition this TQFT is in fact $n+1$-dimensional.

\nn{not so standard, err}

These invariants are local in the following sense.

The vector space $\cA(Y \times I)$, for $Y$ an $n-1$-manifold,

naturally has the structure of a category, with composition given by the gluing map

$I \sqcup I \to I$. Moreover, the vector space $\cA(Y \times I^k)$,

for $Y$ and $n-k$-manifold, has the structure of a $k$-category.

The original $n$-category can be recovered as $\cA(I^n)$.

For the rest of the paragraph, we implicitly drop the factors of $I$.

(So for example the original $n$-category is associated to the point.)

If $Y$ contains $Z$ as a codimension $0$ submanifold of its boundary,

then $\cA(Y)$ is natually a module over $\cA(Z)$. For any $k$-manifold

$Y = Y_1 \cup_Z Y_2$, where $Z$ is a $k-1$-manifold, the category

$\cA(Y)$ can be calculated via a gluing formula,

$$\cA(Y) = \cA(Y_1) \Tensor_{\cA(Z)} \cA(Y_2).$$

In fact, recent work of Lurie on the `cobordism hypothesis' \cite{0905.0465}

shows that all invariants of $n$-manifolds satisfying a certain related locality property

are in a sense TQFT invariants, and in particular determined by

a `fully dualizable object' in some $n+1$-category.

(The discussion above begins with an object in the $n+1$-category of $n$-categories.

The `sufficiently strong duality' mentioned above corresponds roughly to `fully dualizable'.)

This formalism successfully captures Turaev-Viro and Reshetikhin-Turaev invariants

(and indeed invariants based on semisimple categories).

However new invariants on manifolds, particularly those coming from

Seiberg-Witten theory and Ozsv\'{a}th-Szab\'{o} theory, do not fit the framework well.

In particular, they have more complicated gluing formulas, involving derived or

$A_\infty$ tensor products \cite{1003.0598,1005.1248}.

It seems worthwhile to find a more general notion of TQFT that explain these.

While we don't claim to fulfill that goal here, our notions of $n$-category and

of the blob complex are hopefully a step in the right direction,

and provide similar gluing formulas.

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

TQFT invariant via its gluing formulas, replacing tensor products with

derived tensor products. 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 gluing formulas based on $A_\infty$ tensor products.

\nn{Triangulated categories are important; often calculations are via exact sequences,

and the standard TQFT constructions are quotients, which destroy exactness.}

\section{Definitions}

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

\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

\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 \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}

We haven't said precisely what sort of balls we are considering,

because we prefer to let this detail be a parameter in the definition.

It is useful to consider unoriented, oriented, Spin and $\mbox{Pin}_\pm$ balls.

Also useful are more exotic structures, such as balls equipped 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.)

%In fact, the axioms here may easily be varied by considering balls with structure (e.g. $m$ independent vector fields, a map to some target space, etc.). Such variations are useful for axiomatizing categories with less duality, and also as technical tools in proofs.

\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 \todo{} 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 plain (non-$A_\infty$) case, see below.)

\end{axiom}

\begin{axiom}[Strict associativity] \label{nca-assoc}

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 $X_i\sub X$.

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 $Y\sub\bd X$ 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{[plain  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))\ot \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{The blob complex}

\subsubsection{Decompositions of manifolds}

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.

If $X_a$ is some component of $M_0$, its image in $W$ need not be a ball; $\bd X_a$ may have been glued to itself.

A {\it permissible decomposition} of $W$ is a map

\[

	\coprod_a X_a \to W,

\]

which can be completed to a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$.

A permissible decomposition is weaker than a ball decomposition; we forget the order in which the balls

are glued up to yield $W$, and just require that there is some non-pathological way to do this.

Given permissible decompositions $x = \{X_a\}$ and $y = \{Y_b\}$ of $W$, we say that $x$ is a refinement

of $y$, or write $x \le y$, if there is a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$

with $\du_b Y_b = M_i$ for some $i$.

\begin{defn}

The poset $\cell(W)$ has objects the permissible decompositions of $W$, 

and a unique morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.

See Figure \ref{partofJfig} for an example.

\end{defn}

This poset in fact has more structure, since we can glue together permissible decompositions of $W_1$ and $W_2$ to obtain a permissible decomposition of $W_1 \sqcup W_2$. 

An $n$-category $\cC$ determines 

a functor $\psi_{\cC;W}$ from $\cell(W)$ to the category of sets 

(possibly with additional structure if $k=n$).

Each $k$-ball $X$ of a decomposition $y$ of $W$ has its boundary decomposed into $k{-}1$-balls,

and there is a subset $\cC(X)\spl \sub \cC(X)$ of morphisms whose boundaries