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

\title{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}

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

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

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

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\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.

These have more complicated gluing formulas (\cite{1003.0598,1005.1248}, etc); 

it would be nice to give generalized TQFT axioms that encompass these.

Triangulated categories are important; often calculations are via exact sequences,

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

A first attempt to deal with this might be to replace all the tensor products in gluing formulas

with derived tensor products (cite Kh?).

However, in this approach it's probably difficult to prove invariance of constructions,

because they depend on explicit presentations of the manifold.

We'll give a manifestly invariant construction,

and deduce gluing formulas based on derived (actually, $A_\infty$) tensor products.}

\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

\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}

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}

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".

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}

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 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{axiom}

\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}

\todo{

Decide if we need a friendlier, skein-module version.

}

\subsubsection{Examples}

\todo{maps to a space, string diagrams}

\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.

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}

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

are splittable along this decomposition.

\begin{defn}

Define the functor $\psi_{\cC;W} : \cell(W) \to \Set$ as follows.

For a decomposition $x = \bigsqcup_a X_a$ in $\cell(W)$, $\psi_{\cC;W}(x)$ is the subset

\begin{equation*}

%\label{eq:psi-C}

	\psi_{\cC;W}(x) \sub \prod_a \cC(X_a)\spl

\end{equation*}

where the restrictions to the various pieces of shared boundaries amongst the cells

$X_a$ all agree (this is a fibered product of all the labels of $n$-cells over the labels of $n-1$-cells). 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 \nn{a point} of this functor,

that is, a permissible decomposition $x$ of $W$ together with an element of $\psi_{\cC;W}(x)$.

\todo{Mention that the axioms for $n$-categories can be stated in terms of decompositions of balls?}

\subsubsection{Homotopy colimits}

\nn{Motivation: How can we extend an $n$-category from balls to arbitrary manifolds?}

We now define the blob complex $\bc_*(W; \cC)$ of an $n$-manifold $W$

with coefficients in the $n$-category $\cC$ to be the homotopy colimit

of the functor $\psi_{\cC; W}$ described above. 

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.

We say a collection of balls $\{B_i\}$ in a manifold $W$ is \emph{permissible}

if there exists a permissible decomposition $M_0\to\cdots\to M_m = W$ such that

each $B_i$ appears as a connected component of one of the $M_j$. Note that this allows 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

\begin{itemize}

\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 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 signs given by the ordering.

\todo{Say why this really is the homotopy colimit}

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 first three properties are immediate from the definitions.

\begin{property}[Functoriality]

\label{property:functoriality}%

The blob complex is functorial with respect to homeomorphisms.

That is, 

for a fixed $n$-category $\cC$, the association

\begin{equation*}

X \mapsto \bc_*(X; \cC)

\end{equation*}

is a functor from $n$-manifolds and homeomorphisms between them to chain 

complexes and isomorphisms between them.

\end{property}

As a consequence, there is an action of $\Homeo(X)$ on the chain complex $\bc_*(X; \cC)$; 

this action is extended to all of $C_*(\Homeo(X))$ in Theorem \ref{thm:CH} below.

\begin{property}[Disjoint union]

\label{property:disjoint-union}

The blob complex of a disjoint union is naturally isomorphic to the tensor product of the blob complexes.

\begin{equation*}

\bc_*(X_1 \du X_2) \iso \bc_*(X_1) \tensor \bc_*(X_2)

\end{equation*}

\end{property}

If an $n$-manifold $X$ contains $Y \sqcup Y^\text{op}$ (we allow $Y = \eset$) as a codimension $0$ submanifold of its boundary, 

write $X \bigcup_{Y}\selfarrow$ for the manifold obtained by gluing together $Y$ and $Y^\text{op}$.

\begin{property}[Gluing map]

\label{property:gluing-map}%

%If $X_1$ and $X_2$ are $n$-manifolds, with $Y$ a codimension $0$-submanifold of $\bdy X_1$, and $Y^{\text{op}}$ a codimension $0$-submanifold of $\bdy X_2$, there is a chain map

%\begin{equation*}

%\gl_Y: \bc_*(X_1) \tensor \bc_*(X_2) \to \bc_*(X_1 \cup_Y X_2).

%\end{equation*}

Given a gluing $X \to X_\mathrm{gl}$, there is

a map

\[

	\bc_*(X) \to \bc_*(X \bigcup_{Y}\selfarrow),

\]

natural with respect to homeomorphisms, and associative with respect to iterated gluings.

\end{property}

\begin{property}[Contractibility]

\label{property:contractibility}%