%!TEX root = ../blob1.tex

\def\xxpar#1#2{\smallskip\noindent{\bf #1} {\it #2} \smallskip}

\def\mmpar#1#2#3{\smallskip\noindent{\bf #1} (#2). {\it #3} \smallskip}

\section{\texorpdfstring{$n$}{n}-categories and their modules}

\label{sec:ncats}

\subsection{Definition of \texorpdfstring{$n$}{n}-categories}

\label{ss:n-cat-def}

Before proceeding, we need more appropriate definitions of $n$-categories, 

$A_\infty$ $n$-categories, as well as modules for these, and tensor products of these modules.

(As is the case throughout this paper, by ``$n$-category" we mean some notion of

a ``weak" $n$-category with ``strong duality".)

The definitions presented below tie the categories more closely to the topology

and avoid combinatorial questions about, for example, the minimal sufficient

collections of generalized associativity axioms; we prefer maximal sets of axioms to minimal sets.

It is easy to show that examples of topological origin

(e.g.\ categories whose morphisms are maps into spaces or decorated balls), 

satisfy our axioms.

For examples of a more purely algebraic origin, one would typically need the combinatorial

results that we have avoided here.

%\nn{Say something explicit about Lurie's work here? 

%It seems like this was something that Dan Freed wanted explaining when we talked to him in Aspen}

\medskip

There are many existing definitions of $n$-categories, with various intended uses.

In any such definition, there are sets of $k$-morphisms for each $0 \leq k \leq n$.

Generally, these sets are indexed by instances of a certain typical shape. 

Some $n$-category definitions model $k$-morphisms on the standard bihedron (interval, bigon, and so on).

Other definitions have a separate set of 1-morphisms for each interval $[0,l] \sub \r$, 

a separate set of 2-morphisms for each rectangle $[0,l_1]\times [0,l_2] \sub \r^2$,

and so on.

(This allows for strict associativity.)

Still other definitions (see, for example, \cite{MR2094071})

model the $k$-morphisms on more complicated combinatorial polyhedra.

For our definition, we will allow our $k$-morphisms to have any shape, so long as it is homeomorphic to the standard $k$-ball.

Thus we associate a set of $k$-morphisms $\cC_k(X)$ to any $k$-manifold $X$ homeomorphic 

to the standard $k$-ball.

By ``a $k$-ball" we mean any $k$-manifold which is homeomorphic to the 

standard $k$-ball.

We {\it do not} assume that it is equipped with a 

preferred homeomorphism to the standard $k$-ball, and the same applies to ``a $k$-sphere" below.

Given a homeomorphism $f:X\to Y$ between $k$-balls (not necessarily fixed on 

the boundary), we want a corresponding

bijection of sets $f:\cC(X)\to \cC(Y)$.

(This will imply ``strong duality", among other things.) Putting these together, we have

\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: We often omit the subscript $k$.)

We are being deliberately vague about what flavor of $k$-balls

we are considering.

They could be unoriented or oriented or Spin or $\mbox{Pin}_\pm$.

They could be topological or PL or smooth.

%\nn{need to check whether this makes much difference}

(If smooth, ``homeomorphism" should be read ``diffeomorphism", and we would need

to be fussier about corners and boundaries.)

For each flavor of manifold there is a corresponding flavor of $n$-category.

For simplicity, we will concentrate on the case of PL unoriented manifolds.

An ambitious reader may want to keep in mind two other classes of balls.

The first is balls equipped with a map to some other space $Y$ (c.f. \cite{MR2079378}). 

This will be used below (see the end of \S \ref{ss:product-formula}) to describe the blob complex of a fiber bundle with

base space $Y$.

The second is balls equipped with a section of the tangent bundle, or the frame

bundle (i.e.\ framed balls), or more generally some partial flag bundle associated to the tangent bundle.

These can be used to define categories with less than the ``strong" duality we assume here,

though we will not develop that idea fully in this paper.

Next we consider domains and ranges of morphisms (or, as we prefer to say, boundaries

of morphisms).

The 0-sphere is unusual among spheres in that it is disconnected.

Correspondingly, for 1-morphisms it makes sense to distinguish between domain and range.

(Actually, this is only true in the oriented case, with 1-morphisms parameterized

by {\it oriented} 1-balls.)

For $k>1$ and in the presence of strong duality the division into domain and range makes less sense.

For example, in a pivotal tensor category, there are natural isomorphisms $\Hom{}{A}{B \tensor C} \isoto \Hom{}{B^* \tensor A}{C}$, etc. 

(sometimes called ``Frobenius reciprocity''), which canonically identify all the morphism spaces which have the same boundary.

We prefer not to make the distinction in the first place.

Instead, we will combine the domain and range into a single entity which we call the 

boundary of a morphism.

Morphisms are modeled on balls, so their boundaries are modeled on spheres.

In other words, we need to extend the functors $\cC_{k-1}$ from balls to spheres, for 

$1\le k \le n$.

At first it might seem that we need another axiom for this, but in fact once we have

all the axioms in this subsection for $0$ through $k-1$ we can use a colimit

construction, as described in \S\ref{ss:ncat-coend} below, to extend $\cC_{k-1}$

to spheres (and any other manifolds):

\begin{lem}

\label{lem:spheres}

For each $1 \le k \le n$, we have a functor $\cl{\cC}_{k-1}$ from 

the category of $k{-}1$-spheres and 

homeomorphisms to the category of sets and bijections.

\end{lem}

We postpone the proof of this result until after we've actually given all the axioms.

Note that defining this functor for some $k$ only requires the data described in Axiom \ref{axiom:morphisms} at level $k$, 

along with the data described in the other axioms at lower levels. 

%In fact, the functors for spheres are entirely determined by the functors for balls and the subsequent axioms. (In particular, $\cC(S^k)$ is the colimit of $\cC$ applied to decompositions of $S^k$ into balls.) However, it is easiest to think of it as additional data at this point.

\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}

Note that the first ``$\bd$" above is part of the data for the category, 

while the second is the ordinary boundary of manifolds.

Given $c\in\cl{\cC}(\bd(X))$, we will write $\cC(X; c)$ for $\bd^{-1}(c)$, those morphisms with specified boundary $c$.

Most of the examples of $n$-categories we are interested in are enriched in the following sense.

The various sets of $n$-morphisms $\cC(X; c)$, for all $n$-balls $X$ and

all $c\in \cl{\cC}(\bd X)$, have the structure of an object in some auxiliary symmetric monoidal category

with sufficient limits and colimits

(e.g.\ vector spaces, or modules over some ring, or chain complexes),

%\nn{actually, need both disj-union/sum and product/tensor-product; what's the name for this sort of cat?}

and all the structure maps of the $n$-category should be compatible with the auxiliary

category structure.

Note that this auxiliary structure is only in dimension $n$; if $\dim(Y) < n$ then 

$\cC(Y; c)$ is just a plain set.

\medskip

In order to simplify the exposition we have concentrated on the case of 

unoriented PL manifolds and avoided the question of what exactly we mean by 

the boundary of a manifold with extra structure, such as an oriented manifold.

In general, all manifolds of dimension less than $n$ should be equipped with the germ

of a thickening to dimension $n$, and this germ should carry whatever structure we have 

on $n$-manifolds.

In addition, lower dimensional manifolds should be equipped with a framing

of their normal bundle in the thickening; the framing keeps track of which

side (iterated) bounded manifolds lie on.

For example, the boundary of an oriented $n$-ball

should be an $n{-}1$-sphere equipped with an orientation of its once stabilized tangent

bundle and a choice of direction in this bundle indicating

which side the $n$-ball lies on.

\medskip

We have just argued that the boundary of a morphism has no preferred splitting into

domain and range, but the converse meets with our approval.

That is, given compatible domain and range, we should be able to combine them into

the full boundary of a morphism.

The following lemma will follow from the colimit construction used to define $\cl{\cC}_{k-1}$

on spheres.

\begin{lem}[Boundary from domain and range]

\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}

\begin{figure}[!ht] \centering

\begin{tikzpicture}[%every label/.style={green}

]

\node[fill=black, circle, label=below:$E$, inner sep=1.5pt](S) at (0,0) {};

\node[fill=black, circle, label=above:$E$, inner sep=1.5pt](N) at (0,2) {};

\draw (S) arc  (-90:90:1);

\draw (N) arc  (90:270:1);

\node[left] at (-1,1) {$B_1$};

\node[right] at (1,1) {$B_2$};

\end{tikzpicture}

\caption{Combining two balls to get a full boundary.}\label{blah3}\end{figure}

Note that we insist on injectivity above. 

The lemma follows from Definition \ref{def:colim-fields} and Lemma \ref{lem:colim-injective}.

%\nn{we might want a more official looking proof...}

Let $\cl{\cC}(S)_E$ denote the image of $\gl_E$.

We will refer to elements of $\cl{\cC}(S)_E$ as ``splittable along $E$" or ``transverse to $E$". 

If $X$ is a $k$-ball and $E \sub \bd X$ splits $\bd X$ into two $k{-}1$-balls $B_1$ and $B_2$

as above, then we define $\cC(X)_E = \bd^{-1}(\cl{\cC}(\bd X)_E)$.

We will call the projection $\cl{\cC}(S)_E \to \cC(B_i)$

a {\it restriction} map and write $\res_{B_i}(a)$

(or simply $\res(a)$ when there is no ambiguity), for $a\in \cl{\cC}(S)_E$.

More generally, we also include under the rubric ``restriction map"

the boundary maps of Axiom \ref{nca-boundary} above,

another class of maps introduced after Axiom \ref{nca-assoc} below, as well as any composition

of restriction maps.

In particular, we have restriction maps $\cC(X)_E \to \cC(B_i)$

($i = 1, 2$, notation from previous paragraph).

These restriction maps can be thought of as 

domain and range maps, relative to the choice of splitting $\bd X = B_1 \cup_E B_2$.

Next we consider composition of morphisms.

For $n$-categories which lack strong duality, one usually considers

$k$ different types of composition of $k$-morphisms, each associated to a different direction.

(For example, vertical and horizontal composition of 2-morphisms.)

In the presence of strong duality, these $k$ distinct compositions are subsumed into 

one general type of composition which can be in any ``direction".

\begin{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 (domain or range) 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{figure}[!ht] \centering

\begin{tikzpicture}[%every label/.style={green},

				x=1.5cm,y=1.5cm]

\node[fill=black, circle, label=below:$E$, inner sep=2pt](S) at (0,0) {};

\node[fill=black, circle, label=above:$E$, inner sep=2pt](N) at (0,2) {};

\draw (S) arc  (-90:90:1);

\draw (N) arc  (90:270:1);

\draw (N) -- (S);

\node[left] at (-1/4,1) {$B_1$};

\node[right] at (1/4,1) {$B_2$};

\node at (1/6,3/2)  {$Y$};

\end{tikzpicture}

\caption{From two balls to one ball.}\label{blah5}\end{figure}

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

The composition (gluing) maps above are strictly associative.

Given any splitting of a ball $B$ into smaller balls

$$\bigsqcup B_i \to B,$$ 

any sequence of gluings (in the sense of Definition \ref{defn:gluing-decomposition}, where all the intermediate steps are also disjoint unions of balls) yields the same result.

\end{axiom}

\begin{figure}[!ht]

$$\mathfig{.65}{ncat/strict-associativity}$$

\caption{An example of strict associativity.}\label{blah6}\end{figure}

We'll use the notation  $a\bullet b$ for the glued together field $\gl_Y(a, b)$.

In the other direction, we will call the projection from $\cC(B)_E$ to $\cC(B_i)_E$ 

a restriction map (one of many types of map so called) and write $\res_{B_i}(a)$ for $a\in \cC(B)_E$.

%Compositions of boundary and restriction maps will also be called restriction maps.

%For example, if $B$ is a $k$-ball and $Y\sub \bd B$ is a $k{-}1$-ball, there is a

%restriction map from $\cC(B)_{\bd Y}$ to $\cC(Y)$.

We will write $\cC(B)_Y$ for the image of $\gl_Y$ in $\cC(B)$.

We will call elements of $\cC(B)_Y$ morphisms which are 

``splittable along $Y$'' or ``transverse to $Y$''.

We have $\cC(B)_Y \sub \cC(B)_E \sub \cC(B)$.

More generally, let $\alpha$ be a splitting of $X$ into smaller balls.

Let $\cC(X)_\alpha \sub \cC(X)$ denote the image of the iterated gluing maps from 

the smaller balls to $X$.

We  say that elements of $\cC(X)_\alpha$ are morphisms which are ``splittable along $\alpha$".

In situations where the splitting is notationally anonymous, we will write

$\cC(X)\spl$ for the morphisms which are splittable along (a.k.a.\ transverse to)

the unnamed splitting.

If $\beta$ is a ball decomposition of $\bd X$, we define $\cC(X)_\beta \deq \bd\inv(\cl{\cC}(\bd X)_\beta)$;

this can also be denoted $\cC(X)\spl$ if the context contains an anonymous

decomposition of $\bd X$ and no competing splitting of $X$.

The above two composition axioms are equivalent to the following one,

which we state in slightly vague form.

\xxpar{Multi-composition:}

{Given any splitting $B_1 \sqcup \cdots \sqcup B_m \to B$ of a $k$-ball

into small $k$-balls, there is a 

map from an appropriate subset (like a fibered product) 

of $\cC(B_1)\spl\times\cdots\times\cC(B_m)\spl$ to $\cC(B)\spl$,

and these various $m$-fold composition maps satisfy an

operad-type strict associativity condition (Figure \ref{fig:operad-composition}).}

\begin{figure}[!ht]

$$\mathfig{.8}{ncat/operad-composition}$$

\caption{Operad composition and associativity}\label{fig:operad-composition}\end{figure}

The next axiom is related to identity morphisms, though that might not be immediately obvious.

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

For each $k$-ball $X$ and $m$-ball $D$, with $k+m \le n$, there is a map $\cC(X)\to \cC(X\times D)$, 

usually denoted $a\mapsto a\times D$ for $a\in \cC(X)$.

These maps must satisfy the following conditions.

\begin{enumerate}

\item

If $f:X\to X'$ and $\tilde{f}:X\times D \to X'\times D'$ are homeomorphisms such that the diagram

\[ \xymatrix{

	X\times D \ar[r]^{\tilde{f}} \ar[d]_{\pi} & X'\times D' \ar[d]^{\pi} \\

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

} \]

commutes, then we have 

\[

	\tilde{f}(a\times D) = f(a)\times D' .

\]

\item

Product morphisms are compatible with gluing (composition) in both factors:

\[

	(a'\times D)\bullet(a''\times D) = (a'\bullet a'')\times D

\]

and

\[

	(a\times D')\bullet(a\times D'') = a\times (D'\bullet D'') .

\]

\item

Product morphisms are associative:

\[

	(a\times D)\times D' = a\times (D\times D') .

\]

(Here we are implicitly using functoriality and the obvious homeomorphism

$(X\times D)\times D' \to X\times(D\times D')$.)

\item

Product morphisms are compatible with restriction:

\[

	\res_{X\times E}(a\times D) = a\times E

\]

for $E\sub \bd D$ and $a\in \cC(X)$.

\end{enumerate}

\end{axiom}

We will need to strengthen the above preliminary version of the axiom to allow

for products which are ``pinched" in various ways along their boundary.

(See Figure \ref{pinched_prods}.)

\begin{figure}[t]

$$

\begin{tikzpicture}[baseline=0]

\begin{scope}

\path[clip] (0,0) arc (135:45:4) arc (-45:-135:4);

\draw[blue,line width=2pt] (0,0) arc (135:45:4) arc (-45:-135:4);

\foreach \x in {0, 0.5, ..., 6} {

	\draw[green!50!brown] (\x,-2) -- (\x,2);

}

\end{scope}

\draw[blue,line width=1.5pt] (0,-3) -- (5.66,-3);

\draw[->,red,line width=2pt] (2.83,-1.5) -- (2.83,-2.5);

\end{tikzpicture}

\qquad \qquad

\begin{tikzpicture}[baseline=-0.15cm]

\begin{scope}

\path[clip] (0,1) arc (90:135:8 and 4)  arc (-135:-90:8 and 4) -- cycle;

\draw[blue,line width=2pt] (0,1) arc (90:135:8 and 4)  arc (-135:-90:8 and 4) -- cycle;

\foreach \x in {-6, -5.5, ..., 0} {

	\draw[green!50!brown] (\x,-2) -- (\x,2);

}

\end{scope}

\draw[blue,line width=1.5pt] (-5.66,-3.15) -- (0,-3.15);

\draw[->,red,line width=2pt] (-2.83,-1.5) -- (-2.83,-2.5);

\end{tikzpicture}

$$

\caption{Examples of pinched products}\label{pinched_prods}

\end{figure}

(The need for a strengthened version will become apparent in Appendix \ref{sec:comparing-defs}

where we construct a traditional category from a topological category.)

Define a {\it pinched product} to be a map

\[

	\pi: E\to X

\]

such that $E$ is a $k{+}m$-ball, $X$ is a $k$-ball ($m\ge 1$), and $\pi$ is locally modeled

on a standard iterated degeneracy map

\[

	d: \Delta^{k+m}\to\Delta^k .

\]

(We thank Kevin Costello for suggesting this approach.)

Note that for each interior point $x\in X$, $\pi\inv(x)$ is an $m$-ball,

and for each boundary point $x\in\bd X$, $\pi\inv(x)$ is a ball of dimension

$l \le m$, with $l$ depending on $x$.

It is easy to see that a composition of pinched products is again a pinched product.

A {\it sub pinched product} is a sub-$m$-ball $E'\sub E$ such that the restriction

$\pi:E'\to \pi(E')$ is again a pinched product.

A {union} of pinched products is a decomposition $E = \cup_i E_i$

such that each $E_i\sub E$ is a sub pinched product.

(See Figure \ref{pinched_prod_unions}.)

\begin{figure}[t]

$$

\begin{tikzpicture}[baseline=0]

\begin{scope}

\path[clip] (0,0) arc (135:45:4) arc (-45:-135:4);

\draw[blue,line width=2pt] (0,0) arc (135:45:4) arc (-45:-135:4);

\draw[blue] (0,0) -- (5.66,0);

\foreach \x in {0, 0.5, ..., 6} {

	\draw[green!50!brown] (\x,-2) -- (\x,2);

}

\end{scope}

\end{tikzpicture}

\qquad

\begin{tikzpicture}[baseline=0]

\begin{scope}

\path[clip] (0,-1) rectangle (4,1);

\draw[blue,line width=2pt] (0,-1) rectangle (4,1);

\draw[blue] (0,0) -- (5,0);

\foreach \x in {0, 0.5, ..., 6} {

	\draw[green!50!brown] (\x,-2) -- (\x,2);

}

\end{scope}

\end{tikzpicture}

\qquad

\begin{tikzpicture}[baseline=0]

\begin{scope}

\path[clip] (0,0) arc (135:45:4) arc (-45:-135:4);

\draw[blue,line width=2pt] (0,0) arc (135:45:4) arc (-45:-135:4);

\draw[blue] (2.83,3) circle (3);

\foreach \x in {0, 0.5, ..., 6} {

	\draw[green!50!brown] (\x,-2) -- (\x,2);

}

\end{scope}

\end{tikzpicture}

$$

$$

\begin{tikzpicture}[baseline=0]

\begin{scope}

\path[clip] (0,-1) rectangle (4,1);

\draw[blue,line width=2pt] (0,-1) rectangle (4,1);

\draw[blue] (0,-1) -- (4,1);

\foreach \x in {0, 0.5, ..., 6} {

	\draw[green!50!brown] (\x,-2) -- (\x,2);

}

\end{scope}

\end{tikzpicture}

\qquad

\begin{tikzpicture}[baseline=0]

\begin{scope}

\path[clip] (0,-1) rectangle (5,1);

\draw[blue,line width=2pt] (0,-1) rectangle (5,1);

\draw[blue] (1,-1) .. controls  (2,-1) and (3,1) .. (4,1);

\foreach \x in {0, 0.5, ..., 6} {

	\draw[green!50!brown] (\x,-2) -- (\x,2);

}

\end{scope}

\end{tikzpicture}

$$

\caption{Five examples of unions of pinched products}\label{pinched_prod_unions}

\end{figure}

The product axiom will give a map $\pi^*:\cC(X)\to \cC(E)$ for each pinched product

$\pi:E\to X$.

Morphisms in the image of $\pi^*$ will be called product morphisms.

Before stating the axiom, we illustrate it in our two motivating examples of $n$-categories.

In the case where $\cC(X) = \{f: X\to T\}$, we define $\pi^*(f) = f\circ\pi$.

In the case where $\cC(X)$ is the set of all labeled embedded cell complexes $K$ in $X$, 

define $\pi^*(K) = \pi\inv(K)$, with each codimension $i$ cell $\pi\inv(c)$ labeled by the

same (traditional) $i$-morphism as the corresponding codimension $i$ cell $c$.

\addtocounter{axiom}{-1}

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

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