%!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{$n$-categories and their modules}

\label{sec:ncats}

\subsection{Definition of $n$-categories}

\label{ss:n-cat-def}

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

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

(As is the case throughout this paper, by ``$n$-category" we implicitly intend 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.

For examples of topological origin, it is typically easy to show that they

satisfy our axioms.

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

results that we have avoided here.

\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 bihedrons (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 expect to 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 usually omit the subscript $k$.)

We are so far  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.)

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

We will concentrate on the case of PL unoriented manifolds.

(The 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 to describe the blob complex of a fiber bundle with

base space $Y$.

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

bundle (i.e.\ framed balls), or more generally some 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 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 to not 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 the subsection for $0$ through $k-1$ we can use a colimit

construction, as described in Subsection \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 \todo{} 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

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

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 $n$;

$\cC(Y; c)$ is just a plain set if $\dim(Y) < n$.

\medskip

\nn{

%At the moment I'm a little confused about orientations, and more specifically

%about the role of orientation-reversing maps of boundaries when gluing oriented manifolds.

Maybe need a discussion about what the boundary of a manifold with a 

structure (e.g. orientation) means.

Tentatively, I think we need to redefine the oriented boundary of an oriented $n$-manifold.

Instead of an ordinary oriented $(n-1)$-manifold via the inward (or outward) normal 

first (or last) convention, perhaps it is better to define the boundary to be an $(n-1)$-manifold

equipped with an orientation of its once-stabilized tangent bundle.

Similarly, in dimension $n-k$ we would have manifolds equipped with an orientation of 

their $k$ times stabilized tangent bundles.

(cf. \cite{MR2079378}.)

Probably should also have a framing of the stabilized dimensions in order to indicate which 

side the bounded manifold is on.

For the moment just stick with unoriented manifolds.}

\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 follows 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]

$$

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

]

\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);

\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.

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

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 $k < n$ we require that $\gl_Y$ is injective.

(For $k=n$, see below.)

\end{axiom}

\begin{figure}[!ht]

$$

\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.

\end{axiom}

\begin{figure}[!ht]

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

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

We'll use the notations  $a\bullet b$ as well as $a \cup b$ 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 subdivision of a ball $X$ 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 subdivision is notationally anonymous, we will write

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

the unnamed subdivision.

If $\beta$ is a subdivision 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

subdivision of $\bd X$ and no competing subdivision 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 decomposition $B = B_1\cup\cdots\cup B_m$ 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{blah7}).}

\begin{figure}[!ht]

$$\mathfig{.8}{tempkw/blah7}$$

\caption{Operad composition and associativity}\label{blah7}\end{figure}

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

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

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 maps 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}

\nn{need even more subaxioms for product morphisms?}

\nn{Almost certainly we need a little more than the above axiom.

More specifically, in order to bootstrap our way from the top dimension

properties of identity morphisms to low dimensions, we need regular products,

pinched products and even half-pinched products.

I'm not sure what the best way to cleanly axiomatize the properties of these various

products is.

For the moment, I'll assume that all flavors of the product are at

our disposal, and I'll plan on revising the axioms later.}

\nn{current idea for fixing this: make the above axiom a ``preliminary version"

(as we have already done with some of the other axioms), then state the official

axiom for maps $\pi: E \to X$ which are almost fiber bundles.

one option is to restrict E to be a (full/half/not)-pinched product (up to homeo).

the alternative is to give some sort of local criterion for what's allowed.

state a gluing axiom for decomps $E = E'\cup E''$ where all three are of the correct type.

}

All of the axioms listed above hold for both ordinary $n$-categories and $A_\infty$ $n$-categories.

The last axiom (below), concerning actions of 

homeomorphisms in the top dimension $n$, distinguishes the two cases.

We start with the plain $n$-category case.

\begin{axiom}[Isotopy invariance in dimension $n$]{\textup{\textbf{[preliminary]}}}

Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts

to the identity on $\bd X$ and is isotopic (rel boundary) to the identity.

Then $f$ acts trivially on $\cC(X)$; $f(a) = a$ for all $a\in \cC(X)$.

\end{axiom}

This axiom needs to be strengthened to force product morphisms to act as the identity.

Let $X$ be an $n$-ball and $Y\sub\bd X$ be an $n{-}1$-ball.

Let $J$ be a 1-ball (interval).

We have a collaring homeomorphism $s_{Y,J}: X\cup_Y (Y\times J) \to X$.

(Here we use the ``pinched" version of $Y\times J$.

\nn{need notation for this})

We define a map

\begin{eqnarray*}

	\psi_{Y,J}: \cC(X) &\to& \cC(X) \\

	a & \mapsto & s_{Y,J}(a \cup ((a|_Y)\times J)) .

\end{eqnarray*}

(See Figure \ref{glue-collar}.)

\begin{figure}[!ht]

\begin{equation*}

\begin{tikzpicture}

\def\rad{1}

\def\srad{0.75}

\def\gap{4.5}

\foreach \i in {0, 1, 2} {

	\node(\i) at ($\i*(\gap,0)$) [draw, circle through = {($\i*(\gap,0)+(\rad,0)$)}] {};

	\node(\i-small) at (\i.east) [circle through={($(\i.east)+(\srad,0)$)}] {};

	\foreach \n in {1,2} {

		\fill (intersection \n of \i-small and \i) node(\i-intersection-\n) {} circle (2pt);

	}

}

\begin{scope}[decoration={brace,amplitude=10,aspect=0.5}]

	\draw[decorate] (0-intersection-1.east) -- (0-intersection-2.east);

\end{scope}

\node[right=1mm] at (0.east) {$a$};

\draw[->] ($(0.east)+(0.75,0)$) -- ($(1.west)+(-0.2,0)$);

\draw (1-small)  circle (\srad);

\foreach \theta in {90, 72, ..., -90} {

	\draw[blue] (1) -- ($(1)+(\rad,0)+(\theta:\srad)$);

}

\filldraw[fill=white] (1) circle (\rad);

\foreach \n in {1,2} {

	\fill (intersection \n of 1-small and 1) circle (2pt);

}

\node[below] at (1-small.south) {$a \times J$};

\draw[->] ($(1.east)+(1,0)$) -- ($(2.west)+(-0.2,0)$);

\begin{scope}

\path[clip] (2) circle (\rad);

\draw[clip] (2.east) circle (\srad);

\foreach \y in {1, 0.86, ..., -1} {

	\draw[blue] ($(2)+(-1,\y) $)-- ($(2)+(1,\y)$);

}

\end{scope}

\end{tikzpicture}

\end{equation*}

\begin{equation*}

\xymatrix@C+2cm{\cC(X) \ar[r]^(0.45){\text{glue}} & \cC(X \cup \text{collar}) \ar[r]^(0.55){\text{homeo}} & \cC(X)}

\end{equation*}

\caption{Extended homeomorphism.}\label{glue-collar}\end{figure}

We say that $\psi_{Y,J}$ is {\it extended isotopic} to the identity map.

\nn{bad terminology; fix it later}

\nn{also need to make clear that plain old isotopic to the identity implies

extended isotopic}

\nn{maybe remark that in some examples (e.g.\ ones based on sub cell complexes) 

extended isotopies are also plain isotopies, so

no extension necessary}

It can be thought of as the action of the inverse of

a map which projects a collar neighborhood of $Y$ onto $Y$.

The revised axiom is

\addtocounter{axiom}{-1}

\begin{axiom}{\textup{\textbf{[topological  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 is extended isotopic (rel boundary) to the identity.

Then $f$ acts trivially on $\cC(X)$.

\end{axiom}

\nn{need to rephrase this, since extended isotopies don't correspond to homeomorphisms.}

\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.

\addtocounter{axiom}{-1}

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

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) .

\]

Here $C_*$ means singular chains and $\Homeo_\bd(X)$ is the space of homeomorphisms of $X$

which fix $\bd X$.

These action maps are required to be associative up to homotopy

\nn{iterated homotopy?}, and also compatible with composition (gluing) in the sense that

a diagram like the one in Proposition \ref{CHprop} commutes.

\nn{repeat diagram here?}

\nn{restate this with $\Homeo(X\to X')$?  what about boundary fixing property?}

\end{axiom}

We should strengthen the above axiom to apply to families of extended homeomorphisms.

To do this we need to explain how extended homeomorphisms form a topological space.

Roughly, the set of $n{-}1$-balls in the boundary of an $n$-ball has a natural topology,

and we can replace the class of all intervals $J$ with intervals contained in $\r$.

\nn{need to also say something about collaring homeomorphisms.}

\nn{this paragraph needs work.}

Note that if we take homology of chain complexes, we turn an $A_\infty$ $n$-category

into a plain $n$-category (enriched over graded groups).

\nn{say more here?}

In a different direction, if we enrich over topological spaces instead of chain complexes,

we get a space version of an $A_\infty$ $n$-category, with $\Homeo_\bd(X)$ acting 

instead of  $C_*(\Homeo_\bd(X))$.

Taking singular chains converts such a space type $A_\infty$ $n$-category into a chain complex

type $A_\infty$ $n$-category.

\medskip

The alert reader will have already noticed that our definition of a (plain) $n$-category

is extremely similar to our definition of a topological system of fields.

There are two essential differences.

First, for the $n$-category definition we restrict our attention to balls