--- a/text/ncat.tex Wed Jun 22 16:02:27 2011 -0700
+++ b/text/ncat.tex Wed Jun 22 16:02:37 2011 -0700
@@ -33,11 +33,16 @@
\medskip
The axioms for an $n$-category are spread throughout this section.
-Collecting these together, an $n$-category is a gadget satisfying Axioms \ref{axiom:morphisms}, \ref{nca-boundary}, \ref{axiom:composition}, \ref{nca-assoc}, \ref{axiom:product} and \ref{axiom:extended-isotopies}; for an $A_\infty$ $n$-category, we replace Axiom \ref{axiom:extended-isotopies} with Axiom \ref{axiom:families}.
+Collecting these together, an $n$-category is a gadget satisfying Axioms \ref{axiom:morphisms},
+\ref{nca-boundary}, \ref{axiom:composition}, \ref{nca-assoc}, \ref{axiom:product} and
+\ref{axiom:extended-isotopies}; for an $A_\infty$ $n$-category, we replace
+Axiom \ref{axiom:extended-isotopies} with Axiom \ref{axiom:families}.
+\nn{need to revise this after we're done rearranging the a-inf and enriched stuff}
Strictly speaking, before we can state the axioms for $k$-morphisms we need all the axioms
for $k{-}1$-morphisms.
-Readers who prefer things to be presented in a strictly logical order should read this subsection $n+1$ times, first setting $k=0$, then $k=1$, and so on until they reach $k=n$.
+Readers who prefer things to be presented in a strictly logical order should read this
+subsection $n+1$ times, first setting $k=0$, then $k=1$, and so on until they reach $k=n$.
\medskip
@@ -52,7 +57,8 @@
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.
+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
@@ -141,17 +147,6 @@
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
@@ -239,10 +234,10 @@
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.
+$k$ 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".
+one general type of composition which can be in any direction.
\begin{axiom}[Composition]
\label{axiom:composition}
@@ -258,10 +253,9 @@
\]
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,
+If $k < n$
we require that $\gl_Y$ is injective.
-(For $k=n$ in the ordinary (non-$A_\infty$) case, see below.)
+%(For $k=n$ see below.)
\end{axiom}
\begin{figure}[t] \centering
@@ -401,7 +395,7 @@
\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 disk-like category.
+where we construct a traditional 2-category from a disk-like 2-category.
For example, ``half-pinched" products of 1-balls are used to construct weak identities for 1-morphisms
in 2-categories.
We also need fully-pinched products to define collar maps below (see Figure \ref{glue-collar}).
@@ -498,7 +492,11 @@
\caption{Six 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
+Note that $\bd X$ has a (possibly trivial) subdivision according to
+the dimension of $\pi\inv(x)$, $x\in \bd X$.
+Let $\cC(X)\trans{}$ denote the morphisms which are splittable along this subdivision.
+
+The product axiom will give a map $\pi^*:\cC(X)\trans{}\to \cC(E)$ 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.
@@ -512,7 +510,7 @@
\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 $\pi^*:\cC(X)\to \cC(E)$.
+there is a map $\pi^*:\cC(X)\trans{}\to \cC(E)$.
These maps must satisfy the following conditions.
\begin{enumerate}
\item
@@ -535,7 +533,7 @@
but $X_1 \cap X_2$ might not be codimension 1, and indeed we might have $X_1 = X_2 = X$.
We assume that there is a decomposition of $X$ into balls which is compatible with
$X_1$ and $X_2$.
-Let $a\in \cC(X)$, and let $a_i$ denote the restriction of $a$ to $X_i\sub X$.
+Let $a\in \cC(X)\trans{}$, and let $a_i$ denote the restriction of $a$ to $X_i\sub X$.
(We assume that $a$ is splittable with respect to the above decomposition of $X$ into balls.)
Then
\[
@@ -564,16 +562,27 @@
\medskip
-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 ordinary $n$-category case.
+%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 ordinary $n$-category case.
+
+The next axiom says, roughly, that we have strict associativity in dimension $n$,
+even when we reparametrize our $n$-balls.
\begin{axiom}[\textup{\textbf{[preliminary]}} Isotopy invariance in dimension $n$]
-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 $\cC(X)$; that is $f(a) = a$ for all $a\in \cC(X)$.
+Let $X$ be an $n$-ball, $b \in \cC(X)$, and $f: X\to X$ be a homeomorphism which
+acts trivially on the restriction $\bd b$ of $b$ to $\bd X$.
+(Keep in mind the important special case where $f$ restricted to $\bd X$ is the identity.)
+Suppose furthermore that $f$ is isotopic to the identity through homeomorphisms which
+trivially on $\bd b$.
+Then $f(b) = b$.
+In particular, homeomorphisms which are isotopic to the identity rel boundary act trivially on
+all of $\cC(X)$.
\end{axiom}
This axiom needs to be strengthened to force product morphisms to act as the identity.
@@ -585,7 +594,7 @@
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)) .
+ a & \mapsto & s_{Y,J}(a \bullet ((a|_Y)\times J)) .
\end{eqnarray*}
(See Figure \ref{glue-collar}.)
\begin{figure}[t]
@@ -644,50 +653,201 @@
The revised axiom is
%\addtocounter{axiom}{-1}
-\begin{axiom}[\textup{\textbf{[ordinary version]}} Extended isotopy invariance in dimension $n$.]
+\begin{axiom}[\textup{\textbf{[ordinary 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 $\cC(X)$.
+Let $X$ be an $n$-ball, $b \in \cC(X)$, and $f: X\to X$ be a homeomorphism which
+acts trivially on the restriction $\bd b$ of $b$ to $\bd X$.
+Suppose furthermore that $f$ is isotopic to the identity through homeomorphisms which
+act trivially on $\bd b$.
+Then $f(b) = b$.
In addition, collar maps act trivially on $\cC(X)$.
\end{axiom}
-\smallskip
+\medskip
+
+We need one additional axiom, in order to constrain the poset of decompositions of a given morphism.
+We will soon want to take colimits (and homotopy colimits) indexed by such posets, and we want to require
+that these colimits are in some sense locally acyclic.
+Before stating the axiom we need a few preliminary definitions.
+If $P$ is a poset let $P\times I$ denote the product poset, where $I = \{0, 1\}$ with ordering $0\le 1$.
+Let $\Cone(P)$ denote $P$ adjoined an additional object $v$ (the vertex of the cone) with $p\le v$ for all objects $p$ of $P$.
+Finally, let $\vcone(P)$ denote $P\times I \cup \Cone(P)$, where we identify $P\times \{0\}$ with the base of the cone.
+We call $P\times \{1\}$ the base of $\vcone(P)$.
+(See Figure \ref{vcone-fig}.)
+\begin{figure}[t]
+$$\mathfig{.65}{tempkw/vcone}$$
+\caption{(a) $P$, (b) $P\times I$, (c) $\Cone(P)$, (d) $\vcone(P)$}\label{vcone-fig}
+\end{figure}
+
+\nn{maybe call this ``splittings" instead of ``V-cones"?}
+
+\begin{axiom}[V-cones]
+\label{axiom:vcones}
+Let $c\in \cC_k(X)$ and
+let $P$ be a finite poset of splittings of $c$.
+Then we can embed $\vcone(P)$ into the splittings of $c$, with $P$ corresponding to the base of $\vcone(P)$.
+Furthermore, if $q$ is any decomposition of $X$, then we can take the vertex of $\vcone(P)$ to be $q$ up to a small perturbation.
+\end{axiom}
-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.
+\nn{maybe also say that any splitting of $\bd c$ can be extended to a splitting of $c$}
+
+It is easy to see that this axiom holds in our two motivating examples,
+using standard facts about transversality and general position.
+One starts with $q$, perturbs it so that it is in general position with respect to $c$ (in the case of string diagrams)
+and also with respect to each of the decompositions of $P$, then chooses common refinements of each decomposition of $P$
+and the perturbed $q$.
+These common refinements form the middle ($P\times \{0\}$ above) part of $\vcone(P)$.
+
+We note two simple special cases of Axiom \ref{axiom:vcones}.
+If $P$ is the empty poset, then $\vcone(P)$ consists of only the vertex, and the axiom says that any morphism $c$
+can be split along any decomposition of $X$, after a small perturbation.
+If $P$ is the disjoint union of two points, then $\vcone(P)$ looks like a letter W, and the axiom implies that the
+poset of splittings of $c$ is connected.
+Note that we do not require that any two splittings of $c$ have a common refinement (i.e.\ replace the letter W with the letter V).
+Two decompositions of $X$ might intersect in a very messy way, but one can always find a third
+decomposition which has common refinements with each of the original two decompositions.
+
+
+\medskip
+
+This completes the definition of an $n$-category.
+Next we define enriched $n$-categories.
+
+\medskip
-%\addtocounter{axiom}{-1}
+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 appropriate auxiliary category
+(e.g.\ vector spaces, or modules over some ring, or chain complexes),
+and all the structure maps of the $n$-category are 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.
+
+%We will aim for a little bit more generality than we need and not assume that the objects
+%of our auxiliary category are sets with extra structure.
+First we must specify requirements for the auxiliary category.
+It should have a {\it distributive monoidal structure} in the sense of
+\cite{1010.4527}.
+This means that there is a monoidal structure $\otimes$ and also coproduct $\oplus$,
+and these two structures interact in the appropriate way.
+Examples include
+\begin{itemize}
+\item vector spaces (or $R$-modules or chain complexes) with tensor product and direct sum; and
+\item topological spaces with product and disjoint union.
+\end{itemize}
+For convenience, we will also assume that the objects of our auxiliary category are sets with extra structure.
+(Otherwise, stating the axioms for identity morphisms becomes more cumbersome.)
+
+Before stating the revised axioms for an $n$-category enriched in a distributive monoidal category,
+we need a preliminary definition.
+Once we have the above $n$-category axioms for $n{-}1$-morphisms, we can define the
+category $\bbc$ of {\it $n$-balls with boundary conditions}.
+Its objects are pairs $(X, c)$, where $X$ is an $n$-ball and $c \in \cl\cC(\bd X)$ is the ``boundary condition".
+The morphisms from $(X, c)$ to $(X', c')$, denoted $\Homeo(X,c; X', c')$, are
+homeomorphisms $f:X\to X'$ such that $f|_{\bd X}(c) = c'$.
+%Let $\pi_0(\bbc)$ denote
+
+\begin{axiom}[Enriched $n$-categories]
+\label{axiom:enriched}
+Let $\cS$ be a distributive symmetric monoidal category.
+An $n$-category enriched in $\cS$ satisfies the above $n$-category axioms for $k=0,\ldots,n-1$,
+and modifies the axioms for $k=n$ as follows:
+\begin{itemize}
+\item Morphisms. We have a functor $\cC_n$ from $\bbc$ ($n$-balls with boundary conditions) to $\cS$.
+%[already said this above. ack] Furthermore, $\cC_n(f)$ depends only on the path component of a homeomorphism $f: (X, c) \to (X', c')$.
+%In particular, homeomorphisms which are isotopic to the identity rel boundary act trivially
+\item Composition. Let $B = B_1\cup_Y B_2$ as in Axiom \ref{axiom:composition}.
+Let $Y_i = \bd B_i \setmin Y$.
+Note that $\bd B = Y_1\cup Y_2$.
+Let $c_i \in \cC(Y_i)$ with $\bd c_1 = \bd c_2 = d \in \cl\cC(E)$.
+Then we have a map
+\[
+ \gl_Y : \bigoplus_c \cC(B_1; c_1 \bullet c) \otimes \cC(B_2; c_2\bullet c) \to \cC(B; c_1\bullet c_2),
+\]
+where the sum is over $c\in\cC(Y)$ such that $\bd c = d$.
+This map is natural with respect to the action of homeomorphisms and with respect to restrictions.
+%\item Product morphisms. \nn{Hmm... not sure what to say here. maybe we need sets with structure after all.}
+\end{itemize}
+\end{axiom}
+
+\medskip
+
+When the enriching category $\cS$ is chain complexes or topological spaces,
+or more generally an appropriate sort of $\infty$-category,
+we can modify the extended isotopy axiom \ref{axiom:extended-isotopies}
+to require that families of homeomorphisms act
+and obtain an $A_\infty$ $n$-category.
+
+\noop{
+We believe that abstract definitions should be guided by diverse collections
+of concrete examples, and a lack of diversity in our present collection of examples of $A_\infty$ $n$-categories
+makes us reluctant to commit to an all-encompassing general definition.
+Instead, we will give a relatively narrow definition which covers the examples we consider in this paper.
+After stating it, we will briefly discuss ways in which it can be made more general.
+}
+
+Recall the category $\bbc$ of balls with boundary conditions.
+Note that the morphisms $\Homeo(X,c; X', c')$ from $(X, c)$ to $(X', c')$ form a topological space.
+Let $\cS$ be an appropriate $\infty$-category (e.g.\ chain complexes)
+and let $\cJ$ be an $\infty$-functor from topological spaces to $\cS$
+(e.g.\ the singular chain functor $C_*$).
+
\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
+For each pair of $n$-balls $X$ and $X'$ and each pair $c\in \cl{\cC}(\bd X)$ and $c'\in \cl{\cC}(\bd X')$ we have an $\cS$-morphism
+\[
+ \cJ(\Homeo(X,c; X', c')) \ot \cC(X; c) \to \cC(X'; c') .
+\]
+Similarly, we have an $\cS$-morphism
\[
- C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) .
+ \cJ(\Coll(X,c)) \ot \cC(X; c) \to \cC(X; c),
\]
-These action maps are required to be associative up to homotopy,
-%\nn{iterated homotopy?}
+where $\Coll(X,c)$ denotes the space of collar maps.
+(See below for further discussion.)
+These action maps are required to be associative up to coherent homotopy,
and also compatible with composition (gluing) in the sense that
a diagram like the one in Theorem \ref{thm:CH} commutes.
-%\nn{repeat diagram here?}
-%\nn{restate this with $\Homeo(X\to X')$? what about boundary fixing property?}
-On $C_0(\Homeo_\bd(X))\ot \cC(X; c)$ the action should coincide
-with the one coming from Axiom \ref{axiom:morphisms}.
+% say something about compatibility with product morphisms?
\end{axiom}
-We should strengthen the above $A_\infty$ axiom to apply to families of collar maps.
-To do this we need to explain how collar maps form a topological space.
-Roughly, the set of collared $n{-}1$-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 $\r$.
-Having chains on the space of collar maps act gives rise to coherence maps involving
-weak identities.
-We will not pursue this in detail here.
+We now describe the topology on $\Coll(X; c)$.
+We retain notation from the above definition of collar map.
+Each collaring homeomorphism $X \cup (Y\times J) \to X$ determines a map from points $p$ of $\bd X$ to
+(possibly zero-width) embedded intervals in $X$ terminating at $p$.
+If $p \in Y$ this interval is the image of $\{p\}\times J$.
+If $p \notin Y$ then $p$ is assigned the zero-width interval $\{p\}$.
+Such collections of intervals have a natural topology, and $\Coll(X; c)$ inherits its topology from this.
+Note in particular that parts of the collar are allowed to shrink continuously to zero width.
+(This is the real content; if nothing shrinks to zero width then the action of families of collar
+maps follows from the action of families of homeomorphisms and compatibility with gluing.)
-A potential variant on the above axiom would be to drop the ``up to homotopy" and require a strictly associative action. (In fact, the alternative construction of the blob complex described in \S \ref{ss:alt-def} gives $n$-categories as in Example \ref{ex:blob-complexes-of-balls} which satisfy this stronger axiom; since that construction is only homotopy equivalent to the usual one, only the weaker axiom carries across.)
+The $k=n$ case of Axiom \ref{axiom:morphisms} posits a {\it strictly} associative action of {\it sets}
+$\Homeo(X,c; X', c') \times \cC(X; c) \to \cC(X'; c')$, and at first it might seem that this would force the above
+action of $\cJ(\Homeo(X,c; X', c'))$ to be strictly associative as well (assuming the two actions are compatible).
+In fact, compatibility implies less than this.
+For simplicity, assume that $\cJ$ is $C_*$, the singular chains functor.
+(This is the example most relevant to this paper.)
+Then compatibility implies that the action of $C_*(\Homeo(X,c; X', c'))$ agrees with the action
+of $C_0(\Homeo(X,c; X', c'))$ coming from Axiom \ref{axiom:morphisms}, so we only require associativity in degree zero.
+And indeed, this is true for our main example of an $A_\infty$ $n$-category based on the blob construction.
+Stating this sort of compatibility for general $\cS$ and $\cJ$ requires further assumptions,
+such as the forgetful functor from $\cS$ to sets having a left adjoint, and $\cS$ having an internal Hom.
+An alternative (due to Peter Teichner) is to say that Axiom \ref{axiom:families}
+supersedes the $k=n$ case of Axiom \ref{axiom:morphisms}; in dimension $n$ we just have a
+functor $\bbc \to \cS$ of $A_\infty$ 1-categories.
+(This assumes some prior notion of $A_\infty$ 1-category.)
+We are not currently aware of any examples which require this sort of greater generality, so we think it best
+to refrain from settling on a preferred version of the axiom until
+we have a greater variety of examples to guide the choice.
+Another variant of the above axiom would be to drop the ``up to homotopy" and require a strictly associative action.
+In fact, the alternative construction of the blob complex described in \S \ref{ss:alt-def}
+gives $n$-categories as in Example \ref{ex:blob-complexes-of-balls} which satisfy this stronger axiom;
+since that construction is only homotopy equivalent to the usual one, only the weaker axiom carries across.
+
+\noop{
Note that if we take homology of chain complexes, we turn an $A_\infty$ $n$-category
into a ordinary $n$-category (enriched over graded groups).
In a different direction, if we enrich over topological spaces instead of chain complexes,
@@ -695,6 +855,8 @@
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
@@ -704,7 +866,7 @@
\medskip
-The alert reader will have already noticed that our definition of a (ordinary) $n$-category
+The alert reader will have already noticed that our definition of an (ordinary) $n$-category
is extremely similar to our definition of a system of fields.
There are two differences.
First, for the $n$-category definition we restrict our attention to balls
@@ -725,7 +887,7 @@
In the $n$-category axioms above we have intermingled data and properties for expository reasons.
Here's a summary of the definition which segregates the data from the properties.
-An $n$-category consists of the following data:
+An $n$-category consists of the following data: \nn{need to revise this list}
\begin{itemize}
\item functors $\cC_k$ from $k$-balls to sets, $0\le k\le n$ (Axiom \ref{axiom:morphisms});
\item boundary natural transformations $\cC_k \to \cl{\cC}_{k-1} \circ \bd$ (Axiom \ref{nca-boundary});
@@ -936,7 +1098,7 @@
Note in particular that the space for $k=1$ contains a copy of $\Diff(B^n)$, namely
the embeddings of a ``little" ball with image all of the big ball $B^n$.
(But note also that this inclusion is not
-necessarily a homotopy equivalence.)
+necessarily a homotopy equivalence.))
The operad $\cE\cB_n$ is homotopy equivalent to the standard framed little $n$-ball operad:
by shrinking the little balls (precomposing them with dilations),
we see that both operads are homotopic to the space of $k$ framed points
@@ -1001,7 +1163,7 @@
\medskip
-We will first define the ``decomposition" poset $\cell(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$.
+We will first define the {\it decomposition poset} $\cell(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$.
An $n$-category $\cC$ provides a functor from this poset to the category of sets,
and we will define $\cl{\cC}(W)$ as a suitable colimit
(or homotopy colimit in the $A_\infty$ case) of this functor.
@@ -1037,12 +1199,12 @@
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 $i$,
-and with $M_0,\ldots, M_i$ each being a disjoint union of balls.
+and with $M_0, M_1, \ldots, M_i$ each being a disjoint union of balls.
\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.
+See Figure \ref{partofJfig}.
\end{defn}
\begin{figure}[t]
@@ -1056,33 +1218,69 @@
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, as described above, we have a subset $\cC(X)\spl \sub \cC(X)$ of morphisms whose boundaries
-are splittable along this decomposition.
+Let $x = \{X_a\}$ be a permissible decomposition of $W$ (i.e.\ object of $\cD(W)$).
+We will define $\psi_{\cC;W}(x)$ to be a certain subset of $\prod_a \cC(X_a)$.
+Roughly speaking, $\psi_{\cC;W}(x)$ is the subset where the restriction maps from
+$\cC(X_a)$ and $\cC(X_b)$ agree whenever some part of $\bd X_a$ is glued to some part of $\bd X_b$.
+(Keep in mind that perhaps $a=b$.)
+Since we allow decompositions in which the intersection of $X_a$ and $X_b$ might be messy
+(see Example \ref{sin1x-example}), we must define $\psi_{\cC;W}(x)$ in a more roundabout way.
+
+Inductively, we may assume that we have already defined the colimit $\cl\cC(M)$ for $k{-}1$-manifolds $M$.
+(To start the induction, we define $\cl\cC(M)$, where $M = \du_a P_a$ is a 0-manifold and each $P_a$ is
+a 0-ball, to be $\prod_a \cC(P_a)$.)
+We also assume, inductively, that we have gluing and restriction maps for colimits of $k{-}1$-manifolds.
+Gluing and restriction maps for colimits of $k$-manifolds will be defined later in this subsection.
-\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).
-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}
+Let $\du_a X_a = M_0\to\cdots\to M_m = W$ be a ball decomposition compatible with $x$.
+Let $\bd M_i = N_i \cup Y_i \cup Y'_i$, where $Y_i$ and $Y'_i$ are glued together to produce $M_{i+1}$.
+We will define $\psi_{\cC;W}(x)$ be be the subset of $\prod_a \cC(X_a)$ which satisfies a series of conditions
+related to the gluings $M_{i-1} \to M_i$, $1\le i \le m$.
+By Axiom \ref{nca-boundary}, we have a map
+\[
+ \prod_a \cC(X_a) \to \cl\cC(\bd M_0) .
+\]
+The first condition is that the image of $\psi_{\cC;W}(x)$ in $\cl\cC(\bd M_0)$ is splittable
+along $\bd Y_0$ and $\bd Y'_0$, and that the restrictions to $\cl\cC(Y_0)$ and $\cl\cC(Y'_0)$ agree
+(with respect to the identification of $Y_0$ and $Y'_0$ provided by the gluing map).
+
+On the subset of $\prod_a \cC(X_a)$ which satisfies the first condition above, we have a restriction
+map to $\cl\cC(N_0)$ which we can compose with the gluing map
+$\cl\cC(N_0) \to \cl\cC(\bd M_1)$.
+The second condition is that the image of $\psi_{\cC;W}(x)$ in $\cl\cC(\bd M_1)$ is splittable
+along $\bd Y_1$ and $\bd Y'_1$, and that the restrictions to $\cl\cC(Y_1)$ and $\cl\cC(Y'_1)$ agree
+(with respect to the identification of $Y_1$ and $Y'_1$ provided by the gluing map).
+The $i$-th condition is defined similarly.
+Note that these conditions depend on on the boundaries of elements of $\prod_a \cC(X_a)$.
+
+We define $\psi_{\cC;W}(x)$ to be the subset of $\prod_a \cC(X_a)$ which satisfies the
+above conditions for all $i$ and also all
+ball decompositions compatible with $x$.
+(If $x$ is a nice, non-pathological cell decomposition, then it is easy to see that gluing
+compatibility for one ball decomposition implies gluing compatibility for all other ball decompositions.
+Rather than try to prove a similar result for arbitrary
+permissible decompositions, we instead require compatibility with all ways of gluing up the decomposition.)
+
+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 $\cC$.
+This completes the definition of the functor $\psi_{\cC;W}$.
+
+Note that we have constructed, at the last stage of the above procedure,
+a map from $\psi_{\cC;W}(x)$ to $\cl\cC(\bd M_m) = \cl\cC(\bd W)$.
+\nn{need to show at somepoint that this does not depend on choice of ball decomp}
If $k=n$ in the above definition and we are enriching in some auxiliary category,
we need to say a bit more.
-We can rewrite Equation \ref{eq:psi-C} as
-\begin{equation} \label{eq:psi-CC}
+We can rewrite the colimit as
+\[ % \begin{equation} \label{eq:psi-CC}
\psi_{\cC;W}(x) \deq \coprod_\beta \prod_a \cC(X_a; \beta) ,
-\end{equation}
-where $\beta$ runs through labelings of the $k{-}1$-skeleton of the decomposition
-(which are compatible when restricted to the $k{-}2$-skeleton), and $\cC(X_a; \beta)$
-means the subset of $\cC(X_a)$ whose restriction to $\bd X_a$ agress with $\beta$.
+\] % \end{equation}
+where $\beta$ runs through
+boundary conditions on $\du_a X_a$ which are compatible with gluing as specified above
+and $\cC(X_a; \beta)$
+means the subset of $\cC(X_a)$ whose restriction to $\bd X_a$ agrees with $\beta$.
If we are enriching over $\cS$ and $k=n$, then $\cC(X_a; \beta)$ is an object in
-$\cS$ and the coproduct and product in Equation \ref{eq:psi-CC} should be replaced by the approriate
+$\cS$ and the coproduct and product in the above expression should be replaced by the appropriate
operations in $\cS$ (e.g. direct sum and tensor product if $\cS$ is Vect).
Finally, we construct $\cl{\cC}(W)$ as the appropriate colimit of $\psi_{\cC;W}$:
@@ -1184,6 +1382,12 @@
there are well-defined maps $\cl{\cC}(W)\to\cl{\cC}(\bd W)$, and that these maps
comprise a natural transformation of functors.
+
+
+\nn{to do: define splittability and restrictions for colimits}
+
+
+
\begin{lem}
\label{lem:colim-injective}
Let $W$ be a manifold of dimension less than $n$. Then for each
@@ -1281,7 +1485,10 @@
\caption{From manifold with boundary collar to marked ball}\label{blah15}\end{figure}
Define the boundary of a marked $k$-ball $(B, N)$ to be the pair $(\bd B \setmin N, \bd N)$.
-Call such a thing a {marked $k{-}1$-hemisphere}.
+Call such a thing a {\it marked $k{-}1$-hemisphere}.
+(A marked $k{-}1$-hemisphere is, of course, just a $k{-}1$-ball with its entire boundary marked.
+We call it a hemisphere instead of a ball because it plays a role analogous
+to the $k{-}1$-spheres in the $n$-category definition.)
\begin{lem}
\label{lem:hemispheres}
@@ -1320,13 +1527,23 @@
Let $\cl\cM(H)\trans E$ denote the image of $\gl_E$.
We will refer to elements of $\cl\cM(H)\trans E$ as ``splittable along $E$" or ``transverse to $E$".
+\noop{ %%%%%%%
\begin{lem}[Module to category restrictions]
{For each marked $k$-hemisphere $H$ there is a restriction map
-$\cl\cM(H)\to \cC(H)$.
+$\cl\cM(H)\to \cC(H)$.
($\cC(H)$ means apply $\cC$ to the underlying $k$-ball of $H$.)
These maps comprise a natural transformation of functors.}
\end{lem}
+} %%%%%%% end \noop
+It follows from the definition of the colimit $\cl\cM(H)$ that
+given any (unmarked) $k{-}1$-ball $Y$ in the interior of $H$ there is a restriction map
+from a subset $\cl\cM(H)_{\trans{\bdy Y}}$ of $\cl\cM(H)$ to $\cC(Y)$.
+Combining this with the boundary map $\cM(B,N) \to \cl\cM(\bd(B,N))$, we also have a restriction
+map from a subset $\cM(B,N)_{\trans{\bdy Y}}$ of $\cM(B,N)$ to $\cC(Y)$ whenever $Y$ is in the interior of $\bd B \setmin N$.
+This fact will be used below.
+
+\noop{ %%%%
Note that combining the various boundary and restriction maps above
(for both modules and $n$-categories)
we have for each marked $k$-ball $(B, N)$ and each $k{-}1$-ball $Y\sub \bd B \setmin N$
@@ -1334,6 +1551,7 @@
This subset $\cM(B,N)\trans{\bdy Y}$ is the subset of morphisms which are appropriately splittable (transverse to the
cutting submanifolds).
This fact will be used below.
+} %%%%% end \noop
In our example, the various restriction and gluing maps above come from
restricting and gluing maps into $T$.
@@ -2371,3 +2589,99 @@
To define (binary) composition of $n{+}1$-morphisms, choose the obvious common equator
then compose the module maps.
The proof that this composition rule is associative is similar to the proof of Lemma \ref{equator-lemma}.
+
+\medskip
+
+We end this subsection with some remarks about Morita equivalence of disklike $n$-categories.
+Recall that two 1-categories $\cC$ and $\cD$ are Morita equivalent if and only if they are equivalent
+objects in the 2-category of (linear) 1-categories, bimodules, and intertwinors.
+Similarly, we define two disklike $n$-categories to be Morita equivalent if they are equivalent objects in the
+$n{+}1$-category of sphere modules.
+
+Because of the strong duality enjoyed by disklike $n$-categories, the data for such an equivalence lives only in
+dimensions 1 and $n+1$ (the middle dimensions come along for free).
+The $n{+}1$-dimensional part of the data must be invertible and satisfy
+identities corresponding to Morse cancellations in $n$-manifolds.
+We will treat this in detail for the $n=2$ case; the case for general $n$ is very similar.
+
+Let $\cC$ and $\cD$ be (unoriented) disklike 2-categories.
+Let $\cS$ denote the 3-category of 2-category sphere modules.
+The 1-dimensional part of the data for a Morita equivalence between $\cC$ and $\cD$ is a 0-sphere module $\cM = {}_\cC\cM_\cD$
+(categorified bimodule) connecting $\cC$ and $\cD$.
+Because of the full unoriented symmetry, this can also be thought of as a
+0-sphere module ${}_\cD\cM_\cC$ connecting $\cD$ and $\cC$.
+
+We want $\cM$ to be an equivalence, so we need 2-morphisms in $\cS$
+between ${}_\cC\cM_\cD \otimes_\cD {}_\cD\cM_\cC$ and the identity 0-sphere module ${}_\cC\cC_\cC$, and similarly
+with the roles of $\cC$ and $\cD$ reversed.
+These 2-morphisms come for free, in the sense of not requiring additional data, since we can take them to be the labeled
+cell complexes (cups and caps) in $B^2$ shown in Figure \ref{morita-fig-1}.
+\begin{figure}[t]
+$$\mathfig{.65}{tempkw/morita1}$$
+\caption{Cups and caps for free}\label{morita-fig-1}
+\end{figure}
+
+
+We want the 2-morphisms from the previous paragraph to be equivalences, so we need 3-morphisms
+between various compositions of these 2-morphisms and various identity 2-morphisms.
+Recall that the 3-morphisms of $\cS$ are intertwinors between representations of 1-categories associated
+to decorated circles.
+Figure \ref{morita-fig-2}
+\begin{figure}[t]
+$$\mathfig{.55}{tempkw/morita2}$$
+\caption{Intertwinors for a Morita equivalence}\label{morita-fig-2}
+\end{figure}
+shows the intertwinors we need.
+Each decorated 2-ball in that figure determines a representation of the 1-category associated to the decorated circle
+on the boundary.
+This is the 3-dimensional part of the data for the Morita equivalence.
+(Note that, by symmetry, the $c$ and $d$ arrows of Figure \ref{morita-fig-2}
+are the same (up to rotation), as are the $h$ and $g$ arrows.)
+
+In order for these 3-morphisms to be equivalences,
+they must be invertible (i.e.\ $a=b\inv$, $c=d\inv$, $e=f\inv$) and in addition
+they must satisfy identities corresponding to Morse cancellations on 2-manifolds.
+These are illustrated in Figure \ref{morita-fig-3}.
+\begin{figure}[t]
+$$\mathfig{.65}{tempkw/morita3}$$
+\caption{Identities for intertwinors}\label{morita-fig-3}
+\end{figure}
+Each line shows a composition of two intertwinors which we require to be equal to the identity intertwinor.
+
+For general $n$, we start with an $n$-category 0-sphere module $\cM$ which is the data for the 1-dimensional
+part of the Morita equivalence.
+For $2\le k \le n$, the $k$-dimensional parts of the Morita equivalence are various decorated $k$-balls with submanifolds
+labeled by $\cC$, $\cD$ and $\cM$; no additional data is needed for these parts.
+The $n{+}1$-dimensional part of the equivalence is given by certain intertwinors, and these intertwinors must
+be invertible and satisfy
+identities corresponding to Morse cancellations in $n$-manifolds.
+
+\noop{
+One way of thinking of these conditions is as follows.
+Given a decorated $n{+}1$-manifold, with a codimension 1 submanifold labeled by $\cM$ and
+codimension 0 submanifolds labeled by $\cC$ and $\cD$, we can make any local modification we like without
+changing
+}
+
+If $\cC$ and $\cD$ are Morita equivalent $n$-categories, then it is easy to show that for any $n-j$-manifold
+$Y$ the $j$-categories $\cC(Y)$ and $\cD(Y)$ are Morita equivalent.
+When $j=0$ this means that the TQFT Hilbert spaces $\cC(Y)$ and $\cD(Y)$ are isomorphic
+(if we are enriching over vector spaces).
+
+
+
+
+\noop{ % the following doesn't work; need 2^(k+1) different N's, not 2*(k+1)
+More specifically, the 1-dimensional part of the data is a 0-sphere module $\cM = {}_\cCM_\cD$
+(categorified bimodule) connecting $\cC$ and $\cD$.
+From $\cM$ we can construct various $k$-sphere modules $N^k_{j,E}$ for $0 \le k \le n$, $0\le j \le k$, and $E = \cC$ or $\cD$.
+$N^k_{j,E}$ can be thought of as the graph of an index $j$ Morse function on the $k$-ball $B^k$
+(so the graph lives in $B^k\times I = B^{k+1}$).
+The positive side of the graph is labeled by $E$, the negative side by $E'$
+(where $\cC' = \cD$ and $\cD' = \cC$), and the codimension-1
+submanifold separating the positive and negative regions is labeled by $\cM$.
+We think of $N^k_{j,E}$ as a $k{+}1$-morphism connecting
+We plan on treating this in more detail in a future paper.
+\nn{should add a few more details}
+}
+