--- a/text/ncat.tex Sat Jul 03 19:59:25 2010 -0600
+++ b/text/ncat.tex Sun Jul 04 11:56:23 2010 -0600
@@ -11,13 +11,15 @@
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'.)
+(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.
-For examples of topological origin, it is typically easy to show that they
+For examples of topological origin
+(e.g.\ categories whose morphisms are maps into spaces or decorated balls),
+it is 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.
@@ -36,7 +38,7 @@
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
+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.
@@ -58,7 +60,7 @@
(Note: We usually omit the subscript $k$.)
-We are so far being deliberately vague about what flavor of $k$-balls
+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.
@@ -66,13 +68,13 @@
(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.
-We will concentrate on the case of PL unoriented manifolds.
+For simplicity, 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
+The second is balls equipped with a section of 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.)
@@ -94,7 +96,7 @@
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
+all the axioms in this 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):
@@ -107,7 +109,7 @@
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.
+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.
@@ -131,21 +133,21 @@
$\cC(Y; c)$ 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.}
+
+(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 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
@@ -174,8 +176,8 @@
$$
\begin{tikzpicture}[%every label/.style={green}
]
-\node[fill=black, circle, label=below:$E$, inner sep=2pt](S) at (0,0) {};
-\node[fill=black, circle, label=above:$E$, inner sep=2pt](N) at (0,2) {};
+\node[fill=black, circle, label=below:$E$, inner sep=1.5pt](S) at (0,0) {};
+\node[fill=black, circle, label=above:$E$, inner sep=1.5pt](N) at (0,2) {};
\draw (S) arc (-90:90:1);
\draw (N) arc (90:270:1);
\node[left] at (-1,1) {$B_1$};
@@ -184,7 +186,8 @@
$$
\caption{Combining two balls to get a full boundary.}\label{blah3}\end{figure}
-Note that we insist on injectivity above. \todo{Make sure we prove this, as a consequence of the next axiom, later.}
+Note that we insist on injectivity above.
+The lemma follows from Definition \ref{def:colim-fields} and Lemma \ref{lem:colim-injective}.
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 $E$".
@@ -261,7 +264,8 @@
%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 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.
@@ -298,7 +302,7 @@
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
+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'
@@ -363,7 +367,7 @@
$$
\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}
+(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
\[
@@ -525,7 +529,7 @@
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})
+\nn{do we need notation for this?})
We define a map
\begin{eqnarray*}
\psi_{Y,J}: \cC(X) &\to& \cC(X) \\
@@ -577,28 +581,25 @@
\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}
+We call a map of this form a {\it collar map}.
It can be thought of as the action of the inverse of
-a map which projects a collar neighborhood of $Y$ onto $Y$.
+a map which projects a collar neighborhood of $Y$ onto $Y$,
+or as the limit of homeomorphisms $X\to X$ which expand a very thin collar of $Y$
+to a larger collar.
+We call the equivalence relation generated by collar maps and homeomorphisms
+isotopic (rel boundary) to the identity {\it extended isotopy}.
The revised axiom is
\addtocounter{axiom}{-1}
-\begin{axiom}{\textup{\textbf{[topological version]}} Extended isotopy invariance in dimension $n$}
+\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.
+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}
-\nn{need to rephrase this, since extended isotopies don't correspond to homeomorphisms.}
-
\smallskip
For $A_\infty$ $n$-categories, we replace
@@ -961,6 +962,7 @@
Finally, we construct $\cC(W)$ as the appropriate colimit of $\psi_{\cC;W}$.
\begin{defn}[System of fields functor]
+\label{def:colim-fields}
If $\cC$ is an $n$-category enriched in sets or vector spaces, $\cl{\cC}(W)$ is the usual colimit of the functor $\psi_{\cC;W}$.
That is, for each decomposition $x$ there is a map
$\psi_{\cC;W}(x)\to \cl{\cC}(W)$, these maps are compatible with the refinement maps
@@ -1026,7 +1028,14 @@
there are well-defined maps $\cC(W)\to\cC(\bd W)$, and that these maps
comprise a natural transformation of functors.
-\todo{Explicitly say somewhere: `this proves Lemma \ref{lem:domain-and-range}'}
+\begin{lem}
+\label{lem:colim-injective}
+Let $W$ be a manifold of dimension less than $n$. Then for each
+decomposition $x$ of $W$ the natural map $\psi_{\cC;W}(x)\to \cl{\cC}(W)$ is injective.
+\end{lem}
+\begin{proof}
+\nn{...}
+\end{proof}
\nn{need to finish explaining why we have a system of fields;
need to say more about ``homological" fields?