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 \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
+(As is the case throughout this paper, by ``$n$-category" we mean some notion of
-a `weak' $n$-category with `strong duality'.)
+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.
 \medskip
 (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
+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.
 \end{axiom}
 (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.
 %\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.
-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.)
 Next we consider domains and ranges of morphisms (or, as we prefer to say, boundaries
 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
+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):
 \begin{lem}
 \label{lem:spheres}
 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.
+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 $\bd: \cC_k(X)\to \cl{\cC}_{k-1}(\bd X)$.
 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
+(In order to simplify the exposition we have concentrated on the case of
-%about the role of orientation-reversing maps of boundaries when gluing oriented manifolds.
+unoriented PL manifolds and avoided the question of what exactly we mean by
-Maybe need a discussion about what the boundary of a manifold with a
+the boundary a manifold with extra structure, such as an oriented manifold.
-structure (e.g. orientation) means.
+In general, all manifolds of dimension less than $n$ should be equipped with the germ
-Tentatively, I think we need to redefine the oriented boundary of an oriented $n$-manifold.
+of a thickening to dimension $n$, and this germ should carry whatever structure we have
-Instead of an ordinary oriented $(n-1)$-manifold via the inward (or outward) normal
+on $n$-manifolds.
-first (or last) convention, perhaps it is better to define the boundary to be an $(n-1)$-manifold
+In addition, lower dimensional manifolds should be equipped with a framing
-equipped with an orientation of its once-stabilized tangent bundle.
+of their normal bundle in the thickening; the framing keeps track of which
-Similarly, in dimension $n-k$ we would have manifolds equipped with an orientation of
+side (iterated) bounded manifolds lie on.
-their $k$ times stabilized tangent bundles.
+For example, the boundary of an oriented $n$-ball
-(cf. \cite{MR2079378}.)
+should be an $n{-}1$-sphere equipped with an orientation of its once stabilized tangent
-Probably should also have a framing of the stabilized dimensions in order to indicate which
+bundle and a choice of direction in this bundle indicating
-side the bounded manifold is on.
+which side the $n$-ball lies 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
 \begin{figure}[!ht]
 $$
 \begin{tikzpicture}[%every label/.style={green}
 ]
-\node[fill=black, circle, label=below:$E$, inner sep=2pt](S) at (0,0) {};
+\node[fill=black, circle, label=below:$E$, inner sep=1.5pt](S) at (0,0) {};
-\node[fill=black, circle, label=above:$E$, inner sep=2pt](N) at (0,2) {};
+\node[fill=black, circle, label=above:$E$, inner sep=1.5pt](N) at (0,2) {};
 \draw (S) arc  (-90:90:1);
 \draw (N) arc  (90:270:1);
 \node[left] at (-1,1) {$B_1$};
 \node[right] at (1,1) {$B_2$};
 \end{tikzpicture}
 $$
 \caption{Combining two balls to get a full boundary.}\label{blah3}\end{figure}
-Note that we insist on injectivity above. \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 $E$".
 If $X$ is a $k$-ball and $E \sub \bd X$ splits $\bd X$ into two $k{-}1$-balls $B_1$ and $B_2$
 %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 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 $X$.
 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
+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
 \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}
+(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
 \[
 	\pi: E\to X
 \]
 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})
+\nn{do we 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*}
 \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.
+We call a map of this form a {\it collar 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 $Y$.
+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
 isotopy invariance with the requirement that families of homeomorphisms act.
 (i.e. fix an element of the colimit associated to $\bd W$).
 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
 above, and $\cl{\cC}(W)$ is universal with respect to these properties.
 \end{defn}
 \todo{This next sentence is circular: these maps are an axiom, not a consequence of anything. -S} It is easy to see that
 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?
 (actions of homeomorphisms);
 define $k$-cat $\cC(\cdot\times W)$}

changeset 415	8dedd2914d10
parent 411	98b8559b0b7a
child 416	c06a899bd1f0