# HG changeset patch
# User kevin@6e1638ff-ae45-0410-89bd-df963105f760
# Date 1248111470 0
# Node ID 38ceade5cc5d0348cfc1b3a2b5585ca81098120f
# Parent  9c181ef9c5fdbe865932e0bb5149c2aac1b45bc9
...

diff -r 9c181ef9c5fd -r 38ceade5cc5d blob1.tex
--- a/blob1.tex	Sat Jul 18 19:04:15 2009 +0000
+++ b/blob1.tex	Mon Jul 20 17:37:50 2009 +0000
@@ -837,7 +837,7 @@
 
 
 
-
+\input{text/ncat.tex}
 
 \input{text/A-infty.tex}
 
diff -r 9c181ef9c5fd -r 38ceade5cc5d text/A-infty.tex
--- a/text/A-infty.tex	Sat Jul 18 19:04:15 2009 +0000
+++ b/text/A-infty.tex	Mon Jul 20 17:37:50 2009 +0000
@@ -31,6 +31,9 @@
 
 Appendix \ref{sec:comparing-A-infty} explains the translation between this definition and the usual one expressed in terms of `associativity up to higher homotopy', as in \cite{MR1854636}. (In this version of the paper, that appendix is incomplete, however.)
 
+\nn{should say something about objects and restrictions of maps to boundaries of intervals 
+in next paragraph.}
+
 The motivating example is `chains of maps to $M$' for some fixed target space $M$. This is a topological $A_\infty$ category $\Xi_M$ with $\Xi_M(J) = C_*(\Maps(J \to M))$. The gluing maps $\Xi_M(J) \tensor \Xi_M(J') \to \Xi_M(J \cup J')$  takes the product of singular chains, then glues maps to $M$ together; the associativity condition is automatically satisfied. The evaluation map $\ev_{J,J'} : \CD{J \to J'} \tensor \Xi_M(J) \to \Xi_M(J')$ is the composition
 \begin{align*}
 \CD{J \to J'} \tensor C_*(\Maps(J \to M)) & \to C_*(\Diff(J \to J') \times \Maps(J \to M)) \\ & \to C_*(\Maps(J' \to M)),
@@ -68,6 +71,8 @@
 We now define the tensor product of a left module with a right module. The notion of the self-tensor product of a bimodule is a minor variation which we'll leave to the reader.
 Our definition requires choosing a `fixed' interval, and for simplicity we'll use $[0,1]$, but you should note that the definition is equivariant with respect to diffeomorphisms of this interval.
 
+\nn{maybe should do a general interval instead of $[0,1]$.}
+
 \begin{defn}
 The tensor product of a left module $\cM$ and a right module $\cN$ over a topological $A_\infty$ category $\cC$, denoted $\cM \tensor_{\cC} \cN$, is a vector space, which we'll specify as the limit of a certain commutative diagram. This (infinite) diagram has vertices indexed by partitions $$[0,1] = [0,x_1] \cup \cdots \cup [x_k,1]$$ and boundary conditions $$a_1, \ldots, a_k \in \Obj(\cC),$$ and arrows labeled by refinements. At each vertex put the vector space $$\cM([0,x_1],0; a_1) \tensor \cC([x_1,x_2];a_1,a_2]) \tensor \cdots \tensor \cC([x_{k-1},x_k];a_{k-1},a_k) \tensor \cN([x_k,1],1;a_k),$$ and on each arrow the corresponding gluing map. Faces of this diagram commute because the gluing maps compose associatively.
 \end{defn}
diff -r 9c181ef9c5fd -r 38ceade5cc5d text/kw_macros.tex
--- a/text/kw_macros.tex	Sat Jul 18 19:04:15 2009 +0000
+++ b/text/kw_macros.tex	Mon Jul 20 17:37:50 2009 +0000
@@ -23,7 +23,8 @@
 \def\lf{\overline{\cC}}
 \def\ot{\otimes}
 
-\def\nn#1{{{\it \small [#1]}}}
+%\def\nn#1{{{\it \small [#1]}}}
+\def\nn#1{{{\color[rgb]{.2,.5,.6} \small [#1]}}}
 \long\def\noop#1{}
 
 % equations
diff -r 9c181ef9c5fd -r 38ceade5cc5d text/ncat.tex
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/text/ncat.tex	Mon Jul 20 17:37:50 2009 +0000
@@ -0,0 +1,147 @@
+%!TEX root = ../blob1.tex
+
+\def\xxpar#1#2{\smallskip\noindent{\bf #1} {\it #2} \smallskip}
+
+\section{$n$-categories (maybe)}
+\label{sec:ncats}
+
+\nn{experimental section.  maybe this should be rolled into other sections.
+maybe it should be split off into a separate paper.}
+
+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 mean
+a weak $n$-category with strong duality.)
+
+Consider first ordinary $n$-categories.
+We need a set (or sets) of $k$-morphisms for each $0\le k \le n$.
+We must decide on the ``shape" of the $k$-morphisms.
+Some $n$-category definitions model $k$-morphisms on the standard bihedron (interval, bigon, ...).
+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 \nn{need refs for all these; maybe the Leinster book}
+model the $k$-morphisms on more complicated combinatorial polyhedra.
+
+We will allow our $k$-morphisms to have any shape, so long as it is homeomorphic to a $k$-ball.
+In other words,
+
+\xxpar{Morphisms (preliminary version):}{For any $k$-manifold $X$ homeomorphic 
+to a $k$-ball, we have a set of $k$-morphisms
+$\cC(X)$.}
+
+Given a homeomorphism $f:X\to Y$ between such $k$-manifolds, we want a corresponding
+bijection of sets $f:\cC(X)\to \cC(Y)$.
+So we replace the above with
+
+\xxpar{Morphisms:}{For each $0 \le k \le n$, we have a functor $\cC_k$ from 
+the category of manifolds homeomorphic to the $k$-ball and 
+homeomorphisms to the category of sets and bijections.}
+
+(Note: We usually omit the subscript $k$.)
+
+We are being deliberately vague about what flavor of manifolds we are considering.
+They could be unoriented or oriented or Spin or $\mbox{Pin}_\pm$.
+They could be topological or PL or smooth.
+(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 of the case of PL unoriented manifolds.
+
+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.
+For $k>1$ and in the presence of strong duality the domain/range division makes less sense.
+\nn{maybe say more here; rotate disk, Frobenius reciprocity blah blah}
+We prefer to 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:
+
+\xxpar{Boundaries (domain and range), part 1:}
+{For each $0 \le k \le n-1$, we have a functor $\cC_k$ from 
+the category of manifolds homeomorphic to the $k$-sphere and 
+homeomorphisms to the category of sets and bijections.}
+
+(In order to conserve symbols, we use the same symbol $\cC_k$ for both morphisms and boundaries.)
+
+\xxpar{Boundaries, part 2:}
+{For each $X$ homeomorphic to a $k$-ball, we have a map of sets $\bd: \cC(X)\to \cC(\bd X)$.
+These maps, for various $X$, comprise a natural transformation of functors.}
+
+(Note that the first ``$\bd$" above is part of the data for the category, 
+while the second is the ordinary boundary of manifolds.)
+
+Given $c\in\cC(\bd(X))$, let $\cC(X; c) = \bd^{-1}(c)$.
+
+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 $X$ homeomorphic to an $n$-ball and
+all $c\in \cC(\bd X)$, have the structure of an object in some auxiliary 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.
+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.
+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:
+
+\xxpar{Domain $+$ range $\to$ boundary:}
+{Let $S = B_1 \cup_E B_2$, where $S$ is homeomorphic to a $k$-sphere ($0\le k\le n-1$),
+$B_i$ is homeomorphic to a $k$-ball, and $E = B_1\cap B_2$ is homeomorphic to  a $k{-}1$-sphere.
+Let $\cC(B_1) \times_{\cC(E)} \cC(B_2)$ denote the fibered product of the 
+two maps $\bd: \cC(B_i)\to \cC(E)$.
+Then (axiom) we have an injective map
+\[
+	\gl_E : \cC(B_1) \times_{\cC(E)} \cC(B_2) \to \cC(S)
+\]
+which is natural with respect to the actions of homeomorphisms.}
+
+Note that we insist on injectivity above.
+Let $\cC(S)_E$ denote the image of $\gl_E$.
+We have ``restriction" maps $\cC(S)_E \to \cC(B_i)$, which can be thought of as
+domain and range maps, relative to the choice of splitting $S = B_1 \cup_E B_2$.
+
+If $B$ is homeomorphic to a $k$-ball and $E \sub \bd B$ splits $\bd B$ into two $k{-}1$-balls
+as above, then we define $\cC(B)_E = \bd^{-1}(\cC(\bd B)_E)$.
+
+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".
+
+\xxpar{Composition:}
+{Let $B = B_1 \cup_Y B_2$, where $B$, $B_1$ and $B_2$ are homeomorphic to $k$-balls ($0\le k\le n$)
+and $Y = B_1\cap B_2$ is homeomorphic to a $k{-}1$-ball.
+Let $E = \bd Y$, which is homeomorphic to 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. 
+Then (axiom) 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.)}
+
+
+