--- a/text/ncat.tex Wed Jun 23 10:54:29 2010 -0400
+++ b/text/ncat.tex Wed Jun 23 10:54:42 2010 -0400
@@ -64,7 +64,7 @@
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.)
+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.
@@ -1521,36 +1521,37 @@
gluing subintervals together and/or omitting some of the rightmost subintervals.
(See Figure \ref{fig:lmar}.)
\begin{figure}[t]$$
-\begin{tikzpicture}
+\definecolor{arcolor}{rgb}{.75,.4,.1}
+\begin{tikzpicture}[line width=1pt]
\fill (0,0) circle (.1);
\draw (0,0) -- (2,0);
\draw (1,0.1) -- (1,-0.1);
-\draw [->,red] (1,0.25) -- (1,0.75);
+\draw [->, arcolor] (1,0.25) -- (1,0.75);
\fill (0,1) circle (.1);
\draw (0,1) -- (2,1);
\end{tikzpicture}
\qquad
-\begin{tikzpicture}
+\begin{tikzpicture}[line width=1pt]
\fill (0,0) circle (.1);
\draw (0,0) -- (2,0);
\draw (1,0.1) -- (1,-0.1);
-\draw [->,red] (1,0.25) -- (1,0.75);
+\draw [->, arcolor] (1,0.25) -- (1,0.75);
\fill (0,1) circle (.1);
\draw (0,1) -- (1,1);
\end{tikzpicture}
\qquad
-\begin{tikzpicture}
+\begin{tikzpicture}[line width=1pt]
\fill (0,0) circle (.1);
\draw (0,0) -- (3,0);
\foreach \x in {0.5, 1.0, 1.25, 1.5, 2.0, 2.5} {
\draw (\x,0.1) -- (\x,-0.1);
}
-\draw [->,red] (1,0.25) -- (1,0.75);
+\draw [->, arcolor] (1,0.25) -- (1,0.75);
\fill (0,1) circle (.1);
\draw (0,1) -- (2,1);
@@ -1586,8 +1587,7 @@
where $\cbar'$ corresponds to the subintervals of $D_0$ which map to $D_1$ and
$\cbar''$ corresponds to the subintervals
which are dropped off the right side.
-(Either $\cbar'$ or $\cbar''$ might be empty.)
-\nn{surely $\cbar'$ can't be empy: we don't allow $D_1$ to be empty.}
+(If no such subintervals are dropped, then $\cbar''$ is empty.)
Translating from the boundary map for $(\cM_\cC\ot {_\cC\cN})^*$ appearing in Equation \eqref{eq:tensor-product-boundary},
we have
\begin{eqnarray*}
@@ -1645,6 +1645,11 @@
\[
g\ot\id : \cX_\cC \ot {}_\cC\cZ \to \cY_\cC \ot {}_\cC\cZ .
\]
+\nn{...}
+More generally, we have a chain map
+\[
+ \hom_\cC(\cX_\cC \to \cY_\cC) \ot \cX_\cC \ot {}_\cC\cZ \to \cY_\cC \ot {}_\cC\cZ .
+\]
\nn{not sure whether to do low degree examples or try to state the general case; ideally both,
but maybe just low degrees for now.}
@@ -1677,10 +1682,12 @@
whose objects are $n$-categories.
When $n=2$
this is a version of the familiar algebras-bimodules-intertwiners $2$-category.
-While it is clearly appropriate to call an $S^0$ module a bimodule,
+It is clearly appropriate to call an $S^0$ module a bimodule,
but this is much less true for higher dimensional spheres,
so we prefer the term ``sphere module" for the general case.
+For simplicity, we will assume that $n$-categories are enriched over $\c$-vector spaces.
+
The $0$- through $n$-dimensional parts of $\cC$ are various sorts of modules, and we describe
these first.
The $n{+}1$-dimensional part of $\cS$ consists of intertwiners
@@ -1706,7 +1713,7 @@
Another way to say this is that $(X, M)$ is homeomorphic to $B^{k-1}\times([-1,1], \{0\})$.
\begin{figure}[!ht]
-$$\tikz[baseline,line width=2pt]{\draw[blue] (-2,0)--(2,0); \fill[red] (0,0) circle (0.1);} \qquad \qquad \tikz[baseline,line width=2pt]{\draw[blue] (0,0) circle (2 and 1); \draw[red] (0,1)--(0,-1);}$$
+$$\tikz[baseline,line width=2pt]{\draw[blue] (-2,0)--(2,0); \fill[red] (0,0) circle (0.1);} \qquad \qquad \tikz[baseline,line width=2pt]{\draw[blue][fill=blue!30!white] (0,0) circle (2 and 1); \draw[red] (0,1)--(0,-1);}$$
\caption{0-marked 1-ball and 0-marked 2-ball}
\label{feb21a}
\end{figure}
@@ -1731,7 +1738,7 @@
or $\cB_k(X_i)$ (if $X_i$ lies on the $\cB$-labeled side)
or $\cM_k(X_i)$ (if $X_i$ intersects the marking and is therefore a smaller 0-marked ball).
Corresponding to this decomposition we have an action and/or composition map
-from the product of these various sets into $\cM(X)$.
+from the product of these various sets into $\cM_k(X)$.
\medskip
@@ -1756,7 +1763,7 @@
\draw (2,0) -- (4,0) node[below] {$J$};
\fill[red] (3,0) circle (0.1);
-\draw (6,0) node(a) {} arc (135:90:4) node(top) {} arc (90:45:4) node(b) {} arc (-45:-90:4) node(bottom) {} arc(-90:-135:4);
+\draw[fill=blue!30!white] (6,0) node(a) {} arc (135:90:4) node(top) {} arc (90:45:4) node(b) {} arc (-45:-90:4) node(bottom) {} arc(-90:-135:4);
\draw[red] (top.center) -- (bottom.center);
\fill (a) circle (0.1) node[left] {\color{green!50!brown} $a$};
\fill (b) circle (0.1) node[right] {\color{green!50!brown} $b$};
@@ -1831,7 +1838,7 @@
\begin{figure}[!ht]
$$
\begin{tikzpicture}[baseline,line width = 2pt]
-\draw[blue] (0,0) circle (2);
+\draw[blue][fill=blue!15!white] (0,0) circle (2);
\fill[red] (0,0) circle (0.1);
\foreach \qm/\qa/\n in {70/-30/0, 120/95/1, -120/180/2} {
\draw[red] (0,0) -- (\qm:2);
@@ -1871,7 +1878,7 @@
\medskip
-We can now define the $n$- or less dimensional part of our $n{+}1$-category $\cS$.
+We can now define the $n$-or-less-dimensional part of our $n{+}1$-category $\cS$.
Choose some collection of $n$-categories, then choose some collections of bimodules for
these $n$-categories, then choose some collection of 1-sphere modules for the various
possible marked 1-spheres labeled by the $n$-categories and bimodules, and so on.
@@ -1892,7 +1899,7 @@
Thus the $k$-morphisms of $\cS$ (for $k\le n$) can be thought
of as $n$-category $k{-}1$-sphere modules
(generalizations of bimodules).
-On the other hand, we can equally think of the $k$-morphisms as decorations on $k$-balls,
+On the other hand, we can equally well think of the $k$-morphisms as decorations on $k$-balls,
and from this (official) point of view it is clear that they satisfy all of the axioms of an
$n{+}1$-category.
(All of the axioms for the less-than-$n{+}1$-dimensional part of an $n{+}1$-category, that is.)
@@ -1900,12 +1907,40 @@
\medskip
Next we define the $n{+}1$-morphisms of $\cS$.
+The construction of the 0- through $n$-morphisms was easy and tautological, but the
+$n{+}1$-morphisms will require a bit of combinatorial topology effort, as well as addition
+duality assumptions on the lower morphisms.
-
+Let $X$ be an $n{+}1$-ball, and let $c$ be a decoration of its boundary
+by a cell complex labeled by 0- through $n$-morphisms, as above.
+Choose an $n{-}1$-sphere $E\sub \bd X$ which divides
+$\bd X$ into ``incoming" and ``outgoing" boundary $\bd_-X$ and $\bd_+X$.
+Let $E_c$ denote $E$ decorated by the restriction of $c$ to $E$.
+Recall from above the associated 1-category $\cS(E_c)$.
+We can also have $\cS(E_c)$ modules $\cS(\bd_-X_c)$ and $\cS(\bd_+X_c)$.
+Define
+\[
+ \cS(X; c; E) \deq \hom_{\cS(E_c)}(\cS(\bd_-X_c), \cS(\bd_+X_c)) .
+\]
-
+We will show that if the sphere modules are equipped with a compatible family of
+non-degenerate inner products, then there is a coherent family of isomorphisms
+$\cS(X; c; E) \cong \cS(X; c; E')$ for all pairs of choices $E$ and $E'$.
+This will allow us to define $\cS(X; e)$ independently of the choice of $E$.
-
+Let $Y$ be a decorated $n$-ball, and $\ol{Y}$ it's mirror image.
+(We assume we are working in the unoriented category.)
+Let $Y\cup\ol{Y}$ denote the decorated $n$-sphere obtained by gluing $Y$ and $\ol{Y}$
+along their common boundary.
+An {\it inner product} on $\cS(Y)$ is a dual vector
+\[
+ z_Y : \cS(Y\cup\ol{Y}) \to \c.
+\]
+We will also use the notation
+\[
+ \langle a, b\rangle \deq z_Y(a\bullet \ol{b}) \in \c .
+\]
+An inner product is {\it non-degenerate} if
\nn{...}
@@ -1924,10 +1959,7 @@
Stuff that remains to be done (either below or in an appendix or in a separate section or in
a separate paper):
\begin{itemize}
-\item spell out what difference (if any) Top vs PL vs Smooth makes
\item discuss Morita equivalence
-\item morphisms of modules; show that it's adjoint to tensor product
-(need to define dual module for this)
\item functors
\end{itemize}