--- a/blob to-do Sat May 07 08:35:36 2011 -0700
+++ b/blob to-do Sat May 07 09:18:37 2011 -0700
@@ -37,8 +37,6 @@
automorphism, which translates into a involution on objects. Mention
super-stuff.
-* make remarks about defect interpretation of sphere modules
-
colimit subsection:
--- a/blob_changes_v3 Sat May 07 08:35:36 2011 -0700
+++ b/blob_changes_v3 Sat May 07 09:18:37 2011 -0700
@@ -16,4 +16,6 @@
- added brief definition of monoidal n-categories
- fixed statement of compatibility of product morphisms with gluing
- added remark about manifolds which do not admit ball decompositions; restricted product theorem (7.1.1) to apply only to these manifolds
+- added remarks about categories of defects
-
+
--- a/text/ncat.tex Sat May 07 08:35:36 2011 -0700
+++ b/text/ncat.tex Sat May 07 09:18:37 2011 -0700
@@ -1755,21 +1755,28 @@
\subsection{The \texorpdfstring{$n{+}1$}{n+1}-category of sphere modules}
\label{ssec:spherecat}
-In this subsection we define $n{+}1$-categories $\cS$ of ``sphere modules"
-whose objects are $n$-categories.
+In this subsection we define $n{+}1$-categories $\cS$ of ``sphere modules".
+The objects are $n$-categories, the $k$-morphisms are $k{-}1$-sphere modules for $1\le k \le n$,
+and the $n{+}1$-morphisms are intertwinors.
With future applications in mind, we treat simultaneously the big category
of all $n$-categories and all sphere modules and also subcategories thereof.
When $n=1$ this is closely related to familiar $2$-categories consisting of
algebras, bimodules and intertwiners (or a subcategory of that).
+The sphere module $n{+}1$-category is a natural generalization of the
+algebra-bimodule-intertwinor 2-category to higher dimensions.
+
+Another possible name for this $n{+}1$-category is $n{+}1$-category of defects.
+The $n$-categories are thought of as representing field theories, and the
+$0$-sphere modules are codimension 1 defects between adjacent theories.
+In general, $m$-sphere modules are codimension $m{+}1$ defects;
+the link of such a defect is an $m$-sphere decorated with defects of smaller codimension.
+
+\medskip
While it is appropriate to call an $S^0$ module a bimodule,
this is much less true for higher dimensional spheres,
so we prefer the term ``sphere module" for the general case.
-%The results of this subsection are not needed for the rest of the paper,
-%so we will skimp on details in a couple of places. We have included this mostly
-%for the sake of comparing our notion of a disk-like $n$-category to other definitions.
-
For simplicity, we will assume that $n$-categories are enriched over $\c$-vector spaces.
The $0$- through $n$-dimensional parts of $\cS$ are various sorts of modules, and we describe
@@ -1783,7 +1790,7 @@
\medskip
-Our first task is to define an $n$-category $m$-sphere module, for $0\le m \le n-1$.
+Our first task is to define an $n$-category $m$-sphere modules, for $0\le m \le n-1$.
These will be defined in terms of certain classes of marked balls, very similarly
to the definition of $n$-category modules above.
(This, in turn, is very similar to our definition of $n$-category.)
@@ -1814,7 +1821,8 @@
Fix $n$-categories $\cA$ and $\cB$.
These will label the two halves of a $0$-marked $k$-ball.
-An $n$-category $0$-sphere module $\cM$ over the $n$-categories $\cA$ and $\cB$ is a collection of functors $\cM_k$ from the category
+An $n$-category $0$-sphere module $\cM$ over the $n$-categories $\cA$ and $\cB$ is
+a collection of functors $\cM_k$ from the category
of $0$-marked $k$-balls, $1\le k \le n$,
(with the two halves labeled by $\cA$ and $\cB$) to the category of sets.
If $k=n$ these sets should be enriched to the extent $\cA$ and $\cB$ are.
@@ -2024,7 +2032,10 @@
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 some effort and combinatorial topology, as well as additional
-duality assumptions on the lower morphisms. These are required because we define the spaces of $n{+}1$-morphisms by making arbitrary choices of incoming and outgoing boundaries for each $n$-ball. The additional duality assumptions are needed to prove independence of our definition form these choices.
+duality assumptions on the lower morphisms.
+These are required because we define the spaces of $n{+}1$-morphisms by
+making arbitrary choices of incoming and outgoing boundaries for each $n$-ball.
+The additional duality assumptions are needed to prove independence of our definition form these choices.
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.