adding remarks on objects to C.1; this is pretty uninspired writing, so feel free to improve
--- a/text/appendixes/comparing_defs.tex Sat May 28 21:45:13 2011 -0600
+++ b/text/appendixes/comparing_defs.tex Wed Jun 01 15:04:31 2011 -0600
@@ -67,15 +67,16 @@
additional structure on $c(\cX)$.
For 1-categories based on Spin balls,
-the the nontrivial spin homeomorphism from $B^1$ to itself which covers the identity
+the nontrivial spin homeomorphism from $B^1$ to itself which covers the identity
gives an order 2 automorphism of $c(\cX)^1$.
+There is a similar involution on the objects $c(\cX)^0$.
For 1-categories based on $\text{Pin}_-$ balls,
we have an order 4 antiautomorphism of $c(\cX)^1$.
For 1-categories based on $\text{Pin}_+$ balls,
we have an order 2 antiautomorphism and also an order 2 automorphism of $c(\cX)^1$,
and these two maps commute with each other.
-
+In both cases there is a similar map on objects.
\noop{