diff -r ec8587c33c0b -r c5a33223af00 text/appendixes/comparing_defs.tex
--- 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{