--- a/blob to-do	Sun Jun 19 17:31:34 2011 -0600
+++ b/blob to-do	Sun Jun 19 21:35:30 2011 -0600
@@ -1,23 +1,23 @@
+
+====== big ======
 
 * need to change module axioms to follow changes in n-cat axioms; search for and destroy all the "Homeo_\bd"'s, add a v-cone axiom
 
+* probably should go through and refer to new splitting axiom when we need to choose refinements etc.
 
 * Boundary of colimit -- not so easy to see!
 
-* ** new material in colimit section needs a proof-read
-
 
 * framings and duality -- work out what's going on! (alternatively, vague-ify current statement)
 
-* make sure we are clear that boundary = germ
-
-* go through text and remove any disclaimers about continuous (as oppsed to PL) homeos
-
-* review colors in figures
+* make sure we are clear that boundary = germ (perhaps we are already clear enough)
 
 * maybe say something in colimit section about restriction to submanifolds and submanifolds of boundary (we use this in n-cat axioms)
 
 
+
+====== minor/optional ======
+
 * ? define Morita equivalence?
 
 * consider proving the gluing formula for higher codimension manifolds with
@@ -28,9 +28,10 @@
 * should we require, for A-inf n-cats, that families which preserve product morphisms act trivially?  as now defined, this is only true up to homotopy for the blob complex, so maybe best not to open that can of worms
 (but since the strict version of this is true for BT_*, maybe we're OK)
 
-* probably should go through and refer to new splitting axiom when we need to choose refinements etc.
+* review colors in figures
 
 
+====== Scott ======
 
 * SCOTT will go through appendix C.2 and make it better
 
@@ -42,4 +43,4 @@
 
 * SCOTT: add vertical arrow to middle of figure 19 (decomp poset)
 
-* SCOTT: review/proof-read recent KW changes
+* SCOTT: review/proof-read recent KW changes, especially colimit section and n-cat axioms

--- a/text/ncat.tex	Sun Jun 19 17:31:34 2011 -0600
+++ b/text/ncat.tex	Sun Jun 19 21:35:30 2011 -0600
@@ -2587,3 +2587,29 @@
 To define (binary) composition of $n{+}1$-morphisms, choose the obvious common equator
 then compose the module maps.
 The proof that this composition rule is associative is similar to the proof of Lemma \ref{equator-lemma}.
+
+\medskip
+
+We end this subsection with some remarks about Morita equivalence of disklike $n$-categories.
+Recall that two 1-categories $C$ and $D$ are Morita equivalent if and only if they are equivalent
+objects in the 2-category of (linear) 1-categories, bimodules, and intertwinors.
+Similarly, we define two disklike $n$-categories to be Morita equivalent if they are equivalent objects in the
+$n{+}1$-category of sphere modules.
+
+Because of the strong duality enjoyed by disklike $n$-categories, the data for such an equivalence lives only in 
+dimensions 1 and $n+1$ (the middle dimensions come along for free), and this data must satisfy
+identities corresponding to Morse cancellations in $n{+}1$-manifolds.
+\noop{ % the following doesn't work; need 2^(k+1) different N's, not 2*(k+1)
+More specifically, the 1-dimensional part of the data is a 0-sphere module $M = {}_CM_D$ 
+(categorified bimodule) connecting $C$ and $D$.
+From $M$ we can construct various $k$-sphere modules $N^k_{j,E}$ for $0 \le k \le n$, $0\le j \le k$, and $E = C$ or $D$.
+$N^k_{j,E}$ can be thought of as the graph of an index $j$ Morse function on the $k$-ball $B^k$
+(so the graph lives in $B^k\times I = B^{k+1}$).
+The positive side of the graph is labeled by $E$, the negative side by $E'$
+(where $C' = D$ and $D' = C$), and the codimension-1 
+submanifold separating the positive and negative regions is labeled by $M$.
+We think of $N^k_{j,E}$ as a $k{+}1$-morphism connecting 
+}
+We plan on treating this in more detail in a future paper.
+\nn{should add a few more details}
+