corrected statement of module to category restrictions; note that this affects the numbering of items in subsection 6.4
--- a/blob to-do Wed May 25 09:48:01 2011 -0600
+++ b/blob to-do Wed May 25 11:08:16 2011 -0600
@@ -47,11 +47,6 @@
modules:
-* Lemma 6.4.5 needs to actually construct this map! Needs more input! Do
-we actually need this as written?
- - KW will look at it; probably needs to be weakened
-
-
* SCOTT: typo in delfig3a -- upper g should be g^{-1}
* SCOTT: make sure acknowledge list doesn't omit anyone from blob seminar who should be included (I think I have all the speakers; does anyone other than the speakers rate a mention?)
--- a/blob_changes_v3 Wed May 25 09:48:01 2011 -0600
+++ b/blob_changes_v3 Wed May 25 11:08:16 2011 -0600
@@ -20,9 +20,10 @@
- added remarks about categories of defects
- clarified that the "cell complexes" in string diagrams are actually a bit more general
- added remark to insure that the poset of decompositions is a small category
-- rewrote definition of colimit (in "From Balls to Manifolds" subsection) to allow for more general decompositions; also added more details
--
+- corrected statement of module to category restrictions
+INCOMPLETE:
+- rewrote definition of colimit (in "From Balls to Manifolds" subsection) to allow for more general decompositions; also added more details
--- a/text/ncat.tex Wed May 25 09:48:01 2011 -0600
+++ b/text/ncat.tex Wed May 25 11:08:16 2011 -0600
@@ -1365,13 +1365,23 @@
Let $\cl\cM(H)\trans E$ denote the image of $\gl_E$.
We will refer to elements of $\cl\cM(H)\trans E$ as ``splittable along $E$" or ``transverse to $E$".
+\noop{ %%%%%%%
\begin{lem}[Module to category restrictions]
{For each marked $k$-hemisphere $H$ there is a restriction map
-$\cl\cM(H)\to \cC(H)$.
+$\cl\cM(H)\to \cC(H)$.
($\cC(H)$ means apply $\cC$ to the underlying $k$-ball of $H$.)
These maps comprise a natural transformation of functors.}
\end{lem}
+} %%%%%%% end \noop
+It follows from the definition of the colimit $\cl\cM(H)$ that
+given any (unmarked) $k{-}1$-ball $Y$ in the interior of $H$ there is a restriction map
+from a subset $\cl\cM(H)_{\trans{\bdy Y}}$ of $\cl\cM(H)$ to $\cC(Y)$.
+Combining this with the boundary map $\cM(B,N) \to \cl\cM(\bd(B,N))$, we also have a restriction
+map from a subset $\cM(B,N)_{\trans{\bdy Y}}$ of $\cM(B,N)$ to $\cC(Y)$ whenever $Y$ is in the interior of $\bd B \setmin N$.
+This fact will be used below.
+
+\noop{ %%%%
Note that combining the various boundary and restriction maps above
(for both modules and $n$-categories)
we have for each marked $k$-ball $(B, N)$ and each $k{-}1$-ball $Y\sub \bd B \setmin N$
@@ -1379,6 +1389,7 @@
This subset $\cM(B,N)\trans{\bdy Y}$ is the subset of morphisms which are appropriately splittable (transverse to the
cutting submanifolds).
This fact will be used below.
+} %%%%% end \noop
In our example, the various restriction and gluing maps above come from
restricting and gluing maps into $T$.