--- a/blob to-do Fri May 13 20:52:18 2011 -0700
+++ b/blob to-do Fri May 13 21:01:03 2011 -0700
@@ -15,9 +15,6 @@
* Consider moving A_\infty stuff to a subsection
-* (?) dimension n+1, explain the statement and refer to KW's notes. [this was PT's suggestion, but it's sort of already in there. do we need to do more?]
- - add one more sentence: ~"finite dimensional and pos. def"
-
* framings and duality -- work out what's going on! (alternatively, vague-ify current statement)
* consider proving the gluing formula for higher codimension manifolds with
@@ -56,7 +53,7 @@
* SCOTT: typo in delfig3a -- upper g should be g^{-1}
-* SCOTT: make sure acknowledge list doesn't omit anyone from blob seminar (I think I have all the speakers)
+* 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 that the speakers rate a mention?)
* review colors in figures
@@ -67,7 +64,8 @@
* lemma [inject 6.3.5?] assumes more splittablity than the axioms imply (?)
+* consider putting conditions for enriched n-cat all in one place
+
* SCOTT: figure for example 3.1.2 (sin 1/z)
* SCOTT: add vertical arrow to middle of figure 19 (decomp poset)
-
--- a/blob_changes_v3 Fri May 13 20:52:18 2011 -0700
+++ b/blob_changes_v3 Fri May 13 21:01:03 2011 -0700
@@ -5,11 +5,12 @@
Also many typos corrected.
+
The most significant changes are:
- added to acknowledgements
- clarified definition of splittable
-- change to pitchfork notation for splittable subsets of fields
+- changed to pitchfork notation for splittable subsets of fields
- added definition of collaring homeomorphism
- improved definition of bordism n-category
- fixed definition of a refinement of a ball decomposition (intermediate manifolds should also be disjoint unions of balls)
--- a/text/ncat.tex Fri May 13 20:52:18 2011 -0700
+++ b/text/ncat.tex Fri May 13 21:01:03 2011 -0700
@@ -1342,7 +1342,10 @@
\caption{From manifold with boundary collar to marked ball}\label{blah15}\end{figure}
Define the boundary of a marked $k$-ball $(B, N)$ to be the pair $(\bd B \setmin N, \bd N)$.
-Call such a thing a {marked $k{-}1$-hemisphere}.
+Call such a thing a {\it marked $k{-}1$-hemisphere}.
+(A marked $k{-}1$-hemisphere is, of course, just a $k{-}1$-ball with its entire boundary marked.
+We call it a hemisphere instead of a ball because it plays a role analogous
+to the $k{-}1$-spheres in the $n$-category definition.)
\begin{lem}
\label{lem:hemispheres}
--- a/text/tqftreview.tex Fri May 13 20:52:18 2011 -0700
+++ b/text/tqftreview.tex Fri May 13 21:01:03 2011 -0700
@@ -444,11 +444,13 @@
The construction of the $n{+}1$-dimensional part of the theory (the path integral)
requires that the starting data (fields and local relations) satisfy additional
conditions.
-We do not assume these conditions here, so when we say ``TQFT" we mean a decapitated TQFT
+(Specifically, $A(X; c)$ is finite dimensional for all $n$-manifolds $X$ and the inner products
+on $A(B^n; c)$ induced by the path integral of $B^{n+1}$ are positive definite for all $c$.)
+We do not assume these conditions here, so when we say ``TQFT" we mean a ``decapitated" TQFT
that lacks its $n{+}1$-dimensional part.
-Such a ``decapitated'' TQFT is sometimes also called an $n+\epsilon$ or
-$n+\frac{1}{2}$ dimensional TQFT, referring to the fact that it assigns maps to $n{+}1$-dimensional
-mapping cylinders between $n$-manifolds, but nothing to arbitrary $n{+}1$-manifolds.
+Such a decapitated TQFT is sometimes also called an $n{+}\epsilon$ or
+$n{+}\frac{1}{2}$ dimensional TQFT, referring to the fact that it assigns linear maps to $n{+}1$-dimensional
+mapping cylinders between $n$-manifolds, but nothing to general $n{+}1$-manifolds.
Let $Y$ be an $n{-}1$-manifold.
Define a linear 1-category $A(Y)$ as follows.