...
--- a/text/a_inf_blob.tex Wed Oct 07 18:33:41 2009 +0000
+++ b/text/a_inf_blob.tex Tue Oct 13 21:32:06 2009 +0000
@@ -54,8 +54,8 @@
%This defines a filtration degree 0 element of $\bc_*^\cF(Y)$
We will define $\phi$ using a variant of the method of acyclic models.
-Let $a\in S_m$ be a blob diagram on $Y\times F$.
-For $m$ sufficiently small there exist decompositions of $K$ of $Y$ into $k$-balls such that the
+Let $a\in \cS_m$ be a blob diagram on $Y\times F$.
+For $m$ sufficiently small there exists a decomposition $K$ of $Y$ into $k$-balls such that the
codimension 1 cells of $K\times F$ miss the blobs of $a$, and more generally such that $a$ is splittable along $K\times F$.
Let $D(a)$ denote the subcomplex of $\bc_*^\cF(Y)$ generated by all $(a, \bar{K})$
such that each $K_i$ has the aforementioned splittable property
--- a/text/ncat.tex Wed Oct 07 18:33:41 2009 +0000
+++ b/text/ncat.tex Tue Oct 13 21:32:06 2009 +0000
@@ -218,7 +218,7 @@
\[
(a\times D')\bullet(a\times D'') = a\times (D'\bullet D'') .
\]
-\nn{problem: if pinched boundary, then only one factor}
+\nn{if pinched boundary, then remove first case above}
Product morphisms are associative:
\[
(a\times D)\times D' = a\times (D\times D') .
@@ -234,6 +234,14 @@
\nn{need even more subaxioms for product morphisms?}
+\nn{Almost certainly we need a little more than the above axiom.
+More specifically, in order to bootstrap our way from the top dimension
+properties of identity morphisms to low dimensions, we need regular products,
+pinched products and even half-pinched products.
+I'm not sure what the best way to cleanly axiomatize the properties of these various is.
+For the moment, I'll assume that all flavors of the product are at
+our disposal, and I'll plan on revising the axioms later.}
+
All of the axioms listed above hold for both ordinary $n$-categories and $A_\infty$ $n$-categories.
The last axiom (below), concerning actions of
homeomorphisms in the top dimension $n$, distinguishes the two cases.
@@ -260,6 +268,8 @@
Let $X$ be an $n$-ball and $Y\sub\bd X$ be an $n{-}1$-ball.
Let $J$ be a 1-ball (interval).
We have a collaring homeomorphism $s_{Y,J}: X\cup_Y (Y\times J) \to X$.
+(Here we use the ``pinched" version of $Y\times J$.
+\nn{need notation for this})
We define a map
\begin{eqnarray*}
\psi_{Y,J}: \cC(X) &\to& \cC(X) \\
@@ -873,7 +883,7 @@
\item say something about unoriented vs oriented vs spin vs pin for $n=1$ (and $n=2$?)
\item spell out what difference (if any) Top vs PL vs Smooth makes
\item explain relation between old-fashioned blob homology and new-fangled blob homology
-(follows as special case of product formula (product with a point).
+(follows as special case of product formula (product with a point)).
\item define $n{+}1$-cat of $n$-cats (a.k.a.\ $n{+}1$-category of generalized bimodules
a.k.a.\ $n{+}1$-category of sphere modules); discuss Morita equivalence
\end{itemize}