--- a/blob to-do Sat May 28 09:49:30 2011 -0600
+++ b/blob to-do Sat May 28 21:45:13 2011 -0600
@@ -1,71 +1,51 @@
-* We need to be clearer about which types of homeomorphisms the
-"localization" theorem in the appendix works for, in the body of the
-paper. Options here include:
-a) having a better theorem in a separate paper, so we don't actually
-need to worry
-[** currently working on this option]
-b) changing the statements in the paper, for example writing PL-Homeo
-everywhere instead of Homeo
-c) explicitly saying "Homeo means PL-Homeo" everywhere
-c') if we succumb to Peter's suggestion of say "Iso" everywhere,
-perhaps we could adopt the notation that "Iso^*" or similar means one
-of a restricted set of categories, where the appendix works, and using
-this notation in section 5.
+* extend localization lemma to (topological) homeos
+
+* lemma [inject 6.3.5?] assumes more splittablity than the axioms imply (?)
+
* Consider moving A_\infty stuff to a subsection
+* consider putting conditions for enriched n-cat all in one place
+
+* Peter's suggestion for A_inf definition
+
+* Boundary of colimit -- not so easy to see!
+
+* ** new material in colimit section needs a proof-read
+
+* In the appendix on n=1, explain more about orientations. Also say
+what happens on objects for spin manifolds: the unique point has an
+automorphism, which translates into a involution on objects. Mention
+super-stuff. [partly done]
+
+
+* should probably allow product things \pi^*(b) to be defined only when b is appropriately splittable
+
* framings and duality -- work out what's going on! (alternatively, vague-ify current statement)
+* make sure we are clear that boundary = germ
+
+* review colors in figures
+
+* maybe say something in colimit section about restriction to submanifolds and submanifolds of boundary (we use this in n-cat axioms)
+
+
+* ? define Morita equivalence?
+
* consider proving the gluing formula for higher codimension manifolds with
morita equivalence
-* Peter's suggestion for A_inf definition
-
-* enriching in other \infty categories, explaining how "D" should
-interact with coproducts in "S" (break out A_\infty stuff into a
-subsection)
-
* SCOTT will go through appendix C.2 and make it better
-* make sure we are clear that boundary = germ
-
-* In the appendix on n=1, explain more about orientations. Also say
-what happens on objects for spin manifolds: the unique point has an
-automorphism, which translates into a involution on objects. Mention
-super-stuff.
-
-
-colimit subsection:
-
-* Boundary of \cl; not so easy to see!
-
-* new material in colimit section needs a proof-read
-
-
-modules:
-
* 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?)
-
-* review colors in figures
-
-* ? define Morita equivalence?
-
-* 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)
-* maybe say something in colimit section about restriction to submanifolds and submanifolds of boundary
-
* SCOTT: review/proof-read recent KW changes
-
-* should probably allow product things \pi^*(b) to be defined only when b is appropriately splittable
\ No newline at end of file
--- a/blob_changes_v3 Sat May 28 09:49:30 2011 -0600
+++ b/blob_changes_v3 Sat May 28 21:45:13 2011 -0600
@@ -22,6 +22,8 @@
- added remark to insure that the poset of decompositions is a small category
- corrected statement of module to category restrictions
- reduced intermingling for the various n-cat definitions (plain, enriched, A-infinity)
+- strengthened n-cat isotopy invariance axiom to allow for homeomorphisms which act trivially elements on the restriction of an n-morphism to the boundary of the ball
+- more details on axioms for enriched n-cats
-
--- a/text/appendixes/comparing_defs.tex Sat May 28 09:49:30 2011 -0600
+++ b/text/appendixes/comparing_defs.tex Sat May 28 21:45:13 2011 -0600
@@ -43,28 +43,40 @@
Given $a\in c(\cX)^0$, define $\id_a \deq a\times B^1$.
By extended isotopy invariance in $\cX$, this has the expected properties of an identity morphism.
+We have now defined the basic ingredients for the 1-category $c(\cX)$.
+As we explain below, $c(\cX)$ might have additional structure corresponding to the
+unoriented, oriented, Spin, $\text{Pin}_+$ or $\text{Pin}_-$ structure on the 1-balls used to define $\cX$.
-If the underlying manifolds for $\cX$ have further geometric structure, then we obtain certain functors.
-The base case is for oriented manifolds, where we obtain no extra algebraic data.
-
-For 1-categories based on unoriented manifolds,
+For 1-categories based on unoriented balls,
there is a map $\dagger:c(\cX)^1\to c(\cX)^1$
coming from $\cX$ applied to an orientation-reversing homeomorphism (unique up to isotopy)
from $B^1$ to itself.
+(Of course our $B^1$ is unoriented, i.e.\ not equipped with an orientation.
+We mean the homeomorphism which would reverse the orientation if there were one;
+$B^1$ is not oriented, but it is orientable.)
Topological properties of this homeomorphism imply that
$a^{\dagger\dagger} = a$ ($\dagger$ is order 2), $\dagger$ reverses domain and range, and $(ab)^\dagger = b^\dagger a^\dagger$
($\dagger$ is an anti-automorphism).
+Recall that in this context 0-balls should be thought of as equipped with a germ of a 1-dimensional neighborhood.
+There is a unique such 0-ball, up to homeomorphism, but it has a non-identity automorphism corresponding to reversing the
+orientation of the germ.
+Consequently, the objects of $c(\cX)$ are equipped with an involution, also denoted $\dagger$.
+If $a:x\to y$ is a morphism of $c(\cX)$ then $a^\dagger: y^\dagger\to x^\dagger$.
-For 1-categories based on Spin manifolds,
+For 1-categories based on oriented balls, there are no non-trivial homeomorphisms of 0- or 1-balls, and thus no
+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
gives an order 2 automorphism of $c(\cX)^1$.
-For 1-categories based on $\text{Pin}_-$ manifolds,
+For 1-categories based on $\text{Pin}_-$ balls,
we have an order 4 antiautomorphism of $c(\cX)^1$.
-For 1-categories based on $\text{Pin}_+$ manifolds,
+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.
-%\nn{need to also consider automorphisms of $B^0$ / objects}
+
+
\noop{
\medskip