# HG changeset patch # User Kevin Walker # Date 1306640713 21600 # Node ID ec8587c33c0bffbab801362008eaaea8724ef38f # Parent 787914e9e8598621fac737c215d4f5b1b732f448 more details in C.1; reorganized to-do list diff -r 787914e9e859 -r ec8587c33c0b blob to-do --- 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 diff -r 787914e9e859 -r ec8587c33c0b blob_changes_v3 --- 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 - diff -r 787914e9e859 -r ec8587c33c0b text/appendixes/comparing_defs.tex --- 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