# HG changeset patch # User kevin@6e1638ff-ae45-0410-89bd-df963105f760 # Date 1250213407 0 # Node ID a2444aa1ad3137b0b8f72a310032394cebf63208 # Parent 65b291b5e8c8f65bd501c7d0d9c2bd8746128409 ... diff -r 65b291b5e8c8 -r a2444aa1ad31 text/kw_macros.tex --- a/text/kw_macros.tex Sat Aug 08 22:12:58 2009 +0000 +++ b/text/kw_macros.tex Fri Aug 14 01:30:07 2009 +0000 @@ -52,7 +52,7 @@ % \DeclareMathOperator{\pr}{pr} etc. \def\declaremathop#1{\expandafter\DeclareMathOperator\csname #1\endcsname{#1}} -\applytolist{declaremathop}{pr}{im}{gl}{ev}{coinv}{tr}{rot}{Eq}{obj}{mor}{ob}{Rep}{Tet}{cat}{Maps}{Diff}{Homeo}{sign}{supp}{Nbd}; +\applytolist{declaremathop}{pr}{im}{gl}{ev}{coinv}{tr}{rot}{Eq}{obj}{mor}{ob}{Rep}{Tet}{cat}{Maps}{Diff}{Homeo}{sign}{supp}{Nbd}{res}; diff -r 65b291b5e8c8 -r a2444aa1ad31 text/ncat.tex --- a/text/ncat.tex Sat Aug 08 22:12:58 2009 +0000 +++ b/text/ncat.tex Fri Aug 14 01:30:07 2009 +0000 @@ -137,7 +137,10 @@ Let $\cC(S)_E$ denote the image of $\gl_E$. We will refer to elements of $\cC(S)_E$ as ``splittable along $E$" or ``transverse to $E$". -We have ``restriction" maps $\cC(S)_E \to \cC(B_i)$, which can be thought of as +We will call the projection $\cC(S)_E \to \cC(B_i)$ +a {\it restriction} map and write $\res_{B_i}(a)$ +(or simply $\res(a)$ when there is no ambiguity), for $a\in \cC(S)_E$. +These restriction maps can be thought of as domain and range maps, relative to the choice of splitting $S = B_1 \cup_E B_2$. If $B$ is a $k$-ball and $E \sub \bd B$ splits $\bd B$ into two $k{-}1$-balls @@ -170,6 +173,15 @@ {The composition (gluing) maps above are strictly associative.} Notation: $a\bullet b \deq \gl_Y(a, b)$ and/or $a\cup b \deq \gl_Y(a, b)$. +In the other direction, we will call the projection from $\cC(B)_E$ to $\cC(B_i)_E$ +a {\it restriction} map and write $\res_{B_i}(a)$ for $a\in \cC(B)_E$. +Compositions of boundary and restriction maps will also be called restriction maps. +For example, if $B$ is a $k$-ball and $Y\sub \bd B$ is a $k{-}1$-ball, there is a +restriction map from $\cC(B)_{\bd Y}$ to $\cC(Y)$. + +%More notation and terminology: +%We will call the projection from $\cC(B)_E$ to $\cC(B_i)_E$ a {\it restriction} +%map The above two axioms are equivalent to the following axiom, which we state in slightly vague form. @@ -210,11 +222,14 @@ \] (Here we are implicitly using functoriality and the obvious homeomorphism $(X\times D)\times D' \to X\times(D\times D')$.) +Product morphisms are compatible with restriction: +\[ + \res_{X\times E}(a\times D) = a\times E +\] +for $E\sub \bd D$ and $a\in \cC(X)$. } -\nn{need even more subaxioms for product morphisms? -YES: need compatibility with certain restriction maps -in order to prove that dimension less than $n$ identities are act like identities.} +\nn{need even more subaxioms for product morphisms?} All of the axioms listed above hold for both ordinary $n$-categories and $A_\infty$ $n$-categories. The last axiom (below), concerning actions of @@ -453,6 +468,7 @@ a.k.a.\ actions). The definition will be very similar to that of $n$-categories. \nn{** need to make sure all revisions of $n$-cat def are also made to module def.} +\nn{should they be called $n$-modules instead of just modules? probably not, but worth considering.} Our motivating example comes from an $(m{-}n{+}1)$-dimensional manifold $W$ with boundary in the context of an $m{+}1$-dimensional TQFT. @@ -480,6 +496,8 @@ Let $\cC$ be the $n$-category with $\cC(X) \deq \cD(X\times \bd W)$. Let $\cM(B, N) \deq \cD((B\times \bd W)\cup (N\times W))$. (The union is along $N\times \bd W$.) +(If $\cD$ were a general TQFT, we would define $\cM(B, N)$ to be +the subset of $\cD((B\times \bd W)\cup (N\times W))$ which is splittable along $N\times \bd W$.) 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}. @@ -495,7 +513,7 @@ {For each marked $k$-ball $M$ we have a map of sets $\bd: \cM(M)\to \cM(\bd M)$. These maps, for various $M$, comprise a natural transformation of functors.} -Given $c\in\cM(\bd M)$, let $\cM(M; c) = \bd^{-1}(c)$. +Given $c\in\cM(\bd M)$, let $\cM(M; c) \deq \bd^{-1}(c)$. If the $n$-category $\cC$ is enriched over some other category (e.g.\ vector spaces), then $\cM(M; c)$ should be an object in that category for each marked $n$-ball $M$ @@ -512,6 +530,10 @@ \] which is natural with respect to the actions of homeomorphisms.} +Let $\cM(H)_E$ denote the image of $\gl_E$. +We will refer to elements of $\cM(H)_E$ as ``splittable along $E$" or ``transverse to $E$". + + \xxpar{Axiom yet to be named:} {For each marked $k$-hemisphere $H$ there is a restriction map $\cM(H)\to \cC(H)$. @@ -519,10 +541,12 @@ These maps comprise a natural transformation of functors.} 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 $Y\sub \bd B \setmin N$ a natural map from a subset of $\cM(B, N)$ to $\cC(Y)$. +The subset is the subset of morphisms which are appropriately splittable (transverse to the +cutting submanifolds). This fact will be used below. -\nn{need to say more about splitableness/transversality in various places above} In our example, the various restriction and gluing maps above come from restricting and gluing maps into $T$. @@ -572,7 +596,11 @@ \xxpar{Module strict associativity:} {The composition and action maps above are strictly associative.} -The above two axioms are equivalent to the following axiom, +Note that the above associativity axiom applies to mixtures of module composition, +action maps and $n$-category composition. +See Figure xxxx. + +The above three axioms are equivalent to the following axiom, which we state in slightly vague form. \nn{need figure for this} @@ -608,6 +636,8 @@ \nn{Need to say something about compatibility with gluing (of both $M$ and $D$) above.} +\nn{** marker --- resume revising here **} + There are two alternatives for the next axiom, according whether we are defining modules for plain $n$-categories or $A_\infty$ $n$-categories. In the plain case we require