# HG changeset patch # User kevin@6e1638ff-ae45-0410-89bd-df963105f760 # Date 1249769578 0 # Node ID 65b291b5e8c8f65bd501c7d0d9c2bd8746128409 # Parent 631a082cd21bfd1b282445b37eb508fa198d64e2 ... diff -r 631a082cd21b -r 65b291b5e8c8 text/ncat.tex --- a/text/ncat.tex Wed Jul 29 18:23:18 2009 +0000 +++ b/text/ncat.tex Sat Aug 08 22:12:58 2009 +0000 @@ -33,7 +33,7 @@ \xxpar{Morphisms (preliminary version):} {For any $k$-manifold $X$ homeomorphic to the standard $k$-ball, we have a set of $k$-morphisms -$\cC(X)$.} +$\cC_k(X)$.} Terminology: By ``a $k$-ball" we mean any $k$-manifold which is homeomorphic to the standard $k$-ball. @@ -41,8 +41,11 @@ preferred homeomorphism to the standard $k$-ball. The same goes for ``a $k$-sphere" below. -Given a homeomorphism $f:X\to Y$ between $k$-balls, we want a corresponding + +Given a homeomorphism $f:X\to Y$ between $k$-balls (not necessarily fixed on +the boundary), we want a corresponding bijection of sets $f:\cC(X)\to \cC(Y)$. +(This will imply ``strong duality", among other things.) So we replace the above with \xxpar{Morphisms:} @@ -55,6 +58,7 @@ We are being deliberately vague about what flavor of manifolds we are considering. They could be unoriented or oriented or Spin or $\mbox{Pin}_\pm$. They could be topological or PL or smooth. +\nn{need to check whether this makes much difference --- see pseudo-isotopy below} (If smooth, ``homeomorphism" should be read ``diffeomorphism", and we would need to be fussier about corners.) For each flavor of manifold there is a corresponding flavor of $n$-category. @@ -64,7 +68,8 @@ of morphisms). The 0-sphere is unusual among spheres in that it is disconnected. Correspondingly, for 1-morphisms it makes sense to distinguish between domain and range. -(Actually, this is only true in the oriented case.) +(Actually, this is only true in the oriented case, with 1-morphsims parameterized +by oriented 1-balls.) For $k>1$ and in the presence of strong duality the domain/range division makes less sense. \nn{maybe say more here; rotate disk, Frobenius reciprocity blah blah} We prefer to combine the domain and range into a single entity which we call the @@ -85,7 +90,7 @@ (Note that the first ``$\bd$" above is part of the data for the category, while the second is the ordinary boundary of manifolds.) -Given $c\in\cC(\bd(X))$, let $\cC(X; c) = \bd^{-1}(c)$. +Given $c\in\cC(\bd(X))$, let $\cC(X; c) \deq \bd^{-1}(c)$. Most of the examples of $n$-categories we are interested in are enriched in the following sense. The various sets of $n$-morphisms $\cC(X; c)$, for all $n$-balls $X$ and @@ -97,8 +102,11 @@ $\cC(Y; c)$ is just a plain set if $\dim(Y) < n$. \medskip -\nn{At the moment I'm a little confused about orientations, and more specifically -about the role of orientation-reversing maps of boundaries when gluing oriented manifolds. +\nn{ +%At the moment I'm a little confused about orientations, and more specifically +%about the role of orientation-reversing maps of boundaries when gluing oriented manifolds. +Maybe need a discussion about what the boundary of a manifold with a +structure (e.g. orientation) means. Tentatively, I think we need to redefine the oriented boundary of an oriented $n$-manifold. Instead of an ordinary oriented $(n-1)$-manifold via the inward (or outward) normal first (or last) convention, perhaps it is better to define the boundary to be an $(n-1)$-manifold @@ -125,7 +133,10 @@ which is natural with respect to the actions of homeomorphisms.} Note that we insist on injectivity above. + 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 domain and range maps, relative to the choice of splitting $S = B_1 \cup_E B_2$. @@ -158,6 +169,8 @@ \xxpar{Strict associativity:} {The composition (gluing) maps above are strictly associative.} +Notation: $a\bullet b \deq \gl_Y(a, b)$ and/or $a\cup b \deq \gl_Y(a, b)$. + The above two axioms are equivalent to the following axiom, which we state in slightly vague form. @@ -179,9 +192,29 @@ X\times D \ar[r]^{\tilde{f}} \ar[d]_{\pi} & X'\times D' \ar[d]^{\pi} \\ X \ar[r]^{f} & X' } \] -commutes, then we have $\tilde{f}(a\times D) = f(a)\times D'$.} +commutes, then we have +\[ + \tilde{f}(a\times D) = f(a)\times D' . +\] +Product morphisms are compatible with gluing (composition) in both factors: +\[ + (a'\times D)\bullet(a''\times D) = (a'\bullet a'')\times D +\] +and +\[ + (a\times D')\bullet(a\times D'') = a\times (D'\bullet D'') . +\] +Product morphisms are associative: +\[ + (a\times D)\times D' = a\times (D\times D') . +\] +(Here we are implicitly using functoriality and the obvious homeomorphism +$(X\times D)\times D' \to X\times(D\times D')$.) +} -\nn{Need to say something about compatibility with gluing (of both $X$ and $D$) above.} +\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.} All of the axioms listed above hold for both ordinary $n$-categories and $A_\infty$ $n$-categories. The last axiom (below), concerning actions of @@ -254,7 +287,7 @@ \nn{restate this with $\Homeo(X\to X')$? what about boundary fixing property?}} We should strengthen the above axiom to apply to families of extended homeomorphisms. -To do this we need to explain extended homeomorphisms form a space. +To do this we need to explain how extended homeomorphisms form a topological space. Roughly, the set of $n{-}1$-balls in the boundary of an $n$-ball has a natural topology, and we can replace the class of all intervals $J$ with intervals contained in $\r$. \nn{need to also say something about collaring homeomorphisms.} @@ -281,6 +314,7 @@ The $n$-category can be thought of as the local part of the fields. Conversely, given an $n$-category we can construct a system of fields via \nn{gluing, or a universal construction} +\nn{see subsection below} \nn{Next, say something about $A_\infty$ $n$-categories and ``homological" systems of fields. @@ -418,6 +452,7 @@ Next we define [$A_\infty$] $n$-category modules (a.k.a.\ representations, 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.} Our motivating example comes from an $(m{-}n{+}1)$-dimensional manifold $W$ with boundary in the context of an $m{+}1$-dimensional TQFT. @@ -628,6 +663,8 @@ and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to each boundary component $\bd_i W$ of $W$. +\nn{need to generalize to labeling codim 0 submanifolds of the boundary} + We will define a set $\cC(W, \cN)$ using a colimit construction similar to above. \nn{give ref} (If $k = n$ and our $k$-categories are enriched, then @@ -681,9 +718,14 @@ Let $\cM$ and $\cM'$ be modules for an $n$-category $\cC$. (If $k=1$ and manifolds are oriented, then one should be a left module and the other a right module.) -We will define an $n{-}1$-category $\cM\ot_\cC\cM'$, which depend (functorially) +We will define an $n{-}1$-category $\cM\ot_\cC\cM'$, which depends (functorially) on a choice of 1-ball (interval) $J$. + + + + + Define a {\it doubly marked $k$-ball} to be a triple $(B, N, N')$, where $B$ is a $k$-ball and $N$ and $N'$ are disjoint $k{-}1$-balls in $\bd B$.