# HG changeset patch # User kevin@6e1638ff-ae45-0410-89bd-df963105f760 # Date 1250486615 0 # Node ID 0df8bde1c8966db540f5aa50eb34cfda893cc242 # Parent ae196d7a310d4ce55d81712e3caf056c5817dd6d ... diff -r ae196d7a310d -r 0df8bde1c896 text/ncat.tex --- a/text/ncat.tex Sat Aug 15 15:47:52 2009 +0000 +++ b/text/ncat.tex Mon Aug 17 05:23:35 2009 +0000 @@ -444,14 +444,48 @@ (Think fibered product.) If $x$ is a refinement of $y$, define a map $\psi_\cC(x)\to\psi_\cC(y)$ via the composition maps of $\cC$. +(If $\dim(W) = n$ then we need to also make use of the monoidal +product in the enriching category. +\nn{should probably be more explicit here}) Finally, define $\cC(W)$ to be the colimit of $\psi_\cC$. -In other words, for each decomposition $x$ there is a map +In the plain (non-$A_\infty$) case, this means that +for each decomposition $x$ there is a map $\psi_\cC(x)\to \cC(W)$, these maps are compatible with the refinement maps above, and $\cC(W)$ is universal with respect to these properties. -\nn{in A-inf case, need to say more} +In the $A_\infty$ case, it means +\nn{.... need to check if there is a def in the literature before writing this down} + +More concretely, in the plain case enriched over vector spaces, and with $\dim(W) = n$, we can take +\[ + \cC(W) = \left( \oplus_x \psi_\cC(x)\right) \big/ K +\] +where $K$ is generated by all things of the form $a - g(a)$, where +$a\in \psi_\cC(x)$ for some decomposition $x$, $x\le y$, and $g: \psi_\cC(x) +\to \psi_\cC(y)$ is value of $\psi_\cC$ on the antirefinement $x\to y$. -\nn{should give more concrete description (two cases)} +In the $A_\infty$ case enriched over chain complexes, the concrete description of the colimit +is as follows. +Define an $m$-sequence to be a sequence $x_0 \le x_1 \le \dots \le x_{m-1}$ of permissible decompositions. +Such sequences (for all $m$) form a simplicial set. +Let +\[ + V = \bigoplus_{(x_i)} \psi_\cC(x_0) , +\] +where the sum is over all $m$-sequences and all $m$. +We endow $V$ with a differential which is the sum of the differential of the $\psi_\cC(x_0)$ +summands plus another term using the differential of the simplicial set of $m$-sequences. +More specifically, if $(a, \bar{x})$ denotes an element in the $\bar{x}$ +summand of $V$ (with $\bar{x} = (x_0,\dots,x_k)$), define +\[ + \bd (a, \bar{x}) = (\bd a, \bar{x}) \pm (g(a), d_0(\bar{x})) + \sum_{j=1}^k \pm (a, d_j(\bar{x})) , +\] +where $d_j(\bar{x}) = (x_0,\dots,x_{j-1},x_{j+1},\dots,x_k)$ and $g: \psi_\cC(x_0)\to \psi_\cC(x_1)$ +is the usual map. +\nn{need to say this better} +\nn{maybe mention that there is a version that emphasizes minimal gluings (antirefinements) which +combine only two balls at a time; for $n=1$ this version will lead to usual definition +of $A_\infty$ category} $\cC(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. @@ -692,13 +726,12 @@ \subsection{Modules as boundary labels} +\label{moddecss} Let $\cC$ be an [$A_\infty$] $n$-category, let $W$ be a $k$-manifold ($k\le n$), 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 @@ -736,18 +769,21 @@ $\psi(x)\to \cC(W, \cN)$, these maps are compatible with the refinement maps above, and $\cC(W, \cN)$ is universal with respect to these properties. +More generally, each $\cN_i$ could label some codimension zero submanifold of $\bd W$. +\nn{need to say more?} + \nn{boundary restrictions, $k$-cat $\cC(\cdot\times W; N)$ etc.} \subsection{Tensor products} -Next we consider tensor products (or, more generally, self tensor products -or coends). +Next we consider tensor products. + +\nn{what about self tensor products /coends ?} \nn{maybe ``tensor product" is not the best name?} \nn{start with (less general) tensor products; maybe change this later} -** \nn{stuff below needs to be rewritten (shortened), because of new subsections above} Let $\cM$ and $\cM'$ be modules for an $n$-category $\cC$. (If $k=1$ and manifolds are oriented, then one should be @@ -755,97 +791,17 @@ 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 $\bd B$. - -Let $D = (B, N, N')$ be a doubly marked $k$-ball, $1\le k \le n$. -We will define a set $\cM\ot_\cC\cM'(D)$. -(If $k = n$ and our $k$-categories are enriched, then -$\cM\ot_\cC\cM'(D)$ will have additional structure; see below.) -$\cM\ot_\cC\cM'(D)$ will be the colimit of a functor defined on a category $\cJ(D)$, -which we define next. - -Define a permissible decomposition of $D$ to be a decomposition +Let $p$ and $p'$ be the boundary points of $J$. +Given a $k$-ball $X$, let $(X\times J, \cM, \cM')$ denote $X\times J$ with +$X\times\{p\}$ labeled by $\cM$ and $X\times\{p'\}$ labeled by $\cM'$, as in Subsection \ref{moddecss}. +Let \[ - D = (\cup_a X_a) \cup (\cup_b M_b) \cup (\cup_c M'_c) , + \cT(X) \deq \cC(X\times J, \cM, \cM') , \] -Where each $X_a$ is a plain $k$-ball (disjoint from the markings $N$ and $N'$ of $D$), -each $M_b$ is a marked $k$-ball intersecting $N$, and -each $M'_b$ is a marked $k$-ball intersecting $N'$. -Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement -of $y$, or write $x \le y$, if each ball of $y$ is a union of balls of $x$. -This defines a partial ordering $\cJ(D)$, which we will think of as a category. -(The objects of $\cJ(D)$ are permissible decompositions of $D$, and there is a unique -morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.) -\nn{need figures} - -$\cC$, $\cM$ and $\cM'$ determine -a functor $\psi$ from $\cJ(D)$ to the category of sets -(possibly with additional structure if $k=n$). -For a decomposition $x = (X_a, M_b, M'_c)$ in $\cJ(D)$, define $\psi(x)$ to be the subset -\[ - \psi(x) \sub (\prod_a \cC(X_a)) \times (\prod_b \cM(M_b)) \times (\prod_c \cM'(M'_c)) -\] -such that the restrictions to the various pieces of shared boundaries amongst the -$X_a$, $M_b$ and $M'_c$ all agree. -(Think fibered product.) -If $x$ is a refinement of $y$, define a map $\psi(x)\to\psi(y)$ -via the gluing (composition or action) maps from $\cC$, $\cM$ and $\cM'$. +where the right hand side is the colimit construction defined in Subsection \ref{moddecss}. +It is not hard to see that $\cT$ becomes an $n{-}1$-category. +\nn{maybe follows from stuff (not yet written) in previous subsection?} -Finally, define $\cM\ot_\cC\cM'(D)$ to be the colimit of $\psi$. -In other words, for each decomposition $x$ there is a map -$\psi(x)\to \cM\ot_\cC\cM'(D)$, these maps are compatible with the refinement maps -above, and $\cM\ot_\cC\cM'(D)$ is universal with respect to these properties. - -Define a {\it marked $k$-annulus} to be a manifold homeomorphic -to $S^{k-1}\times I$, with its entire boundary ``marked". -Define the boundary of a doubly marked $k$-ball $(B, N, N')$ to be the marked -$k{-}1$-annulus $\bd B \setmin(N\cup N')$. - -Using a colimit construction similar to the one above, we can define a set -$\cM\ot_\cC\cM'(A)$ for any marked $k$-annulus $A$ (for $k < n$). - -$\cM\ot_\cC\cM'$ is (among other things) a functor from the category of -doubly marked $k$-balls ($k\le n$) and homeomorphisms to the category of sets. -We have other functors, also denoted $\cM\ot_\cC\cM'$, from the category of -marked $k$-annuli ($k < n$) and homeomorphisms to the category of sets. - -For each marked $k$-ball $D$ there is a restriction map -\[ - \bd : \cM\ot_\cC\cM(D) \to \cM\ot_\cC\cM(\bd D) . -\] -These maps comprise a natural transformation of functors. -\nn{possible small problem: might need to define $\cM$ of a singly marked annulus} - -For $c \in \cM\ot_\cC\cM(\bd D)$, let -\[ - \cM\ot_\cC\cM(D; c) \deq \bd\inv(c) . -\] - -Note that if $k=n$ and we fix boundary conditions $c$ on the unmarked boundary of $D$, -then $\cM\ot_\cC\cM'(D; c)$ will be an object in the enriching category -(e.g.\ vector space or chain complex). - -Let $J$ be a doubly marked 1-ball (i.e. an interval, where we think of both endpoints -as marked). -For $X$ a plain $k$-ball ($k \le n-1$) or $k$-sphere ($k \le n-2$), define -\[ - \cM\ot_\cC\cM'(X) \deq \cM\ot_\cC\cM'(X\times J) . -\] -We claim that $\cM\ot_\cC\cM'$ has the structure of an $n{-}1$-category. -We have already defined restriction maps $\bd : \cM\ot_\cC\cM'(X) \to -\cM\ot_\cC\cM'(\bd X)$. -The only data for the $n{-}1$-category that we have not defined yet are the product -morphisms. -\nn{so next define those} - -\nn{need to check whether any of the steps in verifying that we have -an $n{-}1$-category are non-trivial.}