# HG changeset patch # User kevin@6e1638ff-ae45-0410-89bd-df963105f760 # Date 1248891798 0 # Node ID 631a082cd21bfd1b282445b37eb508fa198d64e2 # Parent 60bb1039be5096e3346554da46cf32a95b00b209 ... diff -r 60bb1039be50 -r 631a082cd21b text/ncat.tex --- a/text/ncat.tex Tue Jul 28 18:52:39 2009 +0000 +++ b/text/ncat.tex Wed Jul 29 18:23:18 2009 +0000 @@ -8,6 +8,8 @@ \nn{experimental section. maybe this should be rolled into other sections. maybe it should be split off into a separate paper.} +\subsection{Definition of $n$-categories} + Before proceeding, we need more appropriate definitions of $n$-categories, $A_\infty$ $n$-categories, modules for these, and tensor products of these modules. (As is the case throughout this paper, by ``$n$-category" we mean @@ -24,7 +26,8 @@ Still other definitions \nn{need refs for all these; maybe the Leinster book} model the $k$-morphisms on more complicated combinatorial polyhedra. -We will allow our $k$-morphisms to have any shape, so long as it is homeomorphic to a $k$-ball. +We will allow our $k$-morphisms to have any shape, so long as it is homeomorphic to +the standard $k$-ball. In other words, \xxpar{Morphisms (preliminary version):} @@ -351,7 +354,66 @@ \end{itemize} -\medskip + + + + + +\subsection{From $n$-categories to systems of fields} + +We can extend the functors $\cC$ above from $k$-balls to arbitrary $k$-manifolds as follows. + +Let $W$ be a $k$-manifold, $1\le k \le n$. +We will define a set $\cC(W)$. +(If $k = n$ and our $k$-categories are enriched, then +$\cC(W)$ will have additional structure; see below.) +$\cC(W)$ will be the colimit of a functor defined on a category $\cJ(W)$, +which we define next. + +Define a permissible decomposition of $W$ to be a decomposition +\[ + W = \bigcup_a X_a , +\] +where each $X_a$ is a $k$-ball. +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 $\cJ(W)$, which we will think of as a category. +(The objects of $\cJ(W)$ are permissible decompositions of $W$, and there is a unique +morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.) +\nn{need figures} + +$\cC$ determines +a functor $\psi_\cC$ from $\cJ(W)$ to the category of sets +(possibly with additional structure if $k=n$). +For a decomposition $x = (X_a)$ in $\cJ(W)$, define $\psi_\cC(x)$ to be the subset +\[ + \psi_\cC(x) \sub \prod_a \cC(X_a) +\] +such that the restrictions to the various pieces of shared boundaries amongst the +$X_a$ all agree. +(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 $\cC$. + +Finally, define $\cC(W)$ to be the colimit of $\psi_\cC$. +In other words, 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. + +$\cC(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. + +It is easy to see that +there are well-defined maps $\cC(W)\to\cC(\bd W)$, and that these maps +comprise a natural transformation of functors. + +\nn{need to finish explaining why we have a system of fields; +need to say more about ``homological" fields? +(actions of homeomorphisms); +define $k$-cat $\cC(\cdot\times W)$} + + + +\subsection{Modules} Next we define [$A_\infty$] $n$-category modules (a.k.a.\ representations, a.k.a.\ actions). @@ -559,13 +621,63 @@ In all other cases ($k>1$ or unoriented or $\text{Pin}_\pm$), there is no left/right module distinction. -\medskip + +\subsection{Modules as boundary labels} + +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 $W$. + +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 +$\cC(W, \cN)$ will have additional structure; see below.) + +Define a permissible decomposition of $W$ to be a decomposition +\[ + W = (\bigcup_a X_a) \cup (\bigcup_{i,b} M_{ib}) , +\] +where each $X_a$ is a plain $k$-ball (disjoint from $\bd W$) and +each $M_{ib}$ is a marked $k$-ball intersecting $\bd_i W$, +with $M_{ib}\cap\bd_i W$ being the marking. +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 $\cJ(W)$, which we will think of as a category. +(The objects of $\cJ(D)$ are permissible decompositions of $W$, and there is a unique +morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.) +\nn{need figures} + +$\cN$ determines +a functor $\psi_\cN$ from $\cJ(W)$ to the category of sets +(possibly with additional structure if $k=n$). +For a decomposition $x = (X_a, M_{ib})$ in $\cJ(W)$, define $\psi_\cN(x)$ to be the subset +\[ + \psi_\cN(x) \sub (\prod_a \cC(X_a)) \prod (\prod_{ib} \cN_i(M_{ib})) +\] +such that the restrictions to the various pieces of shared boundaries amongst the +$X_a$ and $M_{ib}$ all agree. +(Think fibered product.) +If $x$ is a refinement of $y$, define a map $\psi_\cN(x)\to\psi_\cN(y)$ +via the gluing (composition or action) maps from $\cC$ and the $\cN_i$. + +Finally, define $\cC(W, \cN)$ to be the colimit of $\psi_\cN$. +In other words, for each decomposition $x$ there is a map +$\psi(x)\to \cC(W, \cN)$, these maps are compatible with the refinement maps +above, and $\cC(W, \cN)$ is universal with respect to these properties. + +\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). +\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 a left module and the other a right module.)