1 \nn{This file is obsolete.} |
|
2 |
|
3 \todo{beginning of scott's attempt to write down what fields are...} |
|
4 |
|
5 \newcommand{\manifolds}[1]{\cM_{#1}} |
|
6 \newcommand{\closedManifolds}[1]{\cM_{#1}^{\text{closed}}} |
|
7 \newcommand{\boundaryConditions}[1]{\cM_{#1}^{\bdy}} |
|
8 Let $\manifolds{k}$ be the groupoid of manifolds (possibly with boundary) of dimension $k$ and diffeomorphisms between them. Write |
|
9 $\closedManifolds{k}$ for the subgroupoid of closed manifolds. Taking the boundary gives a functor $\bdy : \manifolds{k} \to \closedManifolds{k-1}$. |
|
10 Both $\manifolds{k}$ and $\closedManifolds{k}$ are symmetric tensor categories under the operation of disjoint union. |
|
11 \begin{defn} |
|
12 A \emph{system of fields} is a collection of functors $\cF_k$ associating a `set of fields' to each manifold of dimension at most $n$. |
|
13 |
|
14 First, there are functors $\cF_k : \closedManifolds{k} \to \Set$ for each $0 \leq k < n$. We ask that these are tensor functors, so they |
|
15 take disjoint unions of manifolds to cartesian products of sets. In particular, this means that $\cF_k(\eset)$ is a point; there's only one field |
|
16 on the empty manifold of any dimension. |
|
17 |
|
18 Define the groupoid $\boundaryConditions{k}$ of `manifolds with boundary conditions' as |
|
19 \begin{equation*} |
|
20 \setc{(Y; c)}{\begin{array}{c} \text{$Y$ a $k$-manifold} \\ c \in \cC_{k-1}(\bdy Y) \end{array}} |
|
21 \xymatrix{ \ar@(ru,rd)@<-1ex>[]} |
|
22 \set{Y \diffeoto Y'} |
|
23 \end{equation*} |
|
24 where we think of $f: Y \diffeoto Y'$ as a morphism $(Y; c) \isoto (Y'; \cC_{k-1}(\restrict{f}{\bdy Y})(c))$. |
|
25 % |
|
26 %The objects are pairs $(Y; c)$ with $Y$ a manifold (possibly with boundary) of dimension $k$ and $c \in \cC_{k-1}(\bdy Y)$ |
|
27 %a field on the boundary of $Y$. A morphism $(Y; c) \to (Y'; c')$ is any diffeomorphism $f: Y \to Y'$ such that $\cC_{k-1}(\restrict{f}{\bdy Y})(c) = c'$. |
|
28 Notice that $\closedManifolds{k}$ is naturally a subgroupoid of $\boundaryConditions{k}$, since a closed manifold has a unique field on its (empty) boundary. |
|
29 |
|
30 We now ask that the functors $\cF_k$ above extend to functors $\cF_k : \boundaryConditions{k} \to \Set$ for each $0 \leq k < n$, |
|
31 and that there is an extra functor at the top level, $\cF_n : \boundaryConditions{n} \to \Vect$. (Notice that for $n$-manifolds we ask for a vector space, not just a set. This isn't essential for the definition, but we will only be interested in this case hereafter.) |
|
32 We still require that these are tensor functors, and so take disjoint unions of manifolds to cartesian products of sets, or tensor products of vector spaces, as appropriate. |
|
33 |
|
34 \scott{Not sure how to say product fields in this setup.} |
|
35 Finally, notice there are functors $- \times I : \manifolds{k} \to \manifolds{k+1}$ |
|
36 Finally (?) we ask for natural transformations $- \times I : \cC_k \to \cC_{k+1} \compose (- \times I)$. Thus for each pair $(Y^k; c)$ we have a map $\cC_k($ |
|
37 \end{defn} |
|
38 \begin{rem} |
|
39 Where the dimension of the manifold is clear, we'll often leave off the subscript on $\cC_k$. |
|
40 \end{rem} |
|
41 |
|
42 \todo{end} |
|