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