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\nn{This file is obsolete.}
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\todo{beginning of scott's attempt to write down what fields are...}
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\newcommand{\manifolds}[1]{\cM_{#1}}
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\newcommand{\closedManifolds}[1]{\cM_{#1}^{\text{closed}}}
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\newcommand{\boundaryConditions}[1]{\cM_{#1}^{\bdy}}
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Let $\manifolds{k}$ be the groupoid of manifolds (possibly with boundary) of dimension $k$ and diffeomorphisms between them. Write
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$\closedManifolds{k}$ for the subgroupoid of closed manifolds. Taking the boundary gives a functor $\bdy : \manifolds{k} \to \closedManifolds{k-1}$.
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Both $\manifolds{k}$ and $\closedManifolds{k}$ are symmetric tensor categories under the operation of disjoint union.
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\begin{defn}
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A \emph{system of fields} is a collection of functors $\cF_k$ associating a `set of fields' to each manifold of dimension at most $n$.
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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
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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
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on the empty manifold of any dimension.
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Define the groupoid $\boundaryConditions{k}$ of `manifolds with boundary conditions' as
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\begin{equation*}
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\setc{(Y; c)}{\begin{array}{c} \text{$Y$ a $k$-manifold} \\ c \in \cC_{k-1}(\bdy Y) \end{array}}
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\xymatrix{ \ar@(ru,rd)@<-1ex>[]}
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\set{Y \diffeoto Y'}
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\end{equation*}
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where we think of $f: Y \diffeoto Y'$ as a morphism $(Y; c) \isoto (Y'; \cC_{k-1}(\restrict{f}{\bdy Y})(c))$.
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%
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%The objects are pairs $(Y; c)$ with $Y$ a manifold (possibly with boundary) of dimension $k$ and $c \in \cC_{k-1}(\bdy Y)$
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%a field on the boundary of $Y$. A morphism $(Y; c) \to (Y'; c')$ is any diffeomorphism $f: Y \to Y'$ such that $\cC_{k-1}(\restrict{f}{\bdy Y})(c) = c'$.
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Notice that $\closedManifolds{k}$ is naturally a subgroupoid of $\boundaryConditions{k}$, since a closed manifold has a unique field on its (empty) boundary.
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We now ask that the functors $\cF_k$ above extend to functors $\cF_k : \boundaryConditions{k} \to \Set$ for each $0 \leq k < n$,
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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.)
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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.
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\scott{Not sure how to say product fields in this setup.}
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Finally, notice there are functors $- \times I : \manifolds{k} \to \manifolds{k+1}$
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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 $\cC_k($
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\end{defn}
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\begin{rem}
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Where the dimension of the manifold is clear, we'll often leave off the subscript on $\cC_k$.
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\end{rem}
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\todo{end} |