
diff -r ead6bc1a703f -r 2625a6f51684 text/fields.tex
--- a/text/fields.tex	Thu Nov 20 21:08:30 2008 +0000
+++ b/text/fields.tex	Tue Feb 03 21:06:04 2009 +0000
@@ -1,36 +1,40 @@
-\newcommand{\manifolds}[1]{\cM_{#1}}
-\newcommand{\closedManifolds}[1]{\cM_{#1}^{\text{closed}}}
-\newcommand{\boundaryConditions}[1]{\cM_{#1}^{\bdy}}
-Let $\manifolds{k}$ be the groupoid of manifolds (possibly with boundary) of dimension $k$ and diffeomorphisms between them. Write
-$\closedManifolds{k}$ for the subgroupoid of closed manifolds. Taking the boundary gives a functor $\bdy : \manifolds{k} \to \closedManifolds{k-1}$.
-Both $\manifolds{k}$ and $\closedManifolds{k}$ are symmetric tensor categories under the operation of disjoint union.
-\begin{defn}
-A \emph{system of fields} is a collection of functors associating a `set of fields' to each manifold of dimension at most $n$.
-
-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
-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
-on the empty manifold of any dimension.
-
-Define the groupoid $\boundaryConditions{k}$ of `manifolds with boundary conditions' as
-\begin{equation*}
-\setc{(Y; c)}{\begin{array}{c} \text{$Y$ a $k$-manifold} \\  c \in \cC_{k-1}(\bdy Y) \end{array}}
-\xymatrix{ \ar@(ru,rd)@<-1ex>[]}
-\set{Y \diffeoto Y'}
-\end{equation*}
-where we think of $f: Y \diffeoto Y'$ as a morphism $(Y; c) \isoto (Y'; \cC_{k-1}(\restrict{f}{\bdy Y})(c))$.
-%
-%The objects are pairs $(Y; c)$ with $Y$ a manifold (possibly with boundary) of dimension $k$ and $c \in \cC_{k-1}(\bdy Y)$
-%a field on the boundary of $Y$. A morphism $(Y; c) \to (Y'; c')$ is any diffeomorphism $f: Y \to Y'$ such that $\cC_{k-1}(\restrict{f}{\bdy Y})(c) = c'$.
-Notice that $\closedManifolds{k}$ is naturally a subgroupoid of $\boundaryConditions{k}$, since a closed manifold has a unique field on its (empty) boundary.
-
-We now ask that the functors $\cC_k$ above extend to functors $\cC_k : \boundaryConditions{k} \to \Set$ for  each $0 \leq k < n$,
-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.)
-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.
-
-\scott{Not sure how to say product fields in this setup.}
-Finally, notice there are functors $- \times I : \manifolds{k} \to \manifolds{k+1}$
-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 $\cC_k($
-\end{defn}
-\begin{rem}
-Where the dimension of the manifold is clear, we'll often leave off the subscript on $\cC_k$.
-\end{rem}
+\todo{beginning of scott's attempt to write down what fields are...}
+
+\newcommand{\manifolds}[1]{\cM_{#1}}
+\newcommand{\closedManifolds}[1]{\cM_{#1}^{\text{closed}}}
+\newcommand{\boundaryConditions}[1]{\cM_{#1}^{\bdy}}
+Let $\manifolds{k}$ be the groupoid of manifolds (possibly with boundary) of dimension $k$ and diffeomorphisms between them. Write
+$\closedManifolds{k}$ for the subgroupoid of closed manifolds. Taking the boundary gives a functor $\bdy : \manifolds{k} \to \closedManifolds{k-1}$.
+Both $\manifolds{k}$ and $\closedManifolds{k}$ are symmetric tensor categories under the operation of disjoint union.
+\begin{defn}
+A \emph{system of fields} is a collection of functors $\cF_k$ associating a `set of fields' to each manifold of dimension at most $n$.
+
+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
+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
+on the empty manifold of any dimension.
+
+Define the groupoid $\boundaryConditions{k}$ of `manifolds with boundary conditions' as
+\begin{equation*}
+\setc{(Y; c)}{\begin{array}{c} \text{$Y$ a $k$-manifold} \\  c \in \cC_{k-1}(\bdy Y) \end{array}}
+\xymatrix{ \ar@(ru,rd)@<-1ex>[]}
+\set{Y \diffeoto Y'}
+\end{equation*}
+where we think of $f: Y \diffeoto Y'$ as a morphism $(Y; c) \isoto (Y'; \cC_{k-1}(\restrict{f}{\bdy Y})(c))$.
+%
+%The objects are pairs $(Y; c)$ with $Y$ a manifold (possibly with boundary) of dimension $k$ and $c \in \cC_{k-1}(\bdy Y)$
+%a field on the boundary of $Y$. A morphism $(Y; c) \to (Y'; c')$ is any diffeomorphism $f: Y \to Y'$ such that $\cC_{k-1}(\restrict{f}{\bdy Y})(c) = c'$.
+Notice that $\closedManifolds{k}$ is naturally a subgroupoid of $\boundaryConditions{k}$, since a closed manifold has a unique field on its (empty) boundary.
+
+We now ask that the functors $\cF_k$ above extend to functors $\cF_k : \boundaryConditions{k} \to \Set$ for  each $0 \leq k < n$,
+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.)
+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.
+
+\scott{Not sure how to say product fields in this setup.}
+Finally, notice there are functors $- \times I : \manifolds{k} \to \manifolds{k+1}$
+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 $\cC_k($
+\end{defn}
+\begin{rem}
+Where the dimension of the manifold is clear, we'll often leave off the subscript on $\cC_k$.
+\end{rem}
+
+\todo{end}
\ No newline at end of file




