blob: comparison blob1.tex

equal deleted inserted replaced

-:ac5c74fa38d7
+:46b5c4f3e83c
 Now for the rest of the definition of system of fields.
 \begin{enumerate}
 \item There are boundary restriction maps $\cC_k(X) \to \cC_{k-1}(\bd X)$,
 and these maps are a natural
 transformation between the functors $\cC_k$ and $\cC_{k-1}\circ\bd$.
+For $c \in \cC_{k-1}(\bd X)$, we will denote by $\cC_k(X; c)$ the subset of
+$\cC(X)$ which restricts to $c$.
+In this context, we will call $c$ a boundary condition.
 \item There are orientation reversal maps $\cC_k(X) \to \cC_k(-X)$, and these maps
 again comprise a natural transformation of functors.
+In addition, the orientation reversal maps are compatible with the boundary restriction maps.
 \item $\cC_k$ is compatible with the symmetric monoidal
 structures on $\cM_k$ and sets: $\cC_k(X \du W) \cong \cC_k(X)\times \cC_k(W)$,
 compatibly with homeomorphisms, restriction to boundary, and orientation reversal.
 \item Gluing without corners.
 Let $\bd X = Y \du -Y \du W$, where $Y$ and $W$ are closed $k{-}1$-manifolds.
 Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity,
 the gluing map is surjective.
 From the point of view of $X\sgl$ and the image $Y \subset X\sgl$ of the
 gluing surface, we say that fields in the image of the gluing map
 are transverse to $Y$ or cuttable along $Y$.
-\item Gluing with corners. \nn{...}
+\item Gluing with corners.
-\item ``product with $I$" maps $\cC(Y)\to \cC(Y\times I)$;
+Let $\bd X = Y \cup -Y \cup W$, where $\pm Y$ and $W$ might intersect along their boundaries.
-fiber-preserving homeos of $Y\times I$ act trivially on image
+Let $X\sgl$ denote $X$ glued to itself along $\pm Y$.
-\nn{...}
+Note that $\bd X\sgl = W\sgl$, where $W\sgl$ denotes $W$ glued to itself
+(without corners) along two copies of $\bd Y$.
+Let $c\sgl \in \cC_{k-1}(W\sgl)$ be a be a cuttable field on $W\sgl$ and let
+$c \in \cC_{k-1}(W)$ be the cut open version of $c\sgl$.
+Let $\cC^c_k(X)$ denote the subset of $\cC(X)$ which restricts to $c$ on $W$.
+(This restriction map uses the gluing without corners map above.)
+Using the boundary restriction, gluing without corners, and (in one case) orientation reversal
+maps, we get two maps $\cC^c_k(X) \to \cC(Y)$, corresponding to the two
+copies of $Y$ in $\bd X$.
+Let $\Eq^c_Y(\cC_k(X))$ denote the equalizer of these two maps.
+Then (here's the axiom/definition part) there is an injective ``gluing" map
+\[
+	\Eq^c_Y(\cC_k(X)) \hookrightarrow \cC_k(X\sgl, c\sgl) ,
+\]
+and this gluing map is compatible with all of the above structure (actions
+of homeomorphisms, boundary restrictions, orientation reversal, disjoint union).
+Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity,
+the gluing map is surjective.
+From the point of view of $X\sgl$ and the image $Y \subset X\sgl$ of the
+gluing surface, we say that fields in the image of the gluing map
+are transverse to $Y$ or cuttable along $Y$.
+\item There are maps $\cC_{k-1}(Y) \to \cC_k(Y \times I)$, denoted
+$c \mapsto c\times I$.
+These maps comprise a natural transformation of functors, and commute appropriately
+with all the structure maps above (disjoint union, boundary restriction, etc.)
+Furthermore, if $f: Y\times I \to Y\times I$ is a fiber-preserving homeomorphism
+covering $\bar{f}:Y\to Y$, then $f(c\times I) = \bar{f}(c)\times I$.
 \end{enumerate}
 \bigskip
 \hrule
 \bigskip
 \input{text/fields.tex}
+\bigskip
+\hrule
+\bigskip
 \nn{note: probably will suppress from notation the distinction
 between fields and their (orientation-reversal) duals}
 \nn{remark that if top dimensional fields are not already linear
 then we will soon linearize them(?)}
-The definition of a system of fields is intended to generalize
+We now describe in more detail systems of fields coming from sub-cell-complexes labeled
-the relevant properties of the following two examples of fields.
+by $n$-category morphisms.
-The first example: Fix a target space $B$ and define $\cC(X)$ (where $X$
-is a manifold of dimension $n$ or less) to be the set of
-all maps from $X$ to $B$.
-The second example will take longer to explain.
 Given an $n$-category $C$ with the right sort of duality
 (e.g. pivotal 2-category, 1-category with duals, star 1-category, disklike $n$-category),
 we can construct a system of fields as follows.
 Roughly speaking, $\cC(X)$ will the set of all embedded cell complexes in $X$
 with codimension $i$ cells labeled by $i$-morphisms of $C$.
 \subsection{Local relations}
 \label{sec:local-relations}
+\nn{the following is not done yet}
 Let $B^n$ denote the standard $n$-ball.
 A {\it local relation} is a collection subspaces $U(B^n; c) \sub \cC_l(B^n; c)$
-(for all $c \in \cC(\bd B^n)$) satisfying the following (three?) properties.
+(for all $c \in \cC(\bd B^n)$) satisfying the following two properties.
+\begin{enumerate}
-\nn{Roughly, these are (1) the local relations imply (extended) isotopy;
+\item local relations imply (extended) isotopy \nn{...}
-(2) $U(B^n; \cdot)$ is an ideal w.r.t.\ gluing; and
+\item $U(B^n; \cdot)$ is an ideal w.r.t.\ gluing \nn{...}
-(3) this ideal is generated by ``small" generators (contained in an open cover of $B^n$).
+\end{enumerate}
-See \cite{kw:tqft} for details.  Need to transfer details to here.}
+See \cite{kw:tqft} for details.
 For maps into spaces, $U(B^n; c)$ is generated by things of the form $a-b \in \cC_l(B^n; c)$,
 where $a$ and $b$ are maps (fields) which are homotopic rel boundary.
 For $n$-category pictures, $U(B^n; c)$ is equal to the kernel of the evaluation map

changeset 60	46b5c4f3e83c
parent 59	ac5c74fa38d7
child 61	4093d7979c56