196 \bc_*(X_1 \sqcup X_2) \iso \bc_*(X_1) \tensor \bc_*(X_2)

   196 \bc_*(X_1 \du X_2) \iso \bc_*(X_1) \tensor \bc_*(X_2)

   292 \nn{maybe should look for better name; but this is the name I use elsewhere}

   299 is a collection of functors $\cC_k$, for $k \le n$, from $\cM_k$ to the

   293 is a collection of functors $\cC$ from manifolds of dimension $n$ or less

   300 category of sets,

   294 to sets.

   301 together with some additional data and satisfying some additional conditions, all specified below.

   295 These functors must satisfy various properties (see \cite{kw:tqft} for details).

   302 

   296 For example:

   303 \nn{refer somewhere to my TQFT notes \cite{kw:tqft}, and possibly also to paper with Chris}

   297 there is a canonical identification $\cC(X \du Y) = \cC(X) \times \cC(Y)$;

   304 

   298 there is a restriction map $\cC(X) \to \cC(\bd X)$;

   305 Before finishing the definition of fields, we give two motivating examples

   299 gluing manifolds corresponds to fibered products of fields;

   306 (actually, families of examples) of systems of fields.

   300 given a field $c \in \cC(Y)$ there is a ``product field"

   307 

   301 $c\times I \in \cC(Y\times I)$; ...

   308 The first examples: Fix a target space $B$, and let $\cC(X)$ be the set of continuous maps

   302 \nn{should eventually include full details of definition of fields.}

   309 from X to $B$.

   310 

   311 The second examples: Fix an $n$-category $C$, and let $\cC(X)$ be 

   312 the set of sub-cell-complexes of $X$ with codimension-$j$ cells labeled by

   313 $j$-morphisms of $C$.

   314 One can think of such sub-cell-complexes as dual to pasting diagrams for $C$.

   315 This is described in more detail below.

   316 

   317 Now for the rest of the definition of system of fields.

   318 \begin{enumerate}

   319 \item There are boundary restriction maps $\cC_k(X) \to \cC_{k-1}(\bd X)$, 

   320 and these maps are a natural

   321 transformation between the functors $\cC_k$ and $\cC_{k-1}\circ\bd$.

   322 \item There are orientation reversal maps $\cC_k(X) \to \cC_k(-X)$, and these maps

   323 again comprise a natural transformation of functors.

   324 \item $\cC_k$ is compatible with the symmetric monoidal

   325 structures on $\cM_k$ and sets: $\cC_k(X \du W) \cong \cC_k(X)\times \cC_k(W)$,

   326 compatibly with homeomorphisms, restriction to boundary, and orientation reversal.

   327 \item Gluing without corners.

   328 Let $\bd X = Y \du -Y \du W$, where $Y$ and $W$ are closed $k{-}1$-manifolds.

   329 Let $X\sgl$ denote $X$ glued to itself along $\pm Y$.

   330 Using the boundary restriction, disjoint union, and (in one case) orientation reversal

   331 maps, we get two maps $\cC_k(X) \to \cC(Y)$, corresponding to the two

   332 copies of $Y$ in $\bd X$.

   333 Let $\Eq_Y(\cC_k(X))$ denote the equalizer of these two maps.

   334 Then (here's the axiom/definition part) there is an injective ``gluing" map

   335 \[

   336 	\Eq_Y(\cC_k(X)) \hookrightarrow \cC_k(X\sgl) ,

   337 \]

   338 and this gluing map is compatible with all of the above structure (actions

   339 of homeomorphisms, boundary restrictions, orientation reversal, disjoint union).

   340 Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity,

   341 the gluing map is surjective.

   342 From the point of view of $X\sgl$ and the image $Y \subset X\sgl$ of the 

   343 gluing surface, we say that fields in the image of the gluing map

   344 are transverse to $Y$ or cuttable along $Y$.

   345 \item Gluing with corners. \nn{...}

   346 \item ``product with $I$" maps $\cC(Y)\to \cC(Y\times I)$; 

   347 fiber-preserving homeos of $Y\times I$ act trivially on image

   348 \nn{...}

   349 \end{enumerate}

   350 

   351 

   352 \bigskip

   353 \hrule

   354 \bigskip