291 $k$ and morphisms homeomorphisms. |
292 $k$ and morphisms homeomorphisms. |
292 (We could equally well work with a different category of manifolds --- |
293 (We could equally well work with a different category of manifolds --- |
293 unoriented, topological, smooth, spin, etc. --- but for definiteness we |
294 unoriented, topological, smooth, spin, etc. --- but for definiteness we |
294 will stick with oriented PL.) |
295 will stick with oriented PL.) |
295 |
296 |
296 Fix a top dimension $n$. |
297 Fix a top dimension $n$, and a symmetric monoidal category $\cS$ whose objects are sets. While reading the definition, you should just think about the case $\cS = \Set$ with cartesian product, until you reach the discussion of a \emph{linear system of fields} later in this section, where $\cS = \Vect$, and \S \ref{sec:homological-fields}, where $\cS = \Kom$. |
297 |
298 |
298 A {\it system of fields} |
299 A $n$-dimensional {\it system of fields} in $\cS$ |
299 is a collection of functors $\cC_k$, for $k \le n$, from $\cM_k$ to the |
300 is a collection of functors $\cC_k : \cM_k \to \Set$ for $0 \leq k \leq n$ |
300 category of sets, |
|
301 together with some additional data and satisfying some additional conditions, all specified below. |
301 together with some additional data and satisfying some additional conditions, all specified below. |
302 |
302 |
303 \nn{refer somewhere to my TQFT notes \cite{kw:tqft}, and possibly also to paper with Chris} |
303 \nn{refer somewhere to my TQFT notes \cite{kw:tqft}, and possibly also to paper with Chris} |
304 |
304 |
305 Before finishing the definition of fields, we give two motivating examples |
305 Before finishing the definition of fields, we give two motivating examples |
320 and these maps are a natural |
320 and these maps are a natural |
321 transformation between the functors $\cC_k$ and $\cC_{k-1}\circ\bd$. |
321 transformation between the functors $\cC_k$ and $\cC_{k-1}\circ\bd$. |
322 For $c \in \cC_{k-1}(\bd X)$, we will denote by $\cC_k(X; c)$ the subset of |
322 For $c \in \cC_{k-1}(\bd X)$, we will denote by $\cC_k(X; c)$ the subset of |
323 $\cC(X)$ which restricts to $c$. |
323 $\cC(X)$ which restricts to $c$. |
324 In this context, we will call $c$ a boundary condition. |
324 In this context, we will call $c$ a boundary condition. |
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325 \item The subset $\cC_n(X;c)$ of top fields with a given boundary condition is an object in our symmetric monoidal category $\cS$. (This condition is of course trivial when $\cS = \Set$.) If the objects are sets with extra structure (e.g. $\cS = \Vect$ or $\Kom$), then this extra structure is considered part of the definition of $\cC_n$. Any maps mentioned below between top level fields must be morphisms in $\cS$. |
325 \item There are orientation reversal maps $\cC_k(X) \to \cC_k(-X)$, and these maps |
326 \item There are orientation reversal maps $\cC_k(X) \to \cC_k(-X)$, and these maps |
326 again comprise a natural transformation of functors. |
327 again comprise a natural transformation of functors. |
327 In addition, the orientation reversal maps are compatible with the boundary restriction maps. |
328 In addition, the orientation reversal maps are compatible with the boundary restriction maps. |
328 \item $\cC_k$ is compatible with the symmetric monoidal |
329 \item $\cC_k$ is compatible with the symmetric monoidal |
329 structures on $\cM_k$ and sets: $\cC_k(X \du W) \cong \cC_k(X)\times \cC_k(W)$, |
330 structures on $\cM_k$, $\Set$ and $\cS$: $\cC_k(X \du W) \cong \cC_k(X)\times \cC_k(W)$, |
330 compatibly with homeomorphisms, restriction to boundary, and orientation reversal. |
331 compatibly with homeomorphisms, restriction to boundary, and orientation reversal. |
331 We will call the projections $\cC(X_1 \du X_2) \to \cC(X_i)$ |
332 We will call the projections $\cC(X_1 \du X_2) \to \cC(X_i)$ |
332 restriction maps. |
333 restriction maps. |
333 \item Gluing without corners. |
334 \item Gluing without corners. |
334 Let $\bd X = Y \du -Y \du W$, where $Y$ and $W$ are closed $k{-}1$-manifolds. |
335 Let $\bd X = Y \du -Y \du W$, where $Y$ and $W$ are closed $k{-}1$-manifolds. |
398 on $M$ generated by isotopy plus all instance of the above construction |
399 on $M$ generated by isotopy plus all instance of the above construction |
399 (for all appropriate $Y$ and $x$). |
400 (for all appropriate $Y$ and $x$). |
400 |
401 |
401 \nn{should also say something about pseudo-isotopy} |
402 \nn{should also say something about pseudo-isotopy} |
402 |
403 |
403 \bigskip |
404 %\bigskip |
404 \hrule |
405 %\hrule |
405 \bigskip |
406 %\bigskip |
406 |
407 % |
407 \input{text/fields.tex} |
408 %\input{text/fields.tex} |
408 |
409 % |
409 |
410 % |
410 \bigskip |
411 %\bigskip |
411 \hrule |
412 %\hrule |
412 \bigskip |
413 %\bigskip |
413 |
414 |
414 \nn{note: probably will suppress from notation the distinction |
415 \nn{note: probably will suppress from notation the distinction |
415 between fields and their (orientation-reversal) duals} |
416 between fields and their (orientation-reversal) duals} |
416 |
417 |
417 \nn{remark that if top dimensional fields are not already linear |
418 \nn{remark that if top dimensional fields are not already linear |
724 \bc_k(X) \deq \bigoplus_{\overline{B}} \bigoplus_{\overline{c}} |
725 \bc_k(X) \deq \bigoplus_{\overline{B}} \bigoplus_{\overline{c}} |
725 \left( \otimes_j U(B_j; c_j)\right) \otimes \lf(X \setmin B^t; c^t) . |
726 \left( \otimes_j U(B_j; c_j)\right) \otimes \lf(X \setmin B^t; c^t) . |
726 \] |
727 \] |
727 Here $\overline{B}$ runs over all configurations of blobs, satisfying the conditions above. |
728 Here $\overline{B}$ runs over all configurations of blobs, satisfying the conditions above. |
728 $\overline{c}$ runs over all boundary conditions, again as described above. |
729 $\overline{c}$ runs over all boundary conditions, again as described above. |
729 $j$ runs over all indices of twig blobs. |
730 $j$ runs over all indices of twig blobs. The final $\lf(X \setmin B^t; c^t)$ must be interpreted as fields which are cuttable along all of the blobs in $\overline{B}$. |
730 |
731 |
731 The boundary map $\bd : \bc_k(X) \to \bc_{k-1}(X)$ is defined as follows. |
732 The boundary map $\bd : \bc_k(X) \to \bc_{k-1}(X)$ is defined as follows. |
732 Let $b = (\{B_i\}, \{u_j\}, r)$ be a $k$-blob diagram. |
733 Let $b = (\{B_i\}, \{u_j\}, r)$ be a $k$-blob diagram. |
733 Let $E_j(b)$ denote the result of erasing the $j$-th blob. |
734 Let $E_j(b)$ denote the result of erasing the $j$-th blob. |
734 If $B_j$ is not a twig blob, this involves only decrementing |
735 If $B_j$ is not a twig blob, this involves only decrementing |