317 Now for the rest of the definition of system of fields. |
317 Now for the rest of the definition of system of fields. |
318 \begin{enumerate} |
318 \begin{enumerate} |
319 \item There are boundary restriction maps $\cC_k(X) \to \cC_{k-1}(\bd X)$, |
319 \item There are boundary restriction maps $\cC_k(X) \to \cC_{k-1}(\bd X)$, |
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 $\cC_{k-1}\circ\bd$. |
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322 For $c \in \cC_{k-1}(\bd X)$, we will denote by $\cC_k(X; c)$ the subset of |
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323 $\cC(X)$ which restricts to $c$. |
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324 In this context, we will call $c$ a boundary condition. |
322 \item There are orientation reversal maps $\cC_k(X) \to \cC_k(-X)$, and these maps |
325 \item There are orientation reversal maps $\cC_k(X) \to \cC_k(-X)$, and these maps |
323 again comprise a natural transformation of functors. |
326 again comprise a natural transformation of functors. |
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327 In addition, the orientation reversal maps are compatible with the boundary restriction maps. |
324 \item $\cC_k$ is compatible with the symmetric monoidal |
328 \item $\cC_k$ is compatible with the symmetric monoidal |
325 structures on $\cM_k$ and sets: $\cC_k(X \du W) \cong \cC_k(X)\times \cC_k(W)$, |
329 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. |
330 compatibly with homeomorphisms, restriction to boundary, and orientation reversal. |
327 \item Gluing without corners. |
331 \item Gluing without corners. |
328 Let $\bd X = Y \du -Y \du W$, where $Y$ and $W$ are closed $k{-}1$-manifolds. |
332 Let $\bd X = Y \du -Y \du W$, where $Y$ and $W$ are closed $k{-}1$-manifolds. |
340 Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity, |
344 Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity, |
341 the gluing map is surjective. |
345 the gluing map is surjective. |
342 From the point of view of $X\sgl$ and the image $Y \subset X\sgl$ of the |
346 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 |
347 gluing surface, we say that fields in the image of the gluing map |
344 are transverse to $Y$ or cuttable along $Y$. |
348 are transverse to $Y$ or cuttable along $Y$. |
345 \item Gluing with corners. \nn{...} |
349 \item Gluing with corners. |
346 \item ``product with $I$" maps $\cC(Y)\to \cC(Y\times I)$; |
350 Let $\bd X = Y \cup -Y \cup W$, where $\pm Y$ and $W$ might intersect along their boundaries. |
347 fiber-preserving homeos of $Y\times I$ act trivially on image |
351 Let $X\sgl$ denote $X$ glued to itself along $\pm Y$. |
348 \nn{...} |
352 Note that $\bd X\sgl = W\sgl$, where $W\sgl$ denotes $W$ glued to itself |
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353 (without corners) along two copies of $\bd Y$. |
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354 Let $c\sgl \in \cC_{k-1}(W\sgl)$ be a be a cuttable field on $W\sgl$ and let |
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355 $c \in \cC_{k-1}(W)$ be the cut open version of $c\sgl$. |
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356 Let $\cC^c_k(X)$ denote the subset of $\cC(X)$ which restricts to $c$ on $W$. |
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357 (This restriction map uses the gluing without corners map above.) |
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358 Using the boundary restriction, gluing without corners, and (in one case) orientation reversal |
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359 maps, we get two maps $\cC^c_k(X) \to \cC(Y)$, corresponding to the two |
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360 copies of $Y$ in $\bd X$. |
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361 Let $\Eq^c_Y(\cC_k(X))$ denote the equalizer of these two maps. |
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362 Then (here's the axiom/definition part) there is an injective ``gluing" map |
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363 \[ |
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364 \Eq^c_Y(\cC_k(X)) \hookrightarrow \cC_k(X\sgl, c\sgl) , |
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365 \] |
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366 and this gluing map is compatible with all of the above structure (actions |
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367 of homeomorphisms, boundary restrictions, orientation reversal, disjoint union). |
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368 Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity, |
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369 the gluing map is surjective. |
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370 From the point of view of $X\sgl$ and the image $Y \subset X\sgl$ of the |
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371 gluing surface, we say that fields in the image of the gluing map |
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372 are transverse to $Y$ or cuttable along $Y$. |
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373 \item There are maps $\cC_{k-1}(Y) \to \cC_k(Y \times I)$, denoted |
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374 $c \mapsto c\times I$. |
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375 These maps comprise a natural transformation of functors, and commute appropriately |
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376 with all the structure maps above (disjoint union, boundary restriction, etc.) |
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377 Furthermore, if $f: Y\times I \to Y\times I$ is a fiber-preserving homeomorphism |
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378 covering $\bar{f}:Y\to Y$, then $f(c\times I) = \bar{f}(c)\times I$. |
349 \end{enumerate} |
379 \end{enumerate} |
350 |
380 |
351 |
381 |
352 \bigskip |
382 \bigskip |
353 \hrule |
383 \hrule |
354 \bigskip |
384 \bigskip |
355 |
385 |
356 \input{text/fields.tex} |
386 \input{text/fields.tex} |
357 |
387 |
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388 |
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389 \bigskip |
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390 \hrule |
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391 \bigskip |
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392 |
358 \nn{note: probably will suppress from notation the distinction |
393 \nn{note: probably will suppress from notation the distinction |
359 between fields and their (orientation-reversal) duals} |
394 between fields and their (orientation-reversal) duals} |
360 |
395 |
361 \nn{remark that if top dimensional fields are not already linear |
396 \nn{remark that if top dimensional fields are not already linear |
362 then we will soon linearize them(?)} |
397 then we will soon linearize them(?)} |
363 |
398 |
364 The definition of a system of fields is intended to generalize |
399 We now describe in more detail systems of fields coming from sub-cell-complexes labeled |
365 the relevant properties of the following two examples of fields. |
400 by $n$-category morphisms. |
366 |
401 |
367 The first example: Fix a target space $B$ and define $\cC(X)$ (where $X$ |
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368 is a manifold of dimension $n$ or less) to be the set of |
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369 all maps from $X$ to $B$. |
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370 |
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371 The second example will take longer to explain. |
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372 Given an $n$-category $C$ with the right sort of duality |
402 Given an $n$-category $C$ with the right sort of duality |
373 (e.g. pivotal 2-category, 1-category with duals, star 1-category, disklike $n$-category), |
403 (e.g. pivotal 2-category, 1-category with duals, star 1-category, disklike $n$-category), |
374 we can construct a system of fields as follows. |
404 we can construct a system of fields as follows. |
375 Roughly speaking, $\cC(X)$ will the set of all embedded cell complexes in $X$ |
405 Roughly speaking, $\cC(X)$ will the set of all embedded cell complexes in $X$ |
376 with codimension $i$ cells labeled by $i$-morphisms of $C$. |
406 with codimension $i$ cells labeled by $i$-morphisms of $C$. |
489 |
519 |
490 |
520 |
491 \subsection{Local relations} |
521 \subsection{Local relations} |
492 \label{sec:local-relations} |
522 \label{sec:local-relations} |
493 |
523 |
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524 \nn{the following is not done yet} |
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525 |
494 Let $B^n$ denote the standard $n$-ball. |
526 Let $B^n$ denote the standard $n$-ball. |
495 A {\it local relation} is a collection subspaces $U(B^n; c) \sub \cC_l(B^n; c)$ |
527 A {\it local relation} is a collection subspaces $U(B^n; c) \sub \cC_l(B^n; c)$ |
496 (for all $c \in \cC(\bd B^n)$) satisfying the following (three?) properties. |
528 (for all $c \in \cC(\bd B^n)$) satisfying the following two properties. |
497 |
529 \begin{enumerate} |
498 \nn{Roughly, these are (1) the local relations imply (extended) isotopy; |
530 \item local relations imply (extended) isotopy \nn{...} |
499 (2) $U(B^n; \cdot)$ is an ideal w.r.t.\ gluing; and |
531 \item $U(B^n; \cdot)$ is an ideal w.r.t.\ gluing \nn{...} |
500 (3) this ideal is generated by ``small" generators (contained in an open cover of $B^n$). |
532 \end{enumerate} |
501 See \cite{kw:tqft} for details. Need to transfer details to here.} |
533 See \cite{kw:tqft} for details. |
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534 |
502 |
535 |
503 For maps into spaces, $U(B^n; c)$ is generated by things of the form $a-b \in \cC_l(B^n; c)$, |
536 For maps into spaces, $U(B^n; c)$ is generated by things of the form $a-b \in \cC_l(B^n; c)$, |
504 where $a$ and $b$ are maps (fields) which are homotopic rel boundary. |
537 where $a$ and $b$ are maps (fields) which are homotopic rel boundary. |
505 |
538 |
506 For $n$-category pictures, $U(B^n; c)$ is equal to the kernel of the evaluation map |
539 For $n$-category pictures, $U(B^n; c)$ is equal to the kernel of the evaluation map |