284 \label{sec:definitions} |
284 \label{sec:definitions} |
285 |
285 |
286 \subsection{Systems of fields} |
286 \subsection{Systems of fields} |
287 \label{sec:fields} |
287 \label{sec:fields} |
288 |
288 |
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289 Let $\cM_k$ denote the category (groupoid, in fact) with objects |
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290 oriented PL manifolds of dimension |
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291 $k$ and morphisms homeomorphisms. |
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292 (We could equally well work with a different category of manifolds --- |
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293 unoriented, topological, smooth, spin, etc. --- but for definiteness we |
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294 will stick with oriented PL.) |
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295 |
289 Fix a top dimension $n$. |
296 Fix a top dimension $n$. |
290 |
297 |
291 A {\it system of fields} |
298 A {\it system of fields} |
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 $B$. |
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310 |
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311 The second examples: Fix an $n$-category $C$, and let $\cC(X)$ be |
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312 the set of sub-cell-complexes of $X$ with codimension-$j$ cells labeled by |
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313 $j$-morphisms of $C$. |
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314 One can think of such sub-cell-complexes as dual to pasting diagrams for $C$. |
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315 This is described in more detail below. |
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316 |
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317 Now for the rest of the definition of system of fields. |
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318 \begin{enumerate} |
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319 \item There are boundary restriction maps $\cC_k(X) \to \cC_{k-1}(\bd X)$, |
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320 and these maps are a natural |
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321 transformation between the functors $\cC_k$ and $\cC_{k-1}\circ\bd$. |
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322 \item There are orientation reversal maps $\cC_k(X) \to \cC_k(-X)$, and these maps |
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323 again comprise a natural transformation of functors. |
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324 \item $\cC_k$ is compatible with the symmetric monoidal |
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325 structures on $\cM_k$ and sets: $\cC_k(X \du W) \cong \cC_k(X)\times \cC_k(W)$, |
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326 compatibly with homeomorphisms, restriction to boundary, and orientation reversal. |
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327 \item Gluing without corners. |
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328 Let $\bd X = Y \du -Y \du W$, where $Y$ and $W$ are closed $k{-}1$-manifolds. |
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329 Let $X\sgl$ denote $X$ glued to itself along $\pm Y$. |
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330 Using the boundary restriction, disjoint union, and (in one case) orientation reversal |
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331 maps, we get two maps $\cC_k(X) \to \cC(Y)$, corresponding to the two |
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332 copies of $Y$ in $\bd X$. |
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333 Let $\Eq_Y(\cC_k(X))$ denote the equalizer of these two maps. |
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334 Then (here's the axiom/definition part) there is an injective ``gluing" map |
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335 \[ |
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336 \Eq_Y(\cC_k(X)) \hookrightarrow \cC_k(X\sgl) , |
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337 \] |
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338 and this gluing map is compatible with all of the above structure (actions |
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339 of homeomorphisms, boundary restrictions, orientation reversal, disjoint union). |
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340 Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity, |
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341 the gluing map is surjective. |
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342 From the point of view of $X\sgl$ and the image $Y \subset X\sgl$ of the |
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343 gluing surface, we say that fields in the image of the gluing map |
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344 are transverse to $Y$ or cuttable along $Y$. |
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345 \item Gluing with corners. \nn{...} |
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346 \item ``product with $I$" maps $\cC(Y)\to \cC(Y\times I)$; |
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347 fiber-preserving homeos of $Y\times I$ act trivially on image |
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348 \nn{...} |
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349 \end{enumerate} |
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350 |
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351 |
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352 \bigskip |
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353 \hrule |
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354 \bigskip |
303 |
355 |
304 \input{text/fields.tex} |
356 \input{text/fields.tex} |
305 |
357 |
306 \nn{note: probably will suppress from notation the distinction |
358 \nn{note: probably will suppress from notation the distinction |
307 between fields and their (orientation-reversal) duals} |
359 between fields and their (orientation-reversal) duals} |