28 together with some additional data and satisfying some additional conditions, all specified below. |
28 together with some additional data and satisfying some additional conditions, all specified below. |
29 |
29 |
30 Before finishing the definition of fields, we give two motivating examples |
30 Before finishing the definition of fields, we give two motivating examples |
31 (actually, families of examples) of systems of fields. |
31 (actually, families of examples) of systems of fields. |
32 |
32 |
33 The first examples: Fix a target space $B$, and let $\cC(X)$ be the set of continuous maps |
33 \begin{example} |
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34 \label{ex:maps-to-a-space(fields)} |
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35 Fix a target space $B$, and let $\cC(X)$ be the set of continuous maps |
34 from X to $B$. |
36 from X to $B$. |
35 |
37 \end{example} |
36 The second examples: Fix an $n$-category $C$, and let $\cC(X)$ be |
38 |
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39 \begin{example} |
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40 \label{ex:traditional-n-categories(fields)} |
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41 Fix an $n$-category $C$, and let $\cC(X)$ be |
37 the set of sub-cell-complexes of $X$ with codimension-$j$ cells labeled by |
42 the set of sub-cell-complexes of $X$ with codimension-$j$ cells labeled by |
38 $j$-morphisms of $C$. |
43 $j$-morphisms of $C$. |
39 One can think of such sub-cell-complexes as dual to pasting diagrams for $C$. |
44 One can think of such sub-cell-complexes as dual to pasting diagrams for $C$. |
40 This is described in more detail below. |
45 This is described in more detail below. |
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46 \end{example} |
41 |
47 |
42 Now for the rest of the definition of system of fields. |
48 Now for the rest of the definition of system of fields. |
43 \begin{enumerate} |
49 \begin{enumerate} |
44 \item There are boundary restriction maps $\cC_k(X) \to \cC_{k-1}(\bd X)$, |
50 \item There are boundary restriction maps $\cC_k(X) \to \cC_{k-1}(\bd X)$, |
45 and these maps are a natural |
51 and these maps are a natural |
260 |
266 |
261 |
267 |
262 |
268 |
263 \subsection{Local relations} |
269 \subsection{Local relations} |
264 \label{sec:local-relations} |
270 \label{sec:local-relations} |
265 |
271 Local relations are certain subspaces of the fields on balls, which form an ideal under gluing. Again, we give the examples first. |
266 |
272 |
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273 \addtocounter{prop}{-2} |
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274 \begin{example}[contd.] |
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275 For maps into spaces, $U(B; c)$ is generated by fields of the form $a-b \in \lf(B; c)$, |
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276 where $a$ and $b$ are maps (fields) which are homotopic rel boundary. |
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277 \end{example} |
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278 |
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279 \begin{example}[contd.] |
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280 For $n$-category pictures, $U(B; c)$ is equal to the kernel of the evaluation map |
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281 $\lf(B; c) \to \mor(c', c'')$, where $(c', c'')$ is some (any) division of $c$ into |
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282 domain and range. |
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283 \end{example} |
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284 |
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285 These motivate the following definition. |
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286 |
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287 \begin{defn} |
267 A {\it local relation} is a collection subspaces $U(B; c) \sub \lf(B; c)$, |
288 A {\it local relation} is a collection subspaces $U(B; c) \sub \lf(B; c)$, |
268 for all $n$-manifolds $B$ which are |
289 for all $n$-manifolds $B$ which are |
269 homeomorphic to the standard $n$-ball and all $c \in \cC(\bd B)$, |
290 homeomorphic to the standard $n$-ball and all $c \in \cC(\bd B)$, |
270 satisfying the following properties. |
291 satisfying the following properties. |
271 \begin{enumerate} |
292 \begin{enumerate} |
275 if $x, y \in \cC(B; c)$ and $x$ is extended isotopic |
296 if $x, y \in \cC(B; c)$ and $x$ is extended isotopic |
276 to $y$, then $x-y \in U(B; c)$. |
297 to $y$, then $x-y \in U(B; c)$. |
277 \item ideal with respect to gluing: |
298 \item ideal with respect to gluing: |
278 if $B = B' \cup B''$, $x\in U(B')$, and $c\in \cC(B'')$, then $x\bullet r \in U(B)$ |
299 if $B = B' \cup B''$, $x\in U(B')$, and $c\in \cC(B'')$, then $x\bullet r \in U(B)$ |
279 \end{enumerate} |
300 \end{enumerate} |
280 See \cite{kw:tqft} for details. |
301 \end{defn} |
281 |
302 See \cite{kw:tqft} for further details. |
282 |
303 |
283 For maps into spaces, $U(B; c)$ is generated by things of the form $a-b \in \lf(B; c)$, |
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284 where $a$ and $b$ are maps (fields) which are homotopic rel boundary. |
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285 |
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286 For $n$-category pictures, $U(B; c)$ is equal to the kernel of the evaluation map |
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287 $\lf(B; c) \to \mor(c', c'')$, where $(c', c'')$ is some (any) division of $c$ into |
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288 domain and range. |
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289 |
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290 \nn{maybe examples of local relations before general def?} |
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291 |
304 |
292 \subsection{Constructing a TQFT} |
305 \subsection{Constructing a TQFT} |
293 \label{sec:constructing-a-tqft} |
306 \label{sec:constructing-a-tqft} |
294 |
307 |
295 In this subsection we briefly review the construction of a TQFT from a system of fields and local relations. |
308 In this subsection we briefly review the construction of a TQFT from a system of fields and local relations. |