366 For convenience we assume that fields are enriched over Vect. |
366 For convenience we assume that fields are enriched over Vect. |
367 |
367 |
368 Local relations are subspaces $U(B; c)\sub \cC(B; c)$ of the fields on balls which form an ideal under gluing. |
368 Local relations are subspaces $U(B; c)\sub \cC(B; c)$ of the fields on balls which form an ideal under gluing. |
369 Again, we give the examples first. |
369 Again, we give the examples first. |
370 |
370 |
371 \addtocounter{prop}{-2} |
371 \addtocounter{subsection}{-2} |
372 \begin{example}[contd.] |
372 \begin{example}[contd.] |
373 For maps into spaces, $U(B; c)$ is generated by fields of the form $a-b \in \lf(B; c)$, |
373 For maps into spaces, $U(B; c)$ is generated by fields of the form $a-b \in \lf(B; c)$, |
374 where $a$ and $b$ are maps (fields) which are homotopic rel boundary. |
374 where $a$ and $b$ are maps (fields) which are homotopic rel boundary. |
375 \end{example} |
375 \end{example} |
376 |
376 |
377 \begin{example}[contd.] |
377 \begin{example}[contd.] |
378 For $n$-category pictures, $U(B; c)$ is equal to the kernel of the evaluation map |
378 For $n$-category pictures, $U(B; c)$ is equal to the kernel of the evaluation map |
379 $\lf(B; c) \to \mor(c', c'')$, where $(c', c'')$ is some (any) division of $c$ into |
379 $\lf(B; c) \to \mor(c', c'')$, where $(c', c'')$ is some (any) division of $c$ into |
380 domain and range. |
380 domain and range. |
381 \end{example} |
381 \end{example} |
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382 \addtocounter{subsection}{2} |
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383 \addtocounter{prop}{-2} |
382 |
384 |
383 These motivate the following definition. |
385 These motivate the following definition. |
384 |
386 |
385 \begin{defn} |
387 \begin{defn} |
386 A {\it local relation} is a collection subspaces $U(B; c) \sub \lf(B; c)$, |
388 A {\it local relation} is a collection subspaces $U(B; c) \sub \lf(B; c)$, |