equal
deleted
inserted
replaced
332 \end{enumerate} |
332 \end{enumerate} |
333 \end{axiom} |
333 \end{axiom} |
334 |
334 |
335 We will need to strengthen the above preliminary version of the axiom to allow |
335 We will need to strengthen the above preliminary version of the axiom to allow |
336 for products which are ``pinched" in various ways along their boundary. |
336 for products which are ``pinched" in various ways along their boundary. |
337 (See Figure xxxx.) |
337 (See Figure \ref{pinched_prods}.) |
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338 \begin{figure}[t] |
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339 $$\mathfig{.8}{tempkw/pinched_prods}$$ |
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340 \caption{Examples of pinched products}\label{pinched_prods} |
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341 \end{figure} |
338 (The need for a strengthened version will become apparent in appendix \ref{sec:comparing-defs} |
342 (The need for a strengthened version will become apparent in appendix \ref{sec:comparing-defs} |
339 where we construct a traditional category from a topological category.) |
343 where we construct a traditional category from a topological category.) |
340 Define a {\it pinched product} to be a map |
344 Define a {\it pinched product} to be a map |
341 \[ |
345 \[ |
342 \pi: E\to X |
346 \pi: E\to X |
357 |
361 |
358 A {\it sub pinched product} is a sub-$m$-ball $E'\sub E$ such that the restriction |
362 A {\it sub pinched product} is a sub-$m$-ball $E'\sub E$ such that the restriction |
359 $\pi:E'\to \pi(E')$ is again a pinched product. |
363 $\pi:E'\to \pi(E')$ is again a pinched product. |
360 A {union} of pinched products is a decomposition $E = \cup_i E_i$ |
364 A {union} of pinched products is a decomposition $E = \cup_i E_i$ |
361 such that each $E_i\sub E$ is a sub pinched product. |
365 such that each $E_i\sub E$ is a sub pinched product. |
362 (See Figure xxxx.) |
366 (See Figure \ref{pinched_prod_unions}.) |
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367 \begin{figure}[t] |
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368 $$\mathfig{.8}{tempkw/pinched_prod_unions}$$ |
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369 \caption{Unions of pinched products}\label{pinched_prod_unions} |
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370 \end{figure} |
363 |
371 |
364 The product axiom will give a map $\pi^*:\cC(X)\to \cC(E)$ for each pinched product |
372 The product axiom will give a map $\pi^*:\cC(X)\to \cC(E)$ for each pinched product |
365 $\pi:E\to X$. |
373 $\pi:E\to X$. |
366 Morphisms in the image of $\pi^*$ will be called product morphisms. |
374 Morphisms in the image of $\pi^*$ will be called product morphisms. |
367 Before stating the axiom, we illustrate it in our two motivating examples of $n$-categories. |
375 Before stating the axiom, we illustrate it in our two motivating examples of $n$-categories. |