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 \ref{pinched_prods}.) |
337 (See Figure \ref{pinched_prods}.) |
338 \begin{figure}[t] |
338 \begin{figure}[t] |
339 $$\mathfig{.8}{tempkw/pinched_prods}$$ |
339 $$ |
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340 \begin{tikzpicture}[baseline=0] |
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341 \begin{scope} |
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342 \path[clip] (0,0) arc (135:45:4) arc (-45:-135:4); |
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343 \draw[blue,line width=2pt] (0,0) arc (135:45:4) arc (-45:-135:4); |
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344 \foreach \x in {0, 0.5, ..., 6} { |
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345 \draw[green!50!brown] (\x,-2) -- (\x,2); |
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346 } |
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347 \end{scope} |
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348 \draw[blue,line width=1.5pt] (0,-3) -- (5.66,-3); |
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349 \draw[->,red,line width=2pt] (2.83,-1.5) -- (2.83,-2.5); |
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350 \end{tikzpicture} |
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351 \qquad \qquad |
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352 \begin{tikzpicture}[baseline=-0.15cm] |
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353 \begin{scope} |
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354 \path[clip] (0,1) arc (90:135:8 and 4) arc (-135:-90:8 and 4) -- cycle; |
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355 \draw[blue,line width=2pt] (0,1) arc (90:135:8 and 4) arc (-135:-90:8 and 4) -- cycle; |
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356 \foreach \x in {-6, -5.5, ..., 0} { |
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357 \draw[green!50!brown] (\x,-2) -- (\x,2); |
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358 } |
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359 \end{scope} |
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360 \draw[blue,line width=1.5pt] (-5.66,-3.15) -- (0,-3.15); |
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361 \draw[->,red,line width=2pt] (-2.83,-1.5) -- (-2.83,-2.5); |
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362 \end{tikzpicture} |
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363 $$ |
340 \caption{Examples of pinched products}\label{pinched_prods} |
364 \caption{Examples of pinched products}\label{pinched_prods} |
341 \end{figure} |
365 \end{figure} |
342 (The need for a strengthened version will become apparent in appendix \ref{sec:comparing-defs} |
366 (The need for a strengthened version will become apparent in appendix \ref{sec:comparing-defs} |
343 where we construct a traditional category from a topological category.) |
367 where we construct a traditional category from a topological category.) |
344 Define a {\it pinched product} to be a map |
368 Define a {\it pinched product} to be a map |
363 $\pi:E'\to \pi(E')$ is again a pinched product. |
387 $\pi:E'\to \pi(E')$ is again a pinched product. |
364 A {union} of pinched products is a decomposition $E = \cup_i E_i$ |
388 A {union} of pinched products is a decomposition $E = \cup_i E_i$ |
365 such that each $E_i\sub E$ is a sub pinched product. |
389 such that each $E_i\sub E$ is a sub pinched product. |
366 (See Figure \ref{pinched_prod_unions}.) |
390 (See Figure \ref{pinched_prod_unions}.) |
367 \begin{figure}[t] |
391 \begin{figure}[t] |
368 $$\mathfig{.8}{tempkw/pinched_prod_unions}$$ |
392 $$ |
369 \caption{Unions of pinched products}\label{pinched_prod_unions} |
393 \begin{tikzpicture}[baseline=0] |
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394 \begin{scope} |
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395 \path[clip] (0,0) arc (135:45:4) arc (-45:-135:4); |
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396 \draw[blue,line width=2pt] (0,0) arc (135:45:4) arc (-45:-135:4); |
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397 \draw[blue] (0,0) -- (5.66,0); |
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398 \foreach \x in {0, 0.5, ..., 6} { |
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399 \draw[green!50!brown] (\x,-2) -- (\x,2); |
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400 } |
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401 \end{scope} |
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402 \end{tikzpicture} |
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403 \qquad |
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404 \begin{tikzpicture}[baseline=0] |
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405 \begin{scope} |
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406 \path[clip] (0,-1) rectangle (4,1); |
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407 \draw[blue,line width=2pt] (0,-1) rectangle (4,1); |
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408 \draw[blue] (0,0) -- (5,0); |
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409 \foreach \x in {0, 0.5, ..., 6} { |
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410 \draw[green!50!brown] (\x,-2) -- (\x,2); |
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411 } |
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412 \end{scope} |
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413 \end{tikzpicture} |
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414 \qquad |
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415 \begin{tikzpicture}[baseline=0] |
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416 \begin{scope} |
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417 \path[clip] (0,0) arc (135:45:4) arc (-45:-135:4); |
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418 \draw[blue,line width=2pt] (0,0) arc (135:45:4) arc (-45:-135:4); |
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419 \draw[blue] (2.83,3) circle (3); |
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420 \foreach \x in {0, 0.5, ..., 6} { |
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421 \draw[green!50!brown] (\x,-2) -- (\x,2); |
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422 } |
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423 \end{scope} |
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424 \end{tikzpicture} |
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425 $$ |
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426 $$ |
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427 \begin{tikzpicture}[baseline=0] |
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428 \begin{scope} |
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429 \path[clip] (0,-1) rectangle (4,1); |
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430 \draw[blue,line width=2pt] (0,-1) rectangle (4,1); |
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431 \draw[blue] (0,-1) -- (4,1); |
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432 \foreach \x in {0, 0.5, ..., 6} { |
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433 \draw[green!50!brown] (\x,-2) -- (\x,2); |
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434 } |
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435 \end{scope} |
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436 \end{tikzpicture} |
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437 \qquad |
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438 \begin{tikzpicture}[baseline=0] |
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439 \begin{scope} |
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440 \path[clip] (0,-1) rectangle (5,1); |
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441 \draw[blue,line width=2pt] (0,-1) rectangle (5,1); |
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442 \draw[blue] (1,-1) .. controls (2,-1) and (3,1) .. (4,1); |
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443 \foreach \x in {0, 0.5, ..., 6} { |
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444 \draw[green!50!brown] (\x,-2) -- (\x,2); |
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445 } |
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446 \end{scope} |
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447 \end{tikzpicture} |
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448 $$ |
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449 \caption{Five examples of unions of pinched products}\label{pinched_prod_unions} |
370 \end{figure} |
450 \end{figure} |
371 |
451 |
372 The product axiom will give a map $\pi^*:\cC(X)\to \cC(E)$ for each pinched product |
452 The product axiom will give a map $\pi^*:\cC(X)\to \cC(E)$ for each pinched product |
373 $\pi:E\to X$. |
453 $\pi:E\to X$. |
374 Morphisms in the image of $\pi^*$ will be called product morphisms. |
454 Morphisms in the image of $\pi^*$ will be called product morphisms. |