381 \begin{figure}[t] |
381 \begin{figure}[t] |
382 $$ |
382 $$ |
383 \begin{tikzpicture}[baseline=0] |
383 \begin{tikzpicture}[baseline=0] |
384 \begin{scope} |
384 \begin{scope} |
385 \path[clip] (0,0) arc (135:45:4) arc (-45:-135:4); |
385 \path[clip] (0,0) arc (135:45:4) arc (-45:-135:4); |
386 \draw[blue,line width=2pt] (0,0) arc (135:45:4) arc (-45:-135:4); |
386 \draw[kw-blue-a,line width=2pt] (0,0) arc (135:45:4) arc (-45:-135:4); |
387 \foreach \x in {0, 0.5, ..., 6} { |
387 \foreach \x in {0, 0.5, ..., 6} { |
388 \draw[green!50!brown] (\x,-2) -- (\x,2); |
388 \draw[green!50!brown] (\x,-2) -- (\x,2); |
389 } |
389 } |
390 \end{scope} |
390 \end{scope} |
391 \draw[blue,line width=1.5pt] (0,-3) -- (5.66,-3); |
391 \draw[kw-blue-a,line width=1.5pt] (0,-3) -- (5.66,-3); |
392 \draw[->,red,line width=2pt] (2.83,-1.5) -- (2.83,-2.5); |
392 \draw[->,red,line width=2pt] (2.83,-1.5) -- (2.83,-2.5); |
393 \end{tikzpicture} |
393 \end{tikzpicture} |
394 \qquad \qquad |
394 \qquad \qquad |
395 \begin{tikzpicture}[baseline=-0.15cm] |
395 \begin{tikzpicture}[baseline=-0.15cm] |
396 \begin{scope} |
396 \begin{scope} |
397 \path[clip] (0,1) arc (90:135:8 and 4) arc (-135:-90:8 and 4) -- cycle; |
397 \path[clip] (0,1) arc (90:135:8 and 4) arc (-135:-90:8 and 4) -- cycle; |
398 \draw[blue,line width=2pt] (0,1) arc (90:135:8 and 4) arc (-135:-90:8 and 4) -- cycle; |
398 \draw[kw-blue-a,line width=2pt] (0,1) arc (90:135:8 and 4) arc (-135:-90:8 and 4) -- cycle; |
399 \foreach \x in {-6, -5.5, ..., 0} { |
399 \foreach \x in {-6, -5.5, ..., 0} { |
400 \draw[green!50!brown] (\x,-2) -- (\x,2); |
400 \draw[green!50!brown] (\x,-2) -- (\x,2); |
401 } |
401 } |
402 \end{scope} |
402 \end{scope} |
403 \draw[blue,line width=1.5pt] (-5.66,-3.15) -- (0,-3.15); |
403 \draw[kw-blue-a,line width=1.5pt] (-5.66,-3.15) -- (0,-3.15); |
404 \draw[->,red,line width=2pt] (-2.83,-1.5) -- (-2.83,-2.5); |
404 \draw[->,red,line width=2pt] (-2.83,-1.5) -- (-2.83,-2.5); |
405 \end{tikzpicture} |
405 \end{tikzpicture} |
406 $$ |
406 $$ |
407 \caption{Examples of pinched products}\label{pinched_prods} |
407 \caption{Examples of pinched products}\label{pinched_prods} |
408 \end{figure} |
408 \end{figure} |
435 \begin{figure}[t] |
435 \begin{figure}[t] |
436 $$ |
436 $$ |
437 \begin{tikzpicture}[baseline=0] |
437 \begin{tikzpicture}[baseline=0] |
438 \begin{scope} |
438 \begin{scope} |
439 \path[clip] (0,0) arc (135:45:4) arc (-45:-135:4); |
439 \path[clip] (0,0) arc (135:45:4) arc (-45:-135:4); |
440 \draw[blue,line width=2pt] (0,0) arc (135:45:4) arc (-45:-135:4); |
440 \draw[kw-blue-a,line width=2pt] (0,0) arc (135:45:4) arc (-45:-135:4); |
441 \draw[blue] (0,0) -- (5.66,0); |
441 \draw[kw-blue-a] (0,0) -- (5.66,0); |
442 \foreach \x in {0, 0.5, ..., 6} { |
442 \foreach \x in {0, 0.5, ..., 6} { |
443 \draw[green!50!brown] (\x,-2) -- (\x,2); |
443 \draw[green!50!brown] (\x,-2) -- (\x,2); |
444 } |
444 } |
445 \end{scope} |
445 \end{scope} |
446 \end{tikzpicture} |
446 \end{tikzpicture} |
447 \qquad |
447 \qquad |
448 \begin{tikzpicture}[baseline=0] |
448 \begin{tikzpicture}[baseline=0] |
449 \begin{scope} |
449 \begin{scope} |
450 \path[clip] (0,-1) rectangle (4,1); |
450 \path[clip] (0,-1) rectangle (4,1); |
451 \draw[blue,line width=2pt] (0,-1) rectangle (4,1); |
451 \draw[kw-blue-a,line width=2pt] (0,-1) rectangle (4,1); |
452 \draw[blue] (0,0) -- (5,0); |
452 \draw[kw-blue-a] (0,0) -- (5,0); |
453 \foreach \x in {0, 0.5, ..., 6} { |
453 \foreach \x in {0, 0.5, ..., 6} { |
454 \draw[green!50!brown] (\x,-2) -- (\x,2); |
454 \draw[green!50!brown] (\x,-2) -- (\x,2); |
455 } |
455 } |
456 \end{scope} |
456 \end{scope} |
457 \end{tikzpicture} |
457 \end{tikzpicture} |
458 \qquad |
458 \qquad |
459 \begin{tikzpicture}[baseline=0] |
459 \begin{tikzpicture}[baseline=0] |
460 \begin{scope} |
460 \begin{scope} |
461 \path[clip] (0,0) arc (135:45:4) arc (-45:-135:4); |
461 \path[clip] (0,0) arc (135:45:4) arc (-45:-135:4); |
462 \draw[blue,line width=2pt] (0,0) arc (135:45:4) arc (-45:-135:4); |
462 \draw[kw-blue-a,line width=2pt] (0,0) arc (135:45:4) arc (-45:-135:4); |
463 \draw[blue] (2.83,3) circle (3); |
463 \draw[kw-blue-a] (2.83,3) circle (3); |
464 \foreach \x in {0, 0.5, ..., 6} { |
464 \foreach \x in {0, 0.5, ..., 6} { |
465 \draw[green!50!brown] (\x,-2) -- (\x,2); |
465 \draw[green!50!brown] (\x,-2) -- (\x,2); |
466 } |
466 } |
467 \end{scope} |
467 \end{scope} |
468 \end{tikzpicture} |
468 \end{tikzpicture} |
469 $$ |
469 $$ |
470 $$ |
470 $$ |
471 \begin{tikzpicture}[baseline=0] |
471 \begin{tikzpicture}[baseline=0] |
472 \begin{scope} |
472 \begin{scope} |
473 \path[clip] (0,-1) rectangle (4,1); |
473 \path[clip] (0,-1) rectangle (4,1); |
474 \draw[blue,line width=2pt] (0,-1) rectangle (4,1); |
474 \draw[kw-blue-a,line width=2pt] (0,-1) rectangle (4,1); |
475 \draw[blue] (0,-1) -- (4,1); |
475 \draw[kw-blue-a] (0,-1) -- (4,1); |
476 \foreach \x in {0, 0.5, ..., 6} { |
476 \foreach \x in {0, 0.5, ..., 6} { |
477 \draw[green!50!brown] (\x,-2) -- (\x,2); |
477 \draw[green!50!brown] (\x,-2) -- (\x,2); |
478 } |
478 } |
479 \end{scope} |
479 \end{scope} |
480 \end{tikzpicture} |
480 \end{tikzpicture} |
481 \qquad |
481 \qquad |
482 \begin{tikzpicture}[baseline=0] |
482 \begin{tikzpicture}[baseline=0] |
483 \begin{scope} |
483 \begin{scope} |
484 \path[clip] (0,-1) rectangle (5,1); |
484 \path[clip] (0,-1) rectangle (5,1); |
485 \draw[blue,line width=2pt] (0,-1) rectangle (5,1); |
485 \draw[kw-blue-a,line width=2pt] (0,-1) rectangle (5,1); |
486 \draw[blue] (1,-1) .. controls (2,-1) and (3,1) .. (4,1); |
486 \draw[kw-blue-a] (1,-1) .. controls (2,-1) and (3,1) .. (4,1); |
487 \foreach \x in {0, 0.5, ..., 6} { |
487 \foreach \x in {0, 0.5, ..., 6} { |
488 \draw[green!50!brown] (\x,-2) -- (\x,2); |
488 \draw[green!50!brown] (\x,-2) -- (\x,2); |
489 } |
489 } |
490 \end{scope} |
490 \end{scope} |
491 \end{tikzpicture} |
491 \end{tikzpicture} |
492 \qquad |
492 \qquad |
493 \begin{tikzpicture}[baseline=0] |
493 \begin{tikzpicture}[baseline=0] |
494 \begin{scope} |
494 \begin{scope} |
495 \path[clip] (0,0) arc (135:45:4) arc (-45:-135:4); |
495 \path[clip] (0,0) arc (135:45:4) arc (-45:-135:4); |
496 \draw[blue,line width=2pt] (0,0) arc (135:45:4) arc (-45:-135:4); |
496 \draw[kw-blue-a,line width=2pt] (0,0) arc (135:45:4) arc (-45:-135:4); |
497 \draw[blue] (2.82,-5) -- (2.83,5); |
497 \draw[kw-blue-a] (2.82,-5) -- (2.83,5); |
498 \foreach \x in {0, 0.5, ..., 6} { |
498 \foreach \x in {0, 0.5, ..., 6} { |
499 \draw[green!50!brown] (\x,-2) -- (\x,2); |
499 \draw[green!50!brown] (\x,-2) -- (\x,2); |
500 } |
500 } |
501 \end{scope} |
501 \end{scope} |
502 \end{tikzpicture} |
502 \end{tikzpicture} |
2533 The set $\cD$ breaks into ``blocks" according to the restrictions to the pinched points of $X\times J$ |
2533 The set $\cD$ breaks into ``blocks" according to the restrictions to the pinched points of $X\times J$ |
2534 (see Figure \ref{feb21b}). |
2534 (see Figure \ref{feb21b}). |
2535 These restrictions are 0-morphisms $(a, b)$ of $\cA$ and $\cB$. |
2535 These restrictions are 0-morphisms $(a, b)$ of $\cA$ and $\cB$. |
2536 |
2536 |
2537 \begin{figure}[t] \centering |
2537 \begin{figure}[t] \centering |
2538 \begin{tikzpicture}[blue,line width=2pt] |
2538 \begin{tikzpicture}[kw-blue-a,line width=2pt] |
2539 \draw (0,1) -- (0,-1) node[below] {$X$}; |
2539 \draw (0,1) -- (0,-1) node[below] {$X$}; |
2540 |
2540 |
2541 \draw (2,0) -- (4,0) node[below] {$J$}; |
2541 \draw (2,0) -- (4,0) node[below] {$J$}; |
2542 \fill[red] (3,0) circle (0.1); |
2542 \fill[red] (3,0) circle (0.1); |
2543 |
2543 |
2544 \draw[fill=blue!30!white] (6,0) node(a) {} arc (135:90:4) node(top) {} arc (90:45:4) node(b) {} arc (-45:-90:4) node(bottom) {} arc(-90:-135:4); |
2544 \draw[fill=kw-blue-a!30!white] (6,0) node(a) {} arc (135:90:4) node(top) {} arc (90:45:4) node(b) {} arc (-45:-90:4) node(bottom) {} arc(-90:-135:4); |
2545 \draw[red] (top.center) -- (bottom.center); |
2545 \draw[red] (top.center) -- (bottom.center); |
2546 \fill (a) circle (0.1) node[left] {\color{green!50!brown} $a$}; |
2546 \fill (a) circle (0.1) node[left] {\color{green!50!brown} $a$}; |
2547 \fill (b) circle (0.1) node[right] {\color{green!50!brown} $b$}; |
2547 \fill (b) circle (0.1) node[right] {\color{green!50!brown} $b$}; |
2548 |
2548 |
2549 \path (bottom) node[below]{$X \times J$}; |
2549 \path (bottom) node[below]{$X \times J$}; |
2561 to obtain an $\cA_0$-$\cA_l$ $0$-sphere module and, forgetfully, an $n{-}1$-category. |
2561 to obtain an $\cA_0$-$\cA_l$ $0$-sphere module and, forgetfully, an $n{-}1$-category. |
2562 This amounts to a definition of taking tensor products of $0$-sphere modules over $n$-categories. |
2562 This amounts to a definition of taking tensor products of $0$-sphere modules over $n$-categories. |
2563 |
2563 |
2564 \begin{figure}[t] \centering |
2564 \begin{figure}[t] \centering |
2565 \begin{tikzpicture}[baseline,line width = 2pt] |
2565 \begin{tikzpicture}[baseline,line width = 2pt] |
2566 \draw[blue] (0,0) -- (6,0); |
2566 \draw[kw-blue-a] (0,0) -- (6,0); |
2567 \foreach \x/\n in {0.5/0,1.5/1,3/2,4.5/3,5.5/4} { |
2567 \foreach \x/\n in {0.5/0,1.5/1,3/2,4.5/3,5.5/4} { |
2568 \path (\x,0) node[below] {\color{green!50!brown}$\cA_{\n}$}; |
2568 \path (\x,0) node[below] {\color{green!50!brown}$\cA_{\n}$}; |
2569 } |
2569 } |
2570 \foreach \x/\n in {1/0,2/1,4/2,5/3} { |
2570 \foreach \x/\n in {1/0,2/1,4/2,5/3} { |
2571 \fill[red] (\x,0) circle (0.1) node[above] {\color{green!50!brown}$\cM_{\n}$}; |
2571 \fill[red] (\x,0) circle (0.1) node[above] {\color{green!50!brown}$\cM_{\n}$}; |
2572 } |
2572 } |
2573 \end{tikzpicture} |
2573 \end{tikzpicture} |
2574 \qquad |
2574 \qquad |
2575 \qquad |
2575 \qquad |
2576 \begin{tikzpicture}[baseline,line width = 2pt] |
2576 \begin{tikzpicture}[baseline,line width = 2pt] |
2577 \draw[blue] (0,0) circle (2); |
2577 \draw[kw-blue-a] (0,0) circle (2); |
2578 \foreach \q/\n in {-45/0,90/1,180/2} { |
2578 \foreach \q/\n in {-45/0,90/1,180/2} { |
2579 \path (\q:2.4) node {\color{green!50!brown}$\cA_{\n}$}; |
2579 \path (\q:2.4) node {\color{green!50!brown}$\cA_{\n}$}; |
2580 } |
2580 } |
2581 \foreach \q/\n in {60/0,120/1,-120/2} { |
2581 \foreach \q/\n in {60/0,120/1,-120/2} { |
2582 \fill[red] (\q:2) circle (0.1); |
2582 \fill[red] (\q:2) circle (0.1); |
2611 (See Figure \ref{subdividing1marked}.) |
2611 (See Figure \ref{subdividing1marked}.) |
2612 We now proceed as in the above module definitions. |
2612 We now proceed as in the above module definitions. |
2613 |
2613 |
2614 \begin{figure}[t] \centering |
2614 \begin{figure}[t] \centering |
2615 \begin{tikzpicture}[baseline,line width = 2pt] |
2615 \begin{tikzpicture}[baseline,line width = 2pt] |
2616 \draw[blue][fill=blue!15!white] (0,0) circle (2); |
2616 \draw[kw-blue-a][fill=kw-blue-a!15!white] (0,0) circle (2); |
2617 \fill[red] (0,0) circle (0.1); |
2617 \fill[red] (0,0) circle (0.1); |
2618 \foreach \qm/\qa/\n in {70/-30/0, 120/95/1, -120/180/2} { |
2618 \foreach \qm/\qa/\n in {70/-30/0, 120/95/1, -120/180/2} { |
2619 \draw[red] (0,0) -- (\qm:2); |
2619 \draw[red] (0,0) -- (\qm:2); |
2620 \path (\qa:1) node {\color{green!50!brown} $\cA_\n$}; |
2620 \path (\qa:1) node {\color{green!50!brown} $\cA_\n$}; |
2621 \path (\qm+20:2.5) node(M\n) {\color{green!50!brown} $\cM_\n$}; |
2621 \path (\qm+20:2.5) node(M\n) {\color{green!50!brown} $\cM_\n$}; |