719:. Note that while cobordisms are cospans, the category of cobordisms is not a "cospan category": it is not the category of all cospans in "the category of manifolds with inclusions on the boundary", but rather a
603:
141:
386:
340:
510:
635:
173:
516:
can be considered a generalised morphism (i.e., where one "inverts the weak equivalences"). Note that this is not the usual point of view taken when dealing with model categories.
436:
838:
548:
474:
93:
345:
299:
772:
691:
64:
56:
777:
480:
245:
87:
608:
146:
48:
743:
687:
52:
409:
858:
834:
262:
826:
767:
44:
28:
75:
466:
852:
71:
720:
59:(and satisfies a small number of other conditions), spans can be considered as
782:
698:
705:
60:
178:
That is, let Λ be the category (-1 ← 0 → +1). Then a span in a category
830:
825:. Lecture Notes in Mathematics. Vol. 47. Springer. pp. 1–77.
241:
183:
17:
795:
273:
598:{\displaystyle \Lambda ^{\text{op}}=(-1\rightarrow 0\leftarrow +1),}
799:
808:
Yoneda, Nobuo (1954). "On the homology theory of modules".
438:
be a morphism in some category. There is a trivial span
136:{\displaystyle \Lambda =(-1\leftarrow 0\rightarrow +1),}
821:
611:
551:
483:
412:
381:{\displaystyle X\times Y{\overset {\pi _{Y}}{\to }}Y}
348:
335:{\displaystyle X\times Y{\overset {\pi _{X}}{\to }}X}
302:
149:
96:
193:. This means that a span consists of three objects
629:
597:
504:
430:
380:
334:
296:is a span, where the maps are the projection maps
167:
135:
715:, where the two maps are the inclusions into
8:
758:with finite limits is also dagger compact.
505:{\displaystyle X\leftarrow Y\rightarrow Z,}
630:{\displaystyle Y\rightarrow X\leftarrow Z}
168:{\displaystyle Y\leftarrow X\rightarrow Z}
610:
556:
550:
533:is a functor K : Λ →
482:
411:
367:
358:
347:
321:
312:
301:
148:
95:
450:, where the left map is the identity on
823:Reports of the Midwest Category Seminar
742:of finite-dimensional cobordisms is a
43:is a generalization of the notion of
7:
731:form a partition of the boundary of
454:and the right map is the given map
391:Any object yields the trivial span
640:Thus it consists of three objects
553:
431:{\displaystyle \phi \colon A\to B}
97:
25:
723:thereof, as the requirement that
810:J. Fac. Sci. Univ. Tokyo Sect. I
746:. More generally, the category
403:where the maps are the identity.
70:The notion of a span is due to
621:
615:
589:
580:
574:
565:
512:where the left morphism is in
493:
487:
422:
360:
314:
159:
153:
127:
118:
112:
103:
1:
680:: it is two maps with common
545:. That is, a diagram of type
477:, then the spans of the form
233:: it is two maps with common
697:An example of a cospan is a
605:i.e., a diagram of the form
143:i.e., a diagram of the form
55:. When the category has all
754:) of spans on any category
189: : Λ →
875:
773:Pullback (category theory)
778:Pushout (category theory)
735:is a global constraint.
744:dagger compact category
631:
599:
506:
432:
382:
336:
261:is a relation between
169:
137:
632:
600:
507:
433:
383:
337:
170:
138:
65:category of fractions
609:
549:
481:
410:
406:More generally, let
346:
300:
147:
94:
831:10.1007/BFb0074299
627:
595:
541:functor from Λ to
537:; equivalently, a
502:
428:
378:
332:
165:
133:
840:978-3-540-35545-8
690:of a cospan is a
559:
475:weak equivalences
373:
327:
82:Formal definition
16:(Redirected from
866:
844:
817:
636:
634:
633:
628:
604:
602:
601:
596:
561:
560:
557:
511:
509:
508:
503:
437:
435:
434:
429:
387:
385:
384:
379:
374:
372:
371:
359:
341:
339:
338:
333:
328:
326:
325:
313:
174:
172:
171:
166:
142:
140:
139:
134:
21:
874:
873:
869:
868:
867:
865:
864:
863:
849:
848:
847:
841:
820:
807:
791:
768:Binary relation
764:
607:
606:
552:
547:
546:
523:
479:
478:
408:
407:
363:
344:
343:
317:
298:
297:
254:
244:of a span is a
145:
144:
92:
91:
84:
29:category theory
23:
22:
15:
12:
11:
5:
872:
870:
862:
861:
851:
850:
846:
845:
839:
818:
805:
792:
790:
787:
786:
785:
780:
775:
770:
763:
760:
656:and morphisms
626:
623:
620:
617:
614:
594:
591:
588:
585:
582:
579:
576:
573:
570:
567:
564:
555:
529:in a category
522:
519:
518:
517:
501:
498:
495:
492:
489:
486:
467:model category
459:
427:
424:
421:
418:
415:
404:
389:
377:
370:
366:
362:
357:
354:
351:
331:
324:
320:
316:
311:
308:
305:
253:
250:
209:and morphisms
164:
161:
158:
155:
152:
132:
129:
126:
123:
120:
117:
114:
111:
108:
105:
102:
99:
83:
80:
41:correspondence
24:
14:
13:
10:
9:
6:
4:
3:
2:
871:
860:
857:
856:
854:
842:
836:
832:
828:
824:
819:
815:
811:
806:
804:
802:
797:
794:
793:
788:
784:
781:
779:
776:
774:
771:
769:
766:
765:
761:
759:
757:
753:
749:
745:
741:
738:The category
736:
734:
730:
726:
722:
718:
714:
710:
707:
703:
700:
695:
693:
689:
684:
683:
679:
676: →
675:
672: :
671:
667:
664: →
663:
660: :
659:
655:
651:
647:
643:
638:
624:
618:
612:
592:
586:
583:
577:
571:
568:
562:
544:
540:
539:contravariant
536:
532:
528:
520:
515:
499:
496:
490:
484:
476:
472:
468:
464:
460:
457:
453:
449:
445:
441:
425:
419:
416:
413:
405:
402:
398:
394:
390:
375:
368:
364:
355:
352:
349:
329:
322:
318:
309:
306:
303:
295:
291:
287:
283:
279:
275:
271:
267:
264:
260:
256:
255:
251:
249:
247:
243:
238:
236:
232:
229: →
228:
225: :
224:
220:
217: →
216:
213: :
212:
208:
204:
200:
196:
192:
188:
185:
181:
176:
162:
156:
150:
130:
124:
121:
115:
109:
106:
100:
89:
81:
79:
77:
73:
68:
66:
62:
58:
54:
50:
46:
42:
38:
34:
30:
19:
822:
813:
809:
800:
755:
751:
747:
739:
737:
732:
728:
724:
716:
712:
708:
704:between two
701:
696:
685:
681:
677:
673:
669:
665:
661:
657:
653:
649:
645:
641:
639:
542:
538:
534:
530:
526:
524:
513:
470:
462:
455:
451:
447:
443:
439:
400:
396:
392:
293:
289:
285:
281:
277:
269:
265:
258:
239:
234:
230:
226:
222:
218:
214:
210:
206:
202:
198:
194:
190:
186:
179:
177:
86:A span is a
85:
76:Jean Bénabou
72:Nobuo Yoneda
69:
47:between two
40:
36:
32:
26:
721:subcategory
473:the set of
74:(1954) and
816:: 193–227.
789:References
783:Cobordism
706:manifolds
699:cobordism
682:codomain.
622:←
616:→
581:←
575:→
569:−
554:Λ
525:A cospan
494:→
488:←
423:→
417::
414:ϕ
365:π
361:→
353:×
319:π
315:→
307:×
160:→
154:←
119:→
113:←
107:−
98:Λ
61:morphisms
57:pullbacks
859:Functors
853:Category
762:See also
692:pullback
284:), then
272:(i.e. a
252:Examples
90:of type
78:(1967).
53:category
45:relation
798:at the
521:Cospans
469:, with
280:×
246:pushout
242:colimit
184:functor
88:diagram
49:objects
837:
274:subset
235:domain
18:Cospan
688:limit
465:is a
182:is a
63:in a
51:of a
835:ISBN
796:span
748:Span
740:nCob
727:and
711:and
686:The
668:and
648:and
342:and
268:and
263:sets
240:The
221:and
201:and
37:roof
33:span
31:, a
827:doi
803:Lab
652:of
461:If
276:of
257:If
205:of
39:or
27:In
855::
833:.
812:.
694:.
644:,
637:.
558:op
514:W,
452:A,
446:→
442:←
401:A,
399:→
395:←
292:→
288:←
248:.
237:.
197:,
175:.
67:.
35:,
843:.
829::
814:7
801:n
756:C
752:C
750:(
733:W
729:N
725:M
717:W
713:N
709:M
702:W
678:X
674:Z
670:g
666:X
662:Y
658:f
654:C
650:Z
646:Y
642:X
625:Z
619:X
613:Y
593:,
590:)
587:1
584:+
578:0
572:1
566:(
563:=
543:C
535:C
531:C
527:K
500:,
497:Z
491:Y
485:X
471:W
463:M
458:.
456:φ
448:B
444:A
440:A
426:B
420:A
397:A
393:A
388:.
376:Y
369:Y
356:Y
350:X
330:X
323:X
310:Y
304:X
294:Y
290:R
286:X
282:Y
278:X
270:Y
266:X
259:R
231:Z
227:X
223:g
219:Y
215:X
211:f
207:C
203:Z
199:Y
195:X
191:C
187:S
180:C
163:Z
157:X
151:Y
131:,
128:)
125:1
122:+
116:0
110:1
104:(
101:=
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.