1424:
81:. Like ordinary categories, they contain objects (the 0-simplices of the simplicial set) and morphisms between these objects (1-simplices). But unlike categories, the composition of two morphisms need not be uniquely defined. All the morphisms that can serve as composition of two given morphisms are related to each other by higher order invertible morphisms (2-simplices thought of as "homotopies"). These higher order morphisms can also be composed, but again the composition is well-defined only up to still higher order invertible morphisms, etc.
1671:
1691:
1681:
84:
The idea of higher category theory (at least, higher category theory when higher morphisms are invertible) is that, as opposed to the standard notion of a category, there should be a mapping space (rather than a mapping set) between two objects. This suggests that a higher category should simply be a
639:
is an example of a quasi-category. In a Kan complex all maps from all horns—not just inner ones—can be filled, which again has the consequence that all morphisms in a Kan complex are invertible. Kan complexes are thus analogues to groupoids - the nerve of a category is a Kan complex iff the category
697:
All of (∞,1)-topos theory can be modeled in terms of sSet-categories. (ToënVezzosi). There is a notion of sSet-site C that models the notion of (∞,1)-site and a model structure on sSet-enriched presheaves on sSet-sites that is a presentation for the ∞-stack (∞,1)-toposes on
577:
is a quasi-category with the extra property that the filling of any inner horn is unique. Conversely a quasi-category such that any inner horn has a unique filling is isomorphic to the nerve of some category. The homotopy category of the nerve of
89:. The model of quasi-categories is, however, better suited to applications than that of topologically enriched categories, though it has been proved by Lurie that the two have natural model structures that are
690:
The notion of homotopy Kan extension and hence in particular that of homotopy limit and homotopy colimit has a direct formulation in terms of Kan-complex-enriched categories. See homotopy Kan extension for
66:
and some of the advanced notions and theorems have their analogues for
Generalization of a category. An elaborate treatise of the theory of Generalization of a category has been expounded by
437:
369:
153:
511:
562:
327:
289:
220:
185:
499:
253:
371:
represents a composable pair. Thus, in a quasi-category, one cannot define a composition law on morphisms, since one can choose many ways to compose maps.
439:
is a trivial Kan fibration. In other words, while the composition law is not uniquely defined, it is unique up to a contractible choice.
1068:
1353:
467:
The morphisms are given by homotopy classes of edges between vertices. Composition is given using the horn filler condition for
909:
762:
853:
377:
1720:
667:
1061:
1265:
1220:
625:
1694:
1634:
1343:
594:
1684:
1470:
1334:
1242:
891:
332:
116:
926:
1643:
1287:
1225:
1148:
522:
1715:
1674:
1630:
1235:
1054:
713:
662: > 1 are equivalences. There are several models of (∞, 1)-categories, including
297:
258:
190:
1230:
1212:
1015:
158:
62:
has much advanced the study of
Generalization of a category showing that most of the usual basic
1437:
1203:
1183:
1106:
723:
477:
48:
44:
1319:
1158:
931:
780:
229:
1013:
Toën, Bertrand; Vezzosi, Gabriele (2005), "Homotopical
Algebraic Geometry I: Topos theory",
985:
1131:
1126:
919:
845:
816:
772:
671:
86:
855:
The theory of
Generalization of a category and its applications, lectures at CRM Barcelona
109:
satisfying the inner Kan conditions (also called weak Kan condition): every inner horn in
8:
1475:
1423:
1349:
1153:
574:
223:
684:
There is a model structure on sSet-categories that presents the (∞,1)-category (∞,1)Cat.
1329:
1324:
1306:
1188:
1163:
1024:
999:
972:
895:
831:
830:, Contemp. Math., vol. 431, Providence, R.I.: Amer. Math. Soc., pp. 277–326,
90:
808:
616:) is a quasi-category in which every morphism is invertible. The homotopy category of
1638:
1575:
1563:
1465:
1390:
1385:
1339:
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1116:
955:
905:
758:
456:
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1314:
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1111:
1034:
804:
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943:
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329:
are supposed to represent commutative triangles (at least up to homotopy). A map
63:
20:
1548:
1527:
1490:
1480:
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718:
708:
663:
106:
78:
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862:
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59:
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1370:
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968:
1292:
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728:
1553:
1533:
1405:
1275:
826:; Tierney, Myles (2007), "Generalization of a category vs Segal spaces",
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636:
67:
1029:
900:
836:
1585:
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754:
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519:
the fundamental category is the same as the homotopy category, i.e.
1648:
1280:
1178:
1004:
1046:
998:
Hinich, Vladimir (2017-09-19). "Lectures on infinity categories".
977:
1618:
1608:
1257:
1168:
745:, Lecture Notes in Mathematics, vol. 347, Berlin, New York:
971:(2011). "Workshop on the homotopy theory of homotopy theories".
1613:
1495:
743:
Homotopy invariant algebraic structures on topological spaces
654:
is a not-necessarily-quasi-category ∞-category in which all
989:
959:
947:
935:
795:(2002), "Generalization of a category and Kan complexes",
828:
Categories in algebra, geometry and mathematical physics
463:. The homotopy category has as objects the vertices of
525:
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232:
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119:
678:. A quasi-category is also an (∞, 1)-category.
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493:
431:
363:
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283:
247:
214:
179:
147:
1707:
890:, Annals of Mathematics Studies, vol. 170,
474:For a general simplicial set there is a functor
187:, has a filler, that is, an extension to a map
54:Generalization of a category were introduced by
47:. The study of such generalizations is known as
432:{\displaystyle C^{\Delta }\to C^{\Lambda ^{1}}}
635:More general than the previous example, every
1062:
451:one can associate to it an ordinary category
822:
740:
55:
1012:
1690:
1680:
1436:
1069:
1055:
374:One consequence of the definition is that
43:) is a generalization of the notion of a
1038:
1028:
1003:
976:
899:
835:
77:Generalization of a category are certain
442:
226:for a definition of the simplicial sets
967:
1708:
997:
1435:
1088:
1050:
885:
851:
791:
782:A short course on infinity-categories
741:Boardman, J. M.; Vogt, R. M. (1973),
71:
1076:
797:Journal of Pure and Applied Algebra
13:
695:Presentation of (∞,1)-topos theory
409:
386:
337:
301:
263:
233:
194:
121:
113:, namely a map of simplicial sets
19:In mathematics, more specifically
14:
1732:
872:
778:
606:fundamental ∞-groupoid of X
364:{\displaystyle \Lambda ^{1}\to C}
148:{\displaystyle \Lambda ^{k}\to C}
1689:
1679:
1670:
1669:
1422:
1089:
101:By definition, a quasi-category
557:{\displaystyle \tau _{1}(C)=hC}
87:topologically enriched category
927:The theory of quasi-categories
668:simplicially enriched category
542:
536:
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130:
1:
809:10.1016/S0022-4049(02)00135-4
734:
294:The idea is that 2-simplices
96:
932:Generalization-of-a-category
512:fundamental category functor
322:{\displaystyle \Delta \to C}
284:{\displaystyle \Lambda ^{k}}
215:{\displaystyle \Delta \to C}
16:Generalization of a category
7:
1364:Constructions on categories
702:
644:
567:
515:, and for a quasi-category
180:{\displaystyle 0<k<n}
10:
1737:
1471:Higher-dimensional algebra
892:Princeton University Press
589:Given a topological space
56:Boardman & Vogt (1973)
1665:
1598:
1562:
1510:
1503:
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1420:
1363:
1305:
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1211:
1202:
1099:
1095:
1084:
1040:10.1016/j.aim.2004.05.004
494:{\displaystyle \tau _{1}}
224:Kan fibration#Definitions
876:Notes on quasicategories
714:Stable infinity category
1281:Cokernels and quotients
1204:Universal constructions
1016:Advances in Mathematics
447:Given a quasi-category
248:{\displaystyle \Delta }
1721:Higher category theory
1438:Higher category theory
1184:Natural transformation
925:Joyal's Catlab entry:
724:Higher category theory
688:Homotopy Kan extension
558:
495:
433:
365:
323:
285:
249:
216:
181:
149:
49:higher category theory
886:Lurie, Jacob (2009),
604:), also known as the
593:, one can define its
559:
496:
443:The homotopy category
434:
366:
324:
286:
250:
217:
182:
150:
1307:Algebraic categories
986:(∞, 1)-category
956:fundamental+category
676:complete Segal space
672:topological category
652:(∞, 1)-category
626:fundamental groupoid
523:
478:
378:
333:
298:
259:
230:
191:
159:
117:
1476:Homotopy hypothesis
1154:Commutative diagram
888:Higher topos theory
575:nerve of a category
1189:Universal property
852:Joyal, A. (2008),
755:10.1007/BFb0068547
554:
491:
429:
361:
319:
281:
245:
212:
177:
145:
91:Quillen equivalent
1703:
1702:
1661:
1660:
1657:
1656:
1639:monoidal category
1594:
1593:
1466:Enriched category
1418:
1417:
1414:
1413:
1391:Quotient category
1386:Opposite category
1301:
1300:
944:infinity-category
911:978-0-691-14049-0
764:978-3-540-06479-4
582:is isomorphic to
457:homotopy category
33:infinity category
29:inner Kan complex
1728:
1693:
1692:
1683:
1682:
1673:
1672:
1508:
1507:
1486:Simplex category
1461:Categorification
1452:
1451:
1433:
1432:
1426:
1396:Product category
1381:Kleisli category
1376:Functor category
1221:Terminal objects
1209:
1208:
1144:Adjoint functors
1097:
1096:
1086:
1085:
1071:
1064:
1057:
1048:
1047:
1043:
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1032:
1009:
1007:
982:
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969:Bergner, Julia E
922:
903:
882:
881:
869:
867:
861:, archived from
860:
848:
839:
819:
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787:
775:
563:
561:
560:
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535:
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500:
498:
497:
492:
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489:
471: = 2.
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129:
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41:Boardman complex
25:weak Kan complex
1736:
1735:
1731:
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1729:
1727:
1726:
1725:
1716:Homotopy theory
1706:
1705:
1704:
1699:
1653:
1623:
1590:
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1410:
1359:
1297:
1266:Initial objects
1252:
1198:
1091:
1080:
1078:Category theory
1075:
1030:math.AG/0207028
912:
901:math.CT/0608040
879:
868:on July 6, 2011
865:
858:
837:math.AT/0607820
785:
779:Groth, Moritz,
765:
747:Springer-Verlag
737:
705:
682:Model structure
658:-morphisms for
647:
570:
530:
526:
524:
521:
520:
509:, known as the
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481:
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192:
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124:
120:
118:
115:
114:
99:
79:simplicial sets
68:Jacob Lurie
64:category theory
21:category theory
17:
12:
11:
5:
1734:
1724:
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1491:String diagram
1488:
1483:
1481:Model category
1478:
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1458:
1456:
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1429:
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1421:
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1408:
1403:
1401:Comma category
1398:
1393:
1388:
1383:
1378:
1373:
1367:
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1361:
1360:
1358:
1357:
1347:
1337:
1335:Abelian groups
1332:
1327:
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1317:
1311:
1309:
1303:
1302:
1299:
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1254:
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1251:
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1215:
1206:
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1199:
1197:
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1186:
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1101:
1093:
1092:
1082:
1081:
1074:
1073:
1066:
1059:
1051:
1045:
1044:
1023:(2): 257–372,
1010:
995:
983:
965:
953:
941:
929:
923:
910:
883:
870:
849:
820:
803:(1): 207–222,
789:
776:
763:
736:
733:
732:
731:
726:
721:
716:
711:
709:Model category
704:
701:
700:
699:
692:
685:
679:
664:Segal category
646:
643:
642:
641:
640:is a groupoid.
633:
587:
569:
566:
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107:simplicial set
98:
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15:
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4:
3:
2:
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1552:
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1547:
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1542:
1540:
1539:Tetracategory
1537:
1535:
1532:
1529:
1528:pseudofunctor
1525:
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1371:Free category
1369:
1368:
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1355:
1354:Vector spaces
1351:
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1259:
1255:
1249:
1248:Inverse limit
1246:
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1234:
1233:
1232:
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1227:
1224:
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1219:
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1207:
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1201:
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1190:
1187:
1185:
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1177:
1175:
1174:Kan extension
1172:
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1165:
1162:
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1157:
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689:
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680:
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649:
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634:
631:
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623:
619:
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611:
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603:
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588:
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551:
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531:
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508:
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486:
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472:
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458:
454:
450:
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421:
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404:
392:
382:
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358:
349:
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316:
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292:
275:
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239:
225:
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174:
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133:
125:
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94:
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88:
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69:
65:
61:
57:
52:
50:
46:
42:
38:
34:
30:
26:
22:
1619:
1600:Categorified
1543:
1504:n-categories
1455:Key concepts
1293:Direct limit
1276:Coequalizers
1194:Yoneda lemma
1100:Key concepts
1090:Key concepts
1020:
1014:
990:
960:
948:
936:
887:
875:
863:the original
854:
827:
824:Joyal, André
800:
796:
793:Joyal, André
781:
742:
729:Globular set
694:
687:
681:
659:
655:
651:
629:
621:
617:
613:
609:
605:
601:
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1568:-categories
1544:Kan complex
1534:Tricategory
1516:-categories
1406:Subcategory
1164:Exponential
1132:Preadditive
1127:Pre-abelian
873:Joyal, A.,
637:Kan complex
455:called the
60:André Joyal
1710:Categories
1586:3-category
1576:2-category
1549:∞-groupoid
1524:Bicategory
1271:Coproducts
1231:Equalizers
1137:Bicategory
1005:1709.06271
735:References
719:∞-groupoid
97:Definition
37:∞-category
1635:Symmetric
1580:2-functor
1320:Relations
1243:Pullbacks
978:1108.2001
624:) is the
528:τ
483:τ
410:Λ
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1695:Glossary
1675:Category
1649:n-monoid
1602:concepts
1258:Colimits
1226:Products
1179:Morphism
1122:Concrete
1117:Additive
1107:Category
703:See also
645:Variants
568:Examples
45:category
1685:Outline
1644:n-group
1609:2-group
1564:Strict
1554:∞-topos
1350:Modules
1288:Pushout
1236:Kernels
1169:Functor
1112:Abelian
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1496:Topos
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880:(PDF)
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