1049:
1296:
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66:, this is an object of the category itself. In the same vein, in a closed category, the (object of) morphisms from one object to another can be seen as lying inside the category. This is the
298:
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90:
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77:
to the category of sets, which in particular takes the internal hom to the external hom.
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594:
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581:
191:{\displaystyle \left:{\mathcal {C}}^{op}\times {\mathcal {C}}\to {\mathcal {C}}}
24:
1210:
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761:
1204:
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16:
Category whose hom objects correspond (di-)naturally to objects in itself
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635:
Proceedings of the
Conference on Categorical Algebra. (La Jolla, 1965
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101:
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569:
are closed categories. The canonical example is the
1223:
1187:
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827:
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603:
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382:
362:
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591:is a closed category. In this case, the object
687:
543:all satisfying certain coherence conditions.
8:
293:{\displaystyle L:\left\to \left\left\right]}
1315:
1305:
1132:
1076:
1057:
833:
721:
710:
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555:are closed categories. In particular, any
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182:
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559:is closed. The canonical example is the
7:
468:{\displaystyle i_{A}:A\cong \left}
14:
1314:
1304:
1295:
1294:
1047:
532:{\displaystyle j_{A}:I\to \left}
637:. Springer. pp. 421–562.
507:
407:{\displaystyle {\mathcal {C}}}
239:
178:
109:{\displaystyle {\mathcal {C}}}
54:) maps a pair of objects to a
1:
643:10.1007/978-3-642-99902-4_22
73:Every closed category has a
989:Constructions on categories
553:Cartesian closed categories
1357:
1096:Higher-dimensional algebra
1290:
1069:
1056:
1045:
720:
709:
567:Compact closed categories
589:monoidal closed category
576:with finite-dimensional
480:dinatural transformation
906:Cokernels and quotients
829:Universal constructions
1063:Higher category theory
809:Natural transformation
605:
533:
469:
408:
384:
364:
340:
320:
294:
192:
110:
40:locally small category
611:is the monoidal unit.
606:
534:
470:
409:
385:
370:, and a fixed object
365:
341:
321:
295:
193:
111:
31:is a special kind of
932:Algebraic categories
595:
587:More generally, any
488:
424:
394:
374:
354:
330:
310:
211:
126:
118:internal Hom functor
96:
89:can be defined as a
1101:Homotopy hypothesis
779:Commutative diagram
631:"Closed categories"
416:natural isomorphism
814:Universal property
601:
529:
465:
404:
380:
360:
336:
316:
290:
188:
106:
1341:Closed categories
1328:
1327:
1286:
1285:
1282:
1281:
1264:monoidal category
1219:
1218:
1091:Enriched category
1043:
1042:
1039:
1038:
1016:Quotient category
1011:Opposite category
926:
925:
652:978-3-642-99902-4
604:{\displaystyle I}
520:
456:
383:{\displaystyle I}
363:{\displaystyle A}
339:{\displaystyle C}
319:{\displaystyle B}
276:
257:
230:
139:
116:with a so-called
75:forgetful functor
1348:
1318:
1317:
1308:
1307:
1298:
1297:
1133:
1111:Simplex category
1086:Categorification
1077:
1058:
1051:
1021:Product category
1006:Kleisli category
1001:Functor category
846:Terminal objects
834:
769:Adjoint functors
722:
711:
696:
689:
682:
673:
656:
610:
608:
607:
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561:category of sets
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64:category of sets
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1052:
1035:
984:
922:
891:Initial objects
877:
823:
716:
705:
703:Category theory
700:
660:Closed category
653:
621:
618:
593:
592:
580:as objects and
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133:
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94:
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87:closed category
83:
29:closed category
21:category theory
17:
12:
11:
5:
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1156:
1145:
1143:
1138:
1130:
1126:
1125:
1123:
1118:
1116:String diagram
1113:
1108:
1106:Model category
1103:
1098:
1093:
1088:
1083:
1081:
1074:
1073:
1070:
1067:
1066:
1061:
1054:
1053:
1046:
1044:
1041:
1040:
1037:
1036:
1034:
1033:
1028:
1026:Comma category
1023:
1018:
1013:
1008:
1003:
998:
992:
990:
986:
985:
983:
982:
972:
962:
960:Abelian groups
957:
952:
947:
942:
936:
934:
928:
927:
924:
923:
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824:
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801:
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684:
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669:
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564:
548:
545:
541:
540:
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509:
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503:
498:
494:
476:
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463:
459:
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379:
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335:
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199:
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185:
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136:
132:
103:
82:
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23:, a branch of
15:
13:
10:
9:
6:
4:
3:
2:
1353:
1342:
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1336:
1321:
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1199:
1198:
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1186:
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1177:
1175:
1172:
1170:
1167:
1165:
1164:Tetracategory
1162:
1160:
1157:
1154:
1153:pseudofunctor
1150:
1147:
1146:
1144:
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1032:
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1027:
1024:
1022:
1019:
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1012:
1009:
1007:
1004:
1002:
999:
997:
996:Free category
994:
993:
991:
987:
980:
979:Vector spaces
976:
973:
970:
966:
963:
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958:
956:
953:
951:
948:
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943:
941:
938:
937:
935:
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929:
919:
916:
914:
911:
907:
904:
903:
902:
899:
897:
894:
892:
889:
888:
886:
884:
880:
874:
873:Inverse limit
871:
869:
866:
862:
859:
858:
857:
854:
852:
849:
847:
844:
843:
841:
839:
835:
832:
830:
826:
820:
817:
815:
812:
810:
807:
805:
802:
800:
799:Kan extension
797:
795:
792:
790:
787:
785:
782:
780:
777:
775:
772:
770:
767:
763:
760:
758:
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748:
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743:
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738:
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734:
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730:
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727:
723:
719:
712:
708:
704:
697:
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690:
685:
683:
678:
677:
674:
668:
666:
661:
658:
654:
648:
644:
640:
636:
632:
628:
624:
623:Eilenberg, S.
620:
619:
615:
598:
590:
586:
584:as morphisms.
583:
579:
578:vector spaces
575:
572:
568:
565:
562:
558:
554:
551:
550:
546:
544:
525:
521:
515:
511:
504:
501:
496:
492:
484:
483:
482:
481:
461:
457:
451:
447:
443:
440:
437:
432:
428:
420:
419:
418:
417:
377:
357:
349:
333:
313:
305:
286:
281:
277:
271:
267:
262:
258:
252:
248:
243:
235:
231:
225:
221:
217:
214:
207:
206:
205:
204:
203:Yoneda arrows
168:
163:
160:
148:
144:
140:
134:
130:
122:
121:
120:
119:
92:
88:
80:
78:
76:
71:
69:
65:
61:
57:
53:
49:
45:
41:
36:
34:
30:
26:
22:
1244:
1225:Categorified
1129:n-categories
1080:Key concepts
918:Direct limit
901:Coequalizers
819:Yoneda lemma
725:Key concepts
715:Key concepts
664:
634:
573:
542:
477:
302:
200:
86:
84:
72:
68:internal hom
67:
62:. So in the
51:
47:
44:external hom
43:
37:
28:
18:
1193:-categories
1169:Kan complex
1159:Tricategory
1141:-categories
1031:Subcategory
789:Exponential
757:Preadditive
752:Pre-abelian
627:Kelly, G.M.
582:linear maps
25:mathematics
1211:3-category
1201:2-category
1174:∞-groupoid
1149:Bicategory
896:Coproducts
856:Equalizers
762:Bicategory
616:References
201:with left
81:Definition
1260:Symmetric
1205:2-functor
945:Relations
868:Pullbacks
629:(2012) .
508:→
444:≅
348:dinatural
240:→
179:→
169:×
141:−
135:−
60:morphisms
1335:Category
1320:Glossary
1300:Category
1274:n-monoid
1227:concepts
883:Colimits
851:Products
804:Morphism
747:Concrete
742:Additive
732:Category
571:category
547:Examples
91:category
33:category
1310:Outline
1269:n-group
1234:2-group
1189:Strict
1179:∞-topos
975:Modules
913:Pushout
861:Kernels
794:Functor
737:Abelian
662:at the
414:with a
304:natural
1256:Traced
1239:2-ring
969:Fields
955:Groups
950:Magmas
838:Limits
649:
574:FdVect
519:
478:and a
455:
275:
256:
229:
138:
42:, the
1250:-ring
1137:Weak
1121:Topos
965:Rings
557:topos
38:In a
940:Sets
647:ISBN
346:and
326:and
27:, a
784:End
774:CCC
667:Lab
639:doi
390:of
350:in
306:in
58:of
56:set
19:In
1337::
1262:)
1258:)(
645:.
633:.
625:;
85:A
70:.
50:,
35:.
1254:(
1247:n
1245:E
1207:)
1203:(
1191:n
1155:)
1151:(
1139:n
981:)
977:(
971:)
967:(
695:e
688:t
681:v
665:n
655:.
641::
599:I
563:.
539:,
526:]
522:A
516:A
512:[
505:I
502::
497:A
493:j
462:]
458:A
452:I
448:[
441:A
438::
433:A
429:i
400:C
378:I
358:A
334:C
314:B
287:]
282:]
278:C
272:A
268:[
263:]
259:B
253:A
249:[
244:[
236:]
232:C
226:B
222:[
218::
215:L
184:C
174:C
164:p
161:o
155:C
149::
145:]
131:[
102:C
52:y
48:x
46:(
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