1349:
1596:
1616:
1606:
373:
799:
864:
839:
285:
993:
942:
660:
986:
1190:
1145:
1619:
1559:
1609:
1395:
1259:
1167:
867:
434:
224:
723:. A wide subcategory is typically not full: the only wide full subcategory of a category is that category itself.
20:
1568:
1212:
1150:
1073:
628:
608:
535:
1640:
1599:
1555:
1160:
979:
50:
750:
1155:
1137:
1362:
1128:
1108:
1031:
879:
39:
1244:
1083:
1056:
1051:
742:
483:
1400:
1348:
1278:
1274:
1078:
588:
464:
905:
1645:
1254:
1249:
1231:
1113:
1088:
449:
445:
438:
1563:
1500:
1488:
1390:
1315:
1310:
1268:
1264:
1046:
1041:
938:
453:
1524:
1410:
1385:
1320:
1305:
1300:
1239:
1068:
1036:
930:
735:
543:
420:
1436:
1002:
31:
1473:
844:
819:
1468:
1452:
1415:
1405:
1325:
922:
884:
368:{\displaystyle \mathrm {Hom} _{\mathcal {S}}(X,Y)=\mathrm {Hom} _{\mathcal {C}}(X,Y).}
1634:
1463:
1295:
1172:
1098:
927:
Proceedings of the
International Conference on Category Theory, Como, Italy (CT 1990)
457:
65:
with the same identities and composition of morphisms. Intuitively, a subcategory of
1217:
1118:
520:
475:
1478:
957:
1458:
1200:
716:
539:
27:
1510:
1448:
1061:
416:
1504:
1195:
639:
516:
427:
929:. Lecture Notes in Mathematics. Vol. 1488. Springer. pp. 95–104.
638:
In some categories, one can also speak of morphisms of the category being
1573:
1205:
1103:
575:
With the definitions of the previous paragraph, for any (full) embedding
58:
1543:
1533:
1182:
1093:
934:
227:
1538:
1420:
971:
16:
Category whose objects and morphisms are inside a bigger category
961:
553:
to be a full and faithful functor that is injective on objects.
211:
is a category in its own right: its collection of objects is ob(
1360:
1013:
975:
478:
forms a full subcategory of the category of (left or right)
426:
The category whose objects are sets and whose morphisms are
341:
303:
564:
is an embedding if it is injective on morphisms. A functor
538:. Such a functor is necessarily injective on objects up to
560:
if it is faithful and injective on objects. Equivalently,
696:. An isomorphism-closed full subcategory is said to be
847:
822:
753:
430:
forms a non-full subcategory of the category of sets.
288:
1523:
1487:
1435:
1428:
1379:
1288:
1230:
1181:
1136:
1127:
1024:
858:
833:
793:
456:) forms a non-full subcategory of the category of
367:
245:which takes objects and morphisms to themselves.
219:), and its identities and composition are as in
623:is not injective on objects then the image of
73:by "removing" some of its objects and arrows.
987:
925:(1991). "Algebraically complete categories".
887:, a full subcategory closed under extensions.
8:
1615:
1605:
1432:
1376:
1357:
1133:
1021:
1010:
994:
980:
972:
572:if it is a full functor and an embedding.
19:For subcategories on Knowledge (XXG), see
846:
821:
752:
340:
339:
328:
302:
301:
290:
287:
794:{\displaystyle 0\to M'\to M\to M''\to 0}
556:Other authors define a functor to be an
398:, there is a unique full subcategory of
378:A full subcategory is one that includes
896:
215:), its collection of morphisms is hom(
7:
719:) if it contains all the objects of
335:
332:
329:
297:
294:
291:
14:
1614:
1604:
1595:
1594:
1347:
730:is a non-empty full subcategory
437:forms a full subcategory of the
419:forms a full subcategory of the
390:. For any collection of objects
111:a subcollection of morphisms of
546:is an embedding in this sense.
515:is both a faithful functor and
252:be a subcategory of a category
785:
774:
768:
757:
359:
347:
321:
309:
100:a subcollection of objects of
1:
207:These conditions ensure that
21:Knowledge (XXG):Subcategories
866:do. This notion arises from
267:if for each pair of objects
179:for every pair of morphisms
69:is a category obtained from
1289:Constructions on categories
402:whose objects are those in
135:), the identity morphism id
1662:
1396:Higher-dimensional algebra
435:category of abelian groups
18:
1590:
1369:
1356:
1345:
1020:
1009:
609:isomorphism of categories
536:full and faithful functor
203:) whenever it is defined.
595:is a (full) subcategory
503:, the inclusion functor
1206:Cokernels and quotients
1129:Universal constructions
906:"Basic category theory"
715:(a term first posed by
549:Some authors define an
530:Some authors define an
527:is a full subcategory.
1363:Higher category theory
1109:Natural transformation
880:Reflective subcategory
860:
835:
795:
646:Types of subcategories
369:
223:. There is an obvious
861:
836:
796:
743:short exact sequences
668:if every isomorphism
448:(whose morphisms are
370:
1232:Algebraic categories
845:
820:
816:if and only if both
751:
542:. For instance, the
495:Given a subcategory
286:
1401:Homotopy hypothesis
1079:Commutative diagram
859:{\displaystyle M''}
386:between objects of
262:full subcategory of
164:), both the source
148:for every morphism
1114:Universal property
935:10.1007/BFb0084215
856:
834:{\displaystyle M'}
831:
791:
741:such that for all
661:isomorphism-closed
519:on objects. It is
470:, the category of
454:ring homomorphisms
439:category of groups
365:
1628:
1627:
1586:
1585:
1582:
1581:
1564:monoidal category
1519:
1518:
1391:Enriched category
1343:
1342:
1339:
1338:
1316:Quotient category
1311:Opposite category
1226:
1225:
944:978-3-540-54706-8
904:Jaap van Oosten.
728:Serre subcategory
703:A subcategory of
568:is then called a
243:inclusion functor
85:be a category. A
77:Formal definition
61:are morphisms in
1653:
1618:
1617:
1608:
1607:
1598:
1597:
1433:
1411:Simplex category
1386:Categorification
1377:
1358:
1351:
1321:Product category
1306:Kleisli category
1301:Functor category
1146:Terminal objects
1134:
1069:Adjoint functors
1022:
1011:
996:
989:
982:
973:
967:
958:Wide subcategory
955:
949:
948:
919:
913:
912:
910:
901:
868:Serre's C-theory
865:
863:
862:
857:
855:
840:
838:
837:
832:
830:
800:
798:
797:
792:
784:
767:
736:abelian category
692:also belongs to
544:Yoneda embedding
444:The category of
421:category of sets
415:The category of
374:
372:
371:
366:
346:
345:
344:
338:
308:
307:
306:
300:
191:) the composite
1661:
1660:
1656:
1655:
1654:
1652:
1651:
1650:
1641:Category theory
1631:
1630:
1629:
1624:
1578:
1548:
1515:
1492:
1483:
1440:
1424:
1375:
1365:
1352:
1335:
1284:
1222:
1191:Initial objects
1177:
1123:
1016:
1005:
1003:Category theory
1000:
970:
956:
952:
945:
921:
920:
916:
908:
903:
902:
898:
894:
876:
848:
843:
842:
823:
818:
817:
777:
760:
749:
748:
648:
523:if and only if
493:
412:
327:
289:
284:
283:
168:and the target
140:
79:
53:are objects in
32:category theory
30:, specifically
24:
17:
12:
11:
5:
1659:
1657:
1649:
1648:
1643:
1633:
1632:
1626:
1625:
1623:
1622:
1612:
1602:
1591:
1588:
1587:
1584:
1583:
1580:
1579:
1577:
1576:
1571:
1566:
1552:
1546:
1541:
1536:
1530:
1528:
1521:
1520:
1517:
1516:
1514:
1513:
1508:
1497:
1495:
1490:
1485:
1484:
1482:
1481:
1476:
1471:
1466:
1461:
1456:
1445:
1443:
1438:
1430:
1426:
1425:
1423:
1418:
1416:String diagram
1413:
1408:
1406:Model category
1403:
1398:
1393:
1388:
1383:
1381:
1374:
1373:
1370:
1367:
1366:
1361:
1354:
1353:
1346:
1344:
1341:
1340:
1337:
1336:
1334:
1333:
1328:
1326:Comma category
1323:
1318:
1313:
1308:
1303:
1298:
1292:
1290:
1286:
1285:
1283:
1282:
1272:
1262:
1260:Abelian groups
1257:
1252:
1247:
1242:
1236:
1234:
1228:
1227:
1224:
1223:
1221:
1220:
1215:
1210:
1209:
1208:
1198:
1193:
1187:
1185:
1179:
1178:
1176:
1175:
1170:
1165:
1164:
1163:
1153:
1148:
1142:
1140:
1131:
1125:
1124:
1122:
1121:
1116:
1111:
1106:
1101:
1096:
1091:
1086:
1081:
1076:
1071:
1066:
1065:
1064:
1059:
1054:
1049:
1044:
1039:
1028:
1026:
1018:
1017:
1014:
1007:
1006:
1001:
999:
998:
991:
984:
976:
969:
968:
950:
943:
914:
895:
893:
890:
889:
888:
885:Exact category
882:
875:
872:
854:
851:
829:
826:
802:
801:
790:
787:
783:
780:
776:
773:
770:
766:
763:
759:
756:
658:is said to be
650:A subcategory
647:
644:
570:full embedding
492:
489:
488:
487:
461:
442:
431:
424:
411:
408:
376:
375:
364:
361:
358:
355:
352:
349:
343:
337:
334:
331:
326:
323:
320:
317:
314:
311:
305:
299:
296:
293:
256:. We say that
205:
204:
177:
146:
136:
121:
120:
115:, denoted hom(
109:
78:
75:
45:is a category
15:
13:
10:
9:
6:
4:
3:
2:
1658:
1647:
1644:
1642:
1639:
1638:
1636:
1621:
1613:
1611:
1603:
1601:
1593:
1592:
1589:
1575:
1572:
1570:
1567:
1565:
1561:
1557:
1553:
1551:
1549:
1542:
1540:
1537:
1535:
1532:
1531:
1529:
1526:
1522:
1512:
1509:
1506:
1502:
1499:
1498:
1496:
1494:
1486:
1480:
1477:
1475:
1472:
1470:
1467:
1465:
1464:Tetracategory
1462:
1460:
1457:
1454:
1453:pseudofunctor
1450:
1447:
1446:
1444:
1442:
1434:
1431:
1427:
1422:
1419:
1417:
1414:
1412:
1409:
1407:
1404:
1402:
1399:
1397:
1394:
1392:
1389:
1387:
1384:
1382:
1378:
1372:
1371:
1368:
1364:
1359:
1355:
1350:
1332:
1329:
1327:
1324:
1322:
1319:
1317:
1314:
1312:
1309:
1307:
1304:
1302:
1299:
1297:
1296:Free category
1294:
1293:
1291:
1287:
1280:
1279:Vector spaces
1276:
1273:
1270:
1266:
1263:
1261:
1258:
1256:
1253:
1251:
1248:
1246:
1243:
1241:
1238:
1237:
1235:
1233:
1229:
1219:
1216:
1214:
1211:
1207:
1204:
1203:
1202:
1199:
1197:
1194:
1192:
1189:
1188:
1186:
1184:
1180:
1174:
1173:Inverse limit
1171:
1169:
1166:
1162:
1159:
1158:
1157:
1154:
1152:
1149:
1147:
1144:
1143:
1141:
1139:
1135:
1132:
1130:
1126:
1120:
1117:
1115:
1112:
1110:
1107:
1105:
1102:
1100:
1099:Kan extension
1097:
1095:
1092:
1090:
1087:
1085:
1082:
1080:
1077:
1075:
1072:
1070:
1067:
1063:
1060:
1058:
1055:
1053:
1050:
1048:
1045:
1043:
1040:
1038:
1035:
1034:
1033:
1030:
1029:
1027:
1023:
1019:
1012:
1008:
1004:
997:
992:
990:
985:
983:
978:
977:
974:
966:
964:
959:
954:
951:
946:
940:
936:
932:
928:
924:
918:
915:
907:
900:
897:
891:
886:
883:
881:
878:
877:
873:
871:
869:
852:
849:
827:
824:
815:
811:
807:
788:
781:
778:
771:
764:
761:
754:
747:
746:
745:
744:
740:
737:
733:
729:
724:
722:
718:
714:
710:
706:
701:
699:
698:strictly full
695:
691:
687:
683:
679:
675:
671:
667:
663:
662:
657:
653:
645:
643:
641:
636:
634:
630:
626:
622:
618:
614:
610:
606:
602:
598:
594:
590:
586:
582:
578:
573:
571:
567:
563:
559:
554:
552:
547:
545:
541:
537:
533:
528:
526:
522:
518:
514:
510:
506:
502:
498:
490:
485:
481:
477:
476:vector spaces
473:
469:
466:
462:
459:
455:
451:
447:
443:
440:
436:
432:
429:
425:
422:
418:
414:
413:
409:
407:
405:
401:
397:
393:
389:
385:
382:morphisms in
381:
362:
356:
353:
350:
324:
318:
315:
312:
282:
281:
280:
278:
274:
270:
266:
263:
259:
255:
251:
246:
244:
241:, called the
240:
236:
232:
229:
226:
222:
218:
214:
210:
202:
198:
194:
190:
186:
182:
178:
175:
171:
167:
163:
159:
155:
151:
147:
144:
139:
134:
130:
126:
125:
124:
118:
114:
110:
107:
104:, denoted ob(
103:
99:
98:
97:
95:
91:
88:
84:
76:
74:
72:
68:
64:
60:
56:
52:
48:
44:
41:
37:
33:
29:
22:
1544:
1525:Categorified
1429:n-categories
1380:Key concepts
1330:
1218:Direct limit
1201:Coequalizers
1119:Yoneda lemma
1025:Key concepts
1015:Key concepts
962:
953:
926:
923:Freyd, Peter
917:
899:
813:
809:
805:
803:
738:
731:
727:
725:
720:
712:
708:
704:
702:
697:
693:
689:
685:
681:
677:
673:
669:
665:
659:
655:
651:
649:
637:
632:
624:
620:
616:
612:
604:
600:
596:
592:
584:
580:
576:
574:
569:
565:
561:
557:
555:
550:
548:
531:
529:
524:
512:
508:
504:
500:
496:
494:
479:
471:
467:
452:-preserving
403:
399:
395:
391:
387:
383:
379:
377:
276:
272:
268:
264:
261:
257:
253:
249:
247:
242:
238:
234:
230:
220:
216:
212:
208:
206:
200:
196:
192:
188:
184:
180:
173:
169:
165:
161:
157:
153:
149:
142:
137:
132:
128:
122:
116:
112:
105:
101:
96:is given by
93:
89:
86:
82:
80:
70:
66:
62:
54:
46:
42:
35:
25:
1493:-categories
1469:Kan complex
1459:Tricategory
1441:-categories
1331:Subcategory
1089:Exponential
1057:Preadditive
1052:Pre-abelian
812:belongs to
717:Peter Freyd
607:induces an
540:isomorphism
417:finite sets
87:subcategory
36:subcategory
28:mathematics
1635:Categories
1511:3-category
1501:2-category
1474:∞-groupoid
1449:Bicategory
1196:Coproducts
1156:Equalizers
1062:Bicategory
892:References
684:such that
640:embeddings
629:equivalent
491:Embeddings
428:bijections
199:is in hom(
172:are in ob(
141:is in hom(
127:for every
123:such that
57:and whose
1646:Hierarchy
1560:Symmetric
1505:2-functor
1245:Relations
1168:Pullbacks
786:→
775:→
769:→
758:→
558:embedding
551:embedding
532:embedding
517:injective
59:morphisms
1620:Glossary
1600:Category
1574:n-monoid
1527:concepts
1183:Colimits
1151:Products
1104:Morphism
1047:Concrete
1042:Additive
1032:Category
874:See also
853:″
828:′
782:″
765:′
672: :
611:between
579: :
534:to be a
507: :
410:Examples
233: :
225:faithful
152: :
40:category
1610:Outline
1569:n-group
1534:2-group
1489:Strict
1479:∞-topos
1275:Modules
1213:Pushout
1161:Kernels
1094:Functor
1037:Abelian
960:at the
666:replete
484:modules
228:functor
187:in hom(
160:in hom(
51:objects
1556:Traced
1539:2-ring
1269:Fields
1255:Groups
1250:Magmas
1138:Limits
941:
734:of an
688:is in
603:, and
463:For a
131:in ob(
49:whose
1550:-ring
1437:Weak
1421:Topos
1265:Rings
909:(PDF)
841:and
619:. If
589:image
465:field
446:rings
260:is a
38:of a
1240:Sets
939:ISBN
713:lluf
709:wide
615:and
587:the
521:full
458:rngs
450:unit
433:The
271:and
248:Let
183:and
81:Let
34:, a
1084:End
1074:CCC
965:Lab
931:doi
804:in
711:or
707:is
680:in
664:or
654:of
631:to
627:is
599:of
591:of
499:of
394:in
380:all
275:of
92:of
26:In
1637::
1562:)
1558:)(
937:.
870:.
808:,
726:A
700:.
676:→
642:.
635:.
583:→
511:→
406:.
279:,
237:→
195:o
176:),
156:→
145:),
119:).
108:),
1554:(
1547:n
1545:E
1507:)
1503:(
1491:n
1455:)
1451:(
1439:n
1281:)
1277:(
1271:)
1267:(
995:e
988:t
981:v
963:n
947:.
933::
911:.
850:M
825:M
814:S
810:M
806:C
789:0
779:M
772:M
762:M
755:0
739:C
732:S
721:C
705:C
694:S
690:S
686:Y
682:C
678:Y
674:X
670:k
656:C
652:S
633:B
625:F
621:F
617:S
613:B
605:F
601:C
597:S
593:F
585:C
581:B
577:F
566:F
562:F
525:S
513:C
509:S
505:I
501:C
497:S
486:.
482:-
480:K
474:-
472:K
468:K
460:.
441:.
423:.
404:A
400:C
396:C
392:A
388:S
384:C
363:.
360:)
357:Y
354:,
351:X
348:(
342:C
336:m
333:o
330:H
325:=
322:)
319:Y
316:,
313:X
310:(
304:S
298:m
295:o
292:H
277:S
273:Y
269:X
265:C
258:S
254:C
250:S
239:C
235:S
231:I
221:C
217:S
213:S
209:S
201:S
197:g
193:f
189:S
185:g
181:f
174:S
170:Y
166:X
162:S
158:Y
154:X
150:f
143:S
138:X
133:S
129:X
117:S
113:C
106:S
102:C
94:C
90:S
83:C
71:C
67:C
63:C
55:C
47:S
43:C
23:.
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