1279:
1526:
1546:
1536:
743:
674:
257:
481:
505:
429:
573:
795:
769:
116:
90:
529:
64:
453:
296:
156:
136:
284:
308:
923:
362:, is symmetric monoidal. The tensor product is the direct product of objects, and any terminal object (empty product) is the unit object.
541:
592:
818:
Fong, Brendan; Spivak, David I. (2018-10-12). "Seven
Sketches in Compositionality: An Invitation to Applied Category Theory".
916:
1570:
1120:
1075:
683:
381:-modules is symmetric monoidal. The latter example class includes the category of all vector spaces over a given field.
1549:
404:(or of the Lie algebra) is a symmetric monoidal category. Here the standard tensor product of representations is used.
359:
625:
208:
355:. Like before, the tensor product is just the cartesian product of groups, and the trivial group is the unit object.
1539:
1325:
1189:
1097:
295:
1498:
1142:
1080:
1003:
798:
170:
159:
1529:
1485:
1090:
909:
611:
603:
508:
1085:
1067:
545:
345:
328:
are the associativity isomorphism, the left unit isomorphism, and the right unit isomorphism respectively.
1292:
1058:
1038:
961:
858:
264:
1174:
1013:
839:
366:
283:
986:
981:
401:
863:
462:
1330:
1278:
1208:
1204:
1008:
580:
163:
490:
414:
307:
1184:
1179:
1161:
1043:
1018:
894:
819:
551:
352:
774:
748:
95:
69:
1493:
1430:
1418:
1320:
1245:
1240:
1198:
1194:
976:
971:
607:
514:
373:
is monoidal (using the ordinary tensor product of modules), but not necessarily symmetric. If
182:
49:
43:
17:
1454:
1340:
1315:
1250:
1235:
1230:
1169:
998:
966:
876:
576:
456:
438:
341:
158:
of the category). One of the prototypical examples of a symmetric monoidal category is the
1366:
932:
596:
31:
1403:
1398:
1382:
1345:
1335:
1255:
141:
121:
1564:
1393:
1225:
1102:
1028:
843:
1147:
1048:
1408:
1388:
1260:
1130:
389:
35:
1440:
1378:
991:
890:
1434:
1125:
1503:
1135:
1033:
1473:
1463:
1112:
1023:
889:
This article incorporates material from
Symmetric monoidal category on
1468:
344:. The tensor product is the set theoretic cartesian product, and any
824:
1350:
901:
336:
Some examples and non-examples of symmetric monoidal categories:
880:
1290:
943:
905:
358:
More generally, any category with finite products, that is, a
27:
Monoidal category where A ⊗ B is naturally equivalent to B ⊗ A
844:"Symmetric Monoidal Categories Model all Connective Spectra"
66:
is defined) such that the tensor product is symmetric (i.e.
622:
In a symmetric monoidal category, the natural isomorphisms
511:
symmetric monoidal category with the internal hom-functor
745:. If we abandon this requirement (but still require that
92:
is, in a certain strict sense, naturally isomorphic to
777:
751:
686:
628:
554:
517:
493:
465:
441:
417:
211:
144:
124:
98:
72:
52:
1453:
1417:
1365:
1358:
1309:
1218:
1160:
1111:
1066:
1057:
954:
595:is a symmetric monoidal category with a compatible
789:
763:
737:
668:
567:
523:
499:
475:
447:
423:
251:
150:
130:
110:
84:
58:
738:{\displaystyle s_{BA}\circ s_{AB}=1_{A\otimes B}}
895:Creative Commons Attribution/Share-Alike License
669:{\displaystyle s_{AB}:A\otimes B\to B\otimes A}
252:{\displaystyle s_{AB}:A\otimes B\to B\otimes A}
275:and such that the following diagrams commute:
917:
46:(i.e. a category in which a "tensor product"
8:
1545:
1535:
1362:
1306:
1287:
1063:
951:
940:
924:
910:
902:
797:), we obtain the more general notion of a
862:
823:
776:
750:
723:
707:
691:
685:
633:
627:
559:
553:
548:) of a symmetric monoidal category is an
516:
492:
468:
467:
466:
464:
440:
416:
216:
210:
143:
123:
97:
71:
51:
810:
483:are symmetric monoidal, and moreover, (
851:Theory and Applications of Categories
377:is commutative, the category of left
7:
181:A symmetric monoidal category is a
593:dagger symmetric monoidal category
560:
25:
1544:
1534:
1525:
1524:
1277:
348:can be fixed as the unit object.
306:
294:
282:
171:tensor product of vector spaces
893:, which is licensed under the
654:
544:(geometric realization of the
476:{\displaystyle {\mathbb {C} }}
237:
1:
614:symmetric monoidal category.
291:The associativity coherence:
18:Symmetric monoidal ∞-category
500:{\displaystyle \circledast }
424:{\displaystyle \circledast }
402:representations of the group
193:) such that, for every pair
1219:Constructions on categories
877:Symmetric monoidal category
771:be naturally isomorphic to
680:inverses in the sense that
568:{\displaystyle E_{\infty }}
360:cartesian monoidal category
40:symmetric monoidal category
1587:
1326:Higher-dimensional algebra
790:{\displaystyle B\otimes A}
764:{\displaystyle A\otimes B}
205:, there is an isomorphism
111:{\displaystyle B\otimes A}
85:{\displaystyle A\otimes B}
1520:
1299:
1286:
1275:
950:
939:
799:braided monoidal category
160:category of vector spaces
524:{\displaystyle \oslash }
59:{\displaystyle \otimes }
1136:Cokernels and quotients
1059:Universal constructions
396:), the category of all
316:In the diagrams above,
1293:Higher category theory
1039:Natural transformation
791:
765:
739:
670:
569:
525:
501:
477:
449:
448:{\displaystyle \odot }
425:
253:
152:
132:
112:
86:
60:
792:
766:
740:
671:
570:
526:
502:
478:
450:
426:
367:category of bimodules
254:
153:
133:
113:
87:
61:
1162:Algebraic categories
775:
749:
684:
626:
552:
515:
491:
463:
439:
415:
279:The unit coherence:
209:
142:
122:
96:
70:
50:
1571:Monoidal categories
1331:Homotopy hypothesis
1009:Commutative diagram
581:infinite loop space
169:using the ordinary
1044:Universal property
787:
761:
735:
666:
565:
521:
497:
473:
445:
421:
388:and a group (or a
353:category of groups
249:
148:
128:
108:
82:
56:
1558:
1557:
1516:
1515:
1512:
1511:
1494:monoidal category
1449:
1448:
1321:Enriched category
1273:
1272:
1269:
1268:
1246:Quotient category
1241:Opposite category
1156:
1155:
542:classifying space
457:stereotype spaces
303:The inverse law:
183:monoidal category
151:{\displaystyle B}
131:{\displaystyle A}
44:monoidal category
16:(Redirected from
1578:
1548:
1547:
1538:
1537:
1528:
1527:
1363:
1341:Simplex category
1316:Categorification
1307:
1288:
1281:
1251:Product category
1236:Kleisli category
1231:Functor category
1076:Terminal objects
1064:
999:Adjoint functors
952:
941:
926:
919:
912:
903:
869:
868:
866:
848:
836:
830:
829:
827:
815:
796:
794:
793:
788:
770:
768:
767:
762:
744:
742:
741:
736:
734:
733:
715:
714:
699:
698:
675:
673:
672:
667:
641:
640:
597:dagger structure
577:group completion
574:
572:
571:
566:
564:
563:
530:
528:
527:
522:
506:
504:
503:
498:
482:
480:
479:
474:
472:
471:
454:
452:
451:
446:
430:
428:
427:
422:
407:The categories (
342:category of sets
310:
298:
286:
258:
256:
255:
250:
224:
223:
162:over some fixed
157:
155:
154:
149:
137:
135:
134:
129:
118:for all objects
117:
115:
114:
109:
91:
89:
88:
83:
65:
63:
62:
57:
21:
1586:
1585:
1581:
1580:
1579:
1577:
1576:
1575:
1561:
1560:
1559:
1554:
1508:
1478:
1445:
1422:
1413:
1370:
1354:
1305:
1295:
1282:
1265:
1214:
1152:
1121:Initial objects
1107:
1053:
946:
935:
933:Category theory
930:
873:
872:
864:10.1.1.501.2534
846:
838:
837:
833:
817:
816:
812:
807:
773:
772:
747:
746:
719:
703:
687:
682:
681:
629:
624:
623:
620:
618:Generalizations
589:
587:Specializations
555:
550:
549:
538:
513:
512:
489:
488:
461:
460:
437:
436:
413:
412:
334:
212:
207:
206:
179:
140:
139:
120:
119:
94:
93:
68:
67:
48:
47:
32:category theory
28:
23:
22:
15:
12:
11:
5:
1584:
1582:
1574:
1573:
1563:
1562:
1556:
1555:
1553:
1552:
1542:
1532:
1521:
1518:
1517:
1514:
1513:
1510:
1509:
1507:
1506:
1501:
1496:
1482:
1476:
1471:
1466:
1460:
1458:
1451:
1450:
1447:
1446:
1444:
1443:
1438:
1427:
1425:
1420:
1415:
1414:
1412:
1411:
1406:
1401:
1396:
1391:
1386:
1375:
1373:
1368:
1360:
1356:
1355:
1353:
1348:
1346:String diagram
1343:
1338:
1336:Model category
1333:
1328:
1323:
1318:
1313:
1311:
1304:
1303:
1300:
1297:
1296:
1291:
1284:
1283:
1276:
1274:
1271:
1270:
1267:
1266:
1264:
1263:
1258:
1256:Comma category
1253:
1248:
1243:
1238:
1233:
1228:
1222:
1220:
1216:
1215:
1213:
1212:
1202:
1192:
1190:Abelian groups
1187:
1182:
1177:
1172:
1166:
1164:
1158:
1157:
1154:
1153:
1151:
1150:
1145:
1140:
1139:
1138:
1128:
1123:
1117:
1115:
1109:
1108:
1106:
1105:
1100:
1095:
1094:
1093:
1083:
1078:
1072:
1070:
1061:
1055:
1054:
1052:
1051:
1046:
1041:
1036:
1031:
1026:
1021:
1016:
1011:
1006:
1001:
996:
995:
994:
989:
984:
979:
974:
969:
958:
956:
948:
947:
944:
937:
936:
931:
929:
928:
921:
914:
906:
900:
899:
886:
871:
870:
840:Thomason, R.W.
831:
809:
808:
806:
803:
786:
783:
780:
760:
757:
754:
732:
729:
726:
722:
718:
713:
710:
706:
702:
697:
694:
690:
665:
662:
659:
656:
653:
650:
647:
644:
639:
636:
632:
619:
616:
588:
585:
575:space, so its
562:
558:
537:
534:
533:
532:
520:
496:
470:
444:
420:
405:
384:Given a field
382:
363:
356:
349:
333:
330:
314:
313:
312:
311:
301:
300:
299:
289:
288:
287:
248:
245:
242:
239:
236:
233:
230:
227:
222:
219:
215:
201:of objects in
178:
175:
147:
127:
107:
104:
101:
81:
78:
75:
55:
34:, a branch of
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1583:
1572:
1569:
1568:
1566:
1551:
1543:
1541:
1533:
1531:
1523:
1522:
1519:
1505:
1502:
1500:
1497:
1495:
1491:
1487:
1483:
1481:
1479:
1472:
1470:
1467:
1465:
1462:
1461:
1459:
1456:
1452:
1442:
1439:
1436:
1432:
1429:
1428:
1426:
1424:
1416:
1410:
1407:
1405:
1402:
1400:
1397:
1395:
1394:Tetracategory
1392:
1390:
1387:
1384:
1383:pseudofunctor
1380:
1377:
1376:
1374:
1372:
1364:
1361:
1357:
1352:
1349:
1347:
1344:
1342:
1339:
1337:
1334:
1332:
1329:
1327:
1324:
1322:
1319:
1317:
1314:
1312:
1308:
1302:
1301:
1298:
1294:
1289:
1285:
1280:
1262:
1259:
1257:
1254:
1252:
1249:
1247:
1244:
1242:
1239:
1237:
1234:
1232:
1229:
1227:
1226:Free category
1224:
1223:
1221:
1217:
1210:
1209:Vector spaces
1206:
1203:
1200:
1196:
1193:
1191:
1188:
1186:
1183:
1181:
1178:
1176:
1173:
1171:
1168:
1167:
1165:
1163:
1159:
1149:
1146:
1144:
1141:
1137:
1134:
1133:
1132:
1129:
1127:
1124:
1122:
1119:
1118:
1116:
1114:
1110:
1104:
1103:Inverse limit
1101:
1099:
1096:
1092:
1089:
1088:
1087:
1084:
1082:
1079:
1077:
1074:
1073:
1071:
1069:
1065:
1062:
1060:
1056:
1050:
1047:
1045:
1042:
1040:
1037:
1035:
1032:
1030:
1029:Kan extension
1027:
1025:
1022:
1020:
1017:
1015:
1012:
1010:
1007:
1005:
1002:
1000:
997:
993:
990:
988:
985:
983:
980:
978:
975:
973:
970:
968:
965:
964:
963:
960:
959:
957:
953:
949:
942:
938:
934:
927:
922:
920:
915:
913:
908:
907:
904:
898:
896:
892:
887:
885:
883:
878:
875:
874:
865:
860:
857:(5): 78–118.
856:
852:
845:
841:
835:
832:
826:
821:
814:
811:
804:
802:
800:
784:
781:
778:
758:
755:
752:
730:
727:
724:
720:
716:
711:
708:
704:
700:
695:
692:
688:
679:
663:
660:
657:
651:
648:
645:
642:
637:
634:
630:
617:
615:
613:
609:
605:
600:
598:
594:
586:
584:
582:
578:
556:
547:
543:
535:
518:
510:
494:
486:
458:
442:
434:
418:
410:
406:
403:
399:
395:
391:
387:
383:
380:
376:
372:
368:
364:
361:
357:
354:
350:
347:
343:
339:
338:
337:
331:
329:
327:
323:
319:
309:
305:
304:
302:
297:
293:
292:
290:
285:
281:
280:
278:
277:
276:
274:
270:
266:
262:
246:
243:
240:
234:
231:
228:
225:
220:
217:
213:
204:
200:
196:
192:
188:
184:
176:
174:
172:
168:
165:
161:
145:
125:
105:
102:
99:
79:
76:
73:
53:
45:
41:
37:
33:
19:
1489:
1474:
1455:Categorified
1359:n-categories
1310:Key concepts
1148:Direct limit
1131:Coequalizers
1049:Yoneda lemma
955:Key concepts
945:Key concepts
888:
881:
854:
850:
834:
813:
677:
621:
601:
590:
539:
484:
432:
408:
397:
393:
385:
378:
374:
370:
369:over a ring
335:
325:
321:
317:
315:
272:
268:
260:
202:
198:
194:
190:
186:
180:
166:
39:
29:
1423:-categories
1399:Kan complex
1389:Tricategory
1371:-categories
1261:Subcategory
1019:Exponential
987:Preadditive
982:Pre-abelian
610:cocomplete
390:Lie algebra
259:called the
36:mathematics
1441:3-category
1431:2-category
1404:∞-groupoid
1379:Bicategory
1126:Coproducts
1086:Equalizers
992:Bicategory
891:PlanetMath
825:1803.05316
805:References
676:are their
536:Properties
177:Definition
1490:Symmetric
1435:2-functor
1175:Relations
1098:Pullbacks
859:CiteSeerX
782:⊗
756:⊗
728:⊗
701:∘
661:⊗
655:→
649:⊗
561:∞
519:⊘
495:⊛
443:⊙
419:⊛
346:singleton
244:⊗
238:→
232:⊗
103:⊗
77:⊗
54:⊗
1565:Category
1550:Glossary
1530:Category
1504:n-monoid
1457:concepts
1113:Colimits
1081:Products
1034:Morphism
977:Concrete
972:Additive
962:Category
842:(1995).
608:complete
400:-linear
332:Examples
267:in both
263:that is
261:swap map
1540:Outline
1499:n-group
1464:2-group
1419:Strict
1409:∞-topos
1205:Modules
1143:Pushout
1091:Kernels
1024:Functor
967:Abelian
879:at the
507:) is a
431:) and (
265:natural
1486:Traced
1469:2-ring
1199:Fields
1185:Groups
1180:Magmas
1068:Limits
861:
612:closed
604:cosmos
579:is an
509:closed
324:, and
1480:-ring
1367:Weak
1351:Topos
1195:Rings
847:(PDF)
820:arXiv
606:is a
546:nerve
459:over
455:) of
392:over
189:, ⊗,
164:field
42:is a
1170:Sets
540:The
365:The
351:The
340:The
271:and
138:and
38:, a
1014:End
1004:CCC
884:Lab
678:own
485:Ste
433:Ste
409:Ste
30:In
1567::
1492:)
1488:)(
853:.
849:.
801:.
602:A
599:.
591:A
583:.
320:,
197:,
173:.
167:k,
1484:(
1477:n
1475:E
1437:)
1433:(
1421:n
1385:)
1381:(
1369:n
1211:)
1207:(
1201:)
1197:(
925:e
918:t
911:v
897:.
882:n
867:.
855:1
828:.
822::
785:A
779:B
759:B
753:A
731:B
725:A
721:1
717:=
712:B
709:A
705:s
696:A
693:B
689:s
664:A
658:B
652:B
646:A
643::
638:B
635:A
631:s
557:E
531:.
487:,
469:C
435:,
411:,
398:k
394:k
386:k
379:R
375:R
371:R
326:r
322:l
318:a
273:B
269:A
247:A
241:B
235:B
229:A
226::
221:B
218:A
214:s
203:C
199:B
195:A
191:I
187:C
185:(
146:B
126:A
106:A
100:B
80:B
74:A
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
