Knowledge (XXG)

Bicategory

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885: 1176: 1132: 1152: 1142: 292: 380: 342: 1217: 148:"morphisms between morphisms" ρ, σ, ... with fixed source and target morphisms (which should have themselves the same source and the same target), called 2- 529: 425:(one 0-cell); this construction is analogous to construction of a small category from a monoid. The objects {T, F} become morphisms, and the morphism 208: 1210: 1236: 462:
Fong, Brendan; Spivak, David I. (2018-10-12). "Seven Sketches in Compositionality: An Invitation to Applied Category Theory".
1241: 522: 1203: 726: 681: 1155: 1095: 1145: 931: 795: 703: 1104: 748: 686: 609: 1135: 1091: 696: 515: 114: 691: 673: 898: 664: 644: 567: 183:) whose objects are the 1-cells and morphisms are the 2-cells. The composition in this category is called 169: 47: 780: 619: 592: 587: 936: 884: 814: 810: 614: 411: 303: 390:, are moreover required to hold: a monoidal category is the same as a bicategory with one 0-cell. 790: 785: 767: 649: 624: 463: 1099: 1036: 1024: 926: 851: 846: 804: 800: 582: 577: 399: 387: 1187: 347: 309: 1060: 946: 921: 856: 841: 836: 775: 604: 572: 430: 1183: 972: 538: 383: 84: 43: 1009: 69: 1004: 988: 951: 941: 861: 1230: 999: 831: 708: 634: 753: 654: 76: 1175: 1014: 414:, T). As a category this is presented with two objects {T, F} and single morphism 994: 866: 736: 287:{\displaystyle *:\mathbf {B} (b,c)\times \mathbf {B} (a,b)\to \mathbf {B} (a,c)} 80: 65: 55: 31: 1046: 92: 1040: 731: 203: 17: 495: 1109: 741: 639: 131: 51: 1079: 1069: 718: 629: 1074: 407: 421:
We can reinterpret this monoid as a bicategory with a single object
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Bicategories may be considered as a weakening of the definition of
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The horizontal composition is required to be associative up to a
499: 896: 549: 511: 485:, Lecture Notes in Mathematics 47, pages 1–77. Springer, 1967. 1191: 481:
J. BĂ©nabou. "Introduction to bicategories, part I". In
350: 312: 211: 141:, ... with fixed source and target objects called 1- 1059: 1023: 971: 964: 915: 824: 766: 717: 672: 663: 560: 374: 336: 286: 79:. A similar process for 3-categories leads to 1211: 523: 50:to handle the cases where the composition of 8: 429:becomes a natural transformation (forming a 1218: 1204: 1151: 1141: 968: 912: 893: 669: 557: 546: 530: 516: 508: 467: 349: 311: 264: 241: 218: 210: 483:Reports of the Midwest Category Seminar 454: 68:. The notion was introduced in 1967 by 7: 1172: 1170: 1190:. You can help Knowledge (XXG) by 394:Example: Boolean monoidal category 25: 1174: 1150: 1140: 1131: 1130: 883: 402:, such as the monoidal preorder 265: 242: 219: 386:, similar to those needed for 363: 351: 331: 319: 281: 269: 261: 258: 246: 235: 223: 1: 46:used to extend the notion of 433:for the single hom-category 825:Constructions on categories 1258: 1169: 932:Higher-dimensional algebra 156:with some more structure: 27:Generalization of category 1126: 905: 892: 881: 556: 545: 83:, and more generally to 742:Cokernels and quotients 665:Universal constructions 375:{\displaystyle (h*g)*f} 337:{\displaystyle h*(g*f)} 106:Formally, a bicategory 58:, but only associative 1237:Higher category theory 1186:-related article is a 899:Higher category theory 645:Natural transformation 376: 338: 296:horizontal composition 288: 1242:Category theory stubs 377: 339: 289: 768:Algebraic categories 348: 310: 306:α between morphisms 209: 190:given three objects 185:vertical composition 937:Homotopy hypothesis 615:Commutative diagram 388:monoidal categories 304:natural isomorphism 650:Universal property 398:Consider a simple 372: 334: 284: 160:given two objects 54:is not (strictly) 42:) is a concept in 1199: 1198: 1164: 1163: 1122: 1121: 1118: 1117: 1100:monoidal category 1055: 1054: 927:Enriched category 879: 878: 875: 874: 852:Quotient category 847:Opposite category 762: 761: 400:monoidal category 16:(Redirected from 1249: 1220: 1213: 1206: 1178: 1171: 1154: 1153: 1144: 1143: 1134: 1133: 969: 947:Simplex category 922:Categorification 913: 894: 887: 857:Product category 842:Kleisli category 837:Functor category 682:Terminal objects 670: 605:Adjoint functors 558: 547: 532: 525: 518: 509: 474: 473: 471: 459: 431:functor category 384:coherence axioms 381: 379: 378: 373: 343: 341: 340: 335: 293: 291: 290: 285: 268: 245: 222: 21: 1257: 1256: 1252: 1251: 1250: 1248: 1247: 1246: 1227: 1226: 1225: 1224: 1184:category theory 1167: 1165: 1160: 1114: 1084: 1051: 1028: 1019: 976: 960: 911: 901: 888: 871: 820: 758: 727:Initial objects 713: 659: 552: 541: 539:Category theory 536: 492: 478: 477: 461: 460: 456: 451: 396: 346: 345: 308: 307: 207: 206: 124:, ... called 0- 104: 44:category theory 40:weak 2-category 28: 23: 22: 15: 12: 11: 5: 1255: 1253: 1245: 1244: 1239: 1229: 1228: 1223: 1222: 1215: 1208: 1200: 1197: 1196: 1179: 1162: 1161: 1159: 1158: 1148: 1138: 1127: 1124: 1123: 1120: 1119: 1116: 1115: 1113: 1112: 1107: 1102: 1088: 1082: 1077: 1072: 1066: 1064: 1057: 1056: 1053: 1052: 1050: 1049: 1044: 1033: 1031: 1026: 1021: 1020: 1018: 1017: 1012: 1007: 1002: 997: 992: 981: 979: 974: 966: 962: 961: 959: 954: 952:String diagram 949: 944: 942:Model category 939: 934: 929: 924: 919: 917: 910: 909: 906: 903: 902: 897: 890: 889: 882: 880: 877: 876: 873: 872: 870: 869: 864: 862:Comma category 859: 854: 849: 844: 839: 834: 828: 826: 822: 821: 819: 818: 808: 798: 796:Abelian groups 793: 788: 783: 778: 772: 770: 764: 763: 760: 759: 757: 756: 751: 746: 745: 744: 734: 729: 723: 721: 715: 714: 712: 711: 706: 701: 700: 699: 689: 684: 678: 676: 667: 661: 660: 658: 657: 652: 647: 642: 637: 632: 627: 622: 617: 612: 607: 602: 601: 600: 595: 590: 585: 580: 575: 564: 562: 554: 553: 550: 543: 542: 537: 535: 534: 527: 520: 512: 506: 505: 491: 490:External links 488: 487: 486: 476: 475: 453: 452: 450: 447: 395: 392: 371: 368: 365: 362: 359: 356: 353: 333: 330: 327: 324: 321: 318: 315: 300: 299: 283: 280: 277: 274: 271: 267: 263: 260: 257: 254: 251: 248: 244: 240: 237: 234: 231: 228: 225: 221: 217: 214: 188: 154: 153: 146: 129: 103: 100: 26: 24: 14: 13: 10: 9: 6: 4: 3: 2: 1254: 1243: 1240: 1238: 1235: 1234: 1232: 1221: 1216: 1214: 1209: 1207: 1202: 1201: 1195: 1193: 1189: 1185: 1180: 1177: 1173: 1168: 1157: 1149: 1147: 1139: 1137: 1129: 1128: 1125: 1111: 1108: 1106: 1103: 1101: 1097: 1093: 1089: 1087: 1085: 1078: 1076: 1073: 1071: 1068: 1067: 1065: 1062: 1058: 1048: 1045: 1042: 1038: 1035: 1034: 1032: 1030: 1022: 1016: 1013: 1011: 1008: 1006: 1003: 1001: 1000:Tetracategory 998: 996: 993: 990: 989:pseudofunctor 986: 983: 982: 980: 978: 970: 967: 963: 958: 955: 953: 950: 948: 945: 943: 940: 938: 935: 933: 930: 928: 925: 923: 920: 918: 914: 908: 907: 904: 900: 895: 891: 886: 868: 865: 863: 860: 858: 855: 853: 850: 848: 845: 843: 840: 838: 835: 833: 832:Free category 830: 829: 827: 823: 816: 815:Vector spaces 812: 809: 806: 802: 799: 797: 794: 792: 789: 787: 784: 782: 779: 777: 774: 773: 771: 769: 765: 755: 752: 750: 747: 743: 740: 739: 738: 735: 733: 730: 728: 725: 724: 722: 720: 716: 710: 709:Inverse limit 707: 705: 702: 698: 695: 694: 693: 690: 688: 685: 683: 680: 679: 677: 675: 671: 668: 666: 662: 656: 653: 651: 648: 646: 643: 641: 638: 636: 635:Kan extension 633: 631: 628: 626: 623: 621: 618: 616: 613: 611: 608: 606: 603: 599: 596: 594: 591: 589: 586: 584: 581: 579: 576: 574: 571: 570: 569: 566: 565: 563: 559: 555: 548: 544: 540: 533: 528: 526: 521: 519: 514: 513: 510: 504: 502: 497: 494: 493: 489: 484: 480: 479: 470: 465: 458: 455: 448: 446: 444: 440: 436: 432: 428: 424: 419: 417: 413: 410:M = ({T, F}, 409: 406:based on the 405: 401: 393: 391: 389: 385: 369: 366: 360: 357: 354: 328: 325: 322: 316: 313: 305: 297: 278: 275: 272: 255: 252: 249: 238: 232: 229: 226: 215: 212: 205: 202:, there is a 201: 197: 193: 189: 186: 182: 178: 174: 171: 167: 163: 159: 158: 157: 151: 147: 144: 140: 136: 133: 130: 127: 123: 119: 116: 113: 112: 111: 110:consists of: 109: 101: 99: 97: 95: 90: 88: 82: 81:tricategories 78: 73: 71: 67: 63: 62: 57: 53: 49: 45: 41: 37: 33: 19: 1192:expanding it 1181: 1166: 1080: 1061:Categorified 984: 965:n-categories 916:Key concepts 754:Direct limit 737:Coequalizers 655:Yoneda lemma 597: 561:Key concepts 551:Key concepts 500: 482: 457: 442: 438: 434: 426: 422: 420: 415: 403: 397: 382:. Some more 301: 295: 199: 195: 191: 184: 180: 176: 172: 165: 161: 155: 149: 142: 138: 134: 125: 121: 117: 107: 105: 93: 86: 77:2-categories 74: 70:Jean BĂ©nabou 59: 39: 35: 29: 18:Bicategories 1029:-categories 1005:Kan complex 995:Tricategory 977:-categories 867:Subcategory 625:Exponential 593:Preadditive 588:Pre-abelian 168:there is a 96:-categories 89:-categories 66:isomorphism 56:associative 32:mathematics 1231:Categories 1047:3-category 1037:2-category 1010:∞-groupoid 985:Bicategory 732:Coproducts 692:Equalizers 598:Bicategory 496:Bicategory 469:1803.05316 449:References 102:Definition 36:bicategory 1096:Symmetric 1041:2-functor 781:Relations 704:Pullbacks 418:: F → T. 367:∗ 358:∗ 326:∗ 317:∗ 262:→ 239:× 213:∗ 204:bifunctor 132:morphisms 52:morphisms 1156:Glossary 1136:Category 1110:n-monoid 1063:concepts 719:Colimits 687:Products 640:Morphism 583:Concrete 578:Additive 568:Category 170:category 48:category 1146:Outline 1105:n-group 1070:2-group 1025:Strict 1015:∞-topos 811:Modules 749:Pushout 697:Kernels 630:Functor 573:Abelian 498:at the 294:called 115:objects 1092:Traced 1075:2-ring 805:Fields 791:Groups 786:Magmas 674:Limits 408:monoid 38:(or a 1182:This 1086:-ring 973:Weak 957:Topos 801:Rings 464:arXiv 150:cells 143:cells 126:cells 85:weak 61:up to 1188:stub 776:Sets 445:)). 404:Bool 344:and 198:and 164:and 91:for 34:, a 620:End 610:CCC 503:Lab 64:an 30:In 1233:: 1098:) 1094:)( 441:, 194:, 179:, 137:, 120:, 98:. 72:. 1219:e 1212:t 1205:v 1194:. 1090:( 1083:n 1081:E 1043:) 1039:( 1027:n 991:) 987:( 975:n 817:) 813:( 807:) 803:( 531:e 524:t 517:v 501:n 472:. 466:: 443:x 439:x 437:( 435:B 427:g 423:x 416:g 412:∧ 370:f 364:) 361:g 355:h 352:( 332:) 329:f 323:g 320:( 314:h 298:. 282:) 279:c 276:, 273:a 270:( 266:B 259:) 256:b 253:, 250:a 247:( 243:B 236:) 233:c 230:, 227:b 224:( 220:B 216:: 200:c 196:b 192:a 187:; 181:b 177:a 175:( 173:B 166:b 162:a 152:; 145:; 139:g 135:f 128:; 122:b 118:a 108:B 94:n 87:n 20:)

Index

Bicategories
mathematics
category theory
category
morphisms
associative
up to
isomorphism
Jean BĂ©nabou
2-categories
tricategories
weak n-categories
n-categories
objects
morphisms
category
bifunctor
natural isomorphism
coherence axioms
monoidal categories
monoidal category
monoid
∧
functor category
arXiv
1803.05316
Bicategory
nLab
v
t

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