1175:
1422:
1442:
1432:
535:) can have at most one element. In applications to logic, this then looks like a very general description of negation (that is, proofs run in the opposite direction). For example, if we take the opposite of a
393:
329:
454:
287:
161:
with objects and morphisms as distinct sorts, together with the relations of an object being the source or target of a morphism and a symbol for composing two morphisms.
221:
195:
422:
355:
84:. Duality, as such, is the assertion that truth is invariant under this operation on statements. In other words, if a statement is true about
819:
1104:
649:
779:
527:
This example on orders is a special case, since partial orders correspond to a certain kind of category in which Hom(
17:
752:
812:
735:
717:
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1016:
971:
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1385:
730:
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1094:
1435:
1221:
1085:
993:
707:
585:
360:
296:
1471:
1394:
1038:
976:
899:
135:
427:
260:
1466:
1425:
1381:
986:
805:
689:
981:
963:
563:
559:
464:. In short, the property of being a monomorphism is dual to the property of being an epimorphism.
1188:
954:
934:
857:
552:
200:
174:
118:
need not be a category that arises from mathematical practice. In this case, another category
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909:
725:
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31:
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158:
77:
65:
398:
80:
two morphisms, a corresponding dual statement is obtained regarding the opposite category
8:
1226:
1174:
1100:
904:
548:
334:
1080:
1075:
1057:
939:
914:
577:
226:
Informally, these conditions state that the dual of a statement is formed by reversing
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171:
Interchange the order of composing morphisms. That is, replace each occurrence of
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598:
581:
38:
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1241:
1231:
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536:
472:
is a monomorphism if and only if the reverse morphism in the opposite category
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998:
924:
789:
762:
481:
492:
and ≤ a partial order relation, we can define a new partial order relation ≤
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164:
Let σ be any statement in this language. We form the dual σ as follows:
42:
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157:
We define the elementary language of category theory as the two-sorted
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1031:
929:
480:
An example comes from reversing the direction of inequalities in a
227:
73:
69:
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919:
774:(2nd ed.). Oxford: Oxford University Press. pp. 53–55.
747:(Second ed.). New York, NY: Springer New York. p. 33.
27:
Correspondence between properties of a category and its opposite
1364:
468:
Applying duality, this means that a morphism in some category
1246:
547:
have their roles interchanged. This is an abstract form of
357:. Performing the dual operation, we get the statement that
168:
Interchange each occurrence of "source" in σ with "target".
49:
is a correspondence between the properties of a category
670:
658:
637:
430:
401:
363:
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203:
177:
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is the observation that σ is true for some category
30:For general notions of duality in mathematics, see
448:
416:
387:
349:
323:
281:
215:
189:
110:, it is often the case that the opposite category
1458:
631:
584:. In this context, the duality is often called
149:are equivalent, such a category is self-dual.
813:
641:Locally Presentable and Accessible Categories
60:. Given a statement regarding the category
1441:
1431:
1187:
820:
806:
644:. Cambridge University Press. p. 62.
745:Categories for the Working Mathematician
742:
664:
88:, then its dual statement is true about
14:
1459:
769:
676:
456:, this is precisely what it means for
96:, then its dual has to be false about
92:. Also, if a statement is false about
76:as well as interchanging the order of
1186:
839:
801:
122:is also termed to be in duality with
152:
827:
24:
25:
1483:
388:{\displaystyle g\circ f=h\circ f}
324:{\displaystyle f\circ g=f\circ h}
1440:
1430:
1421:
1420:
1173:
840:
638:Jiří Adámek; J. Rosicky (1994).
576:are examples of dual notions in
53:and the dual properties of the
449:{\displaystyle f\colon B\to A}
440:
282:{\displaystyle f\colon A\to B}
273:
13:
1:
624:
244:if and only if σ is true for
7:
1115:Constructions on categories
743:Mac Lane, Saunders (1978).
731:Encyclopedia of Mathematics
713:Encyclopedia of Mathematics
695:Encyclopedia of Mathematics
592:
251:
10:
1488:
1222:Higher-dimensional algebra
29:
1416:
1349:
1313:
1261:
1254:
1205:
1195:
1182:
1171:
1114:
1056:
1007:
962:
953:
850:
846:
835:
216:{\displaystyle f\circ g}
190:{\displaystyle g\circ f}
136:equivalent as categories
1032:Cokernels and quotients
955:Universal constructions
64:, by interchanging the
1189:Higher category theory
935:Natural transformation
770:Awodey, Steve (2010).
586:Eckmann–Hilton duality
450:
418:
389:
351:
325:
283:
217:
191:
609:Duality (mathematics)
555:applied to lattices.
451:
419:
390:
352:
326:
284:
218:
192:
32:Duality (mathematics)
1058:Algebraic categories
539:, we will find that
428:
417:{\displaystyle g=h.}
399:
361:
335:
297:
261:
201:
175:
159:first order language
114:per se is abstract.
1227:Homotopy hypothesis
905:Commutative diagram
708:"Duality principle"
476:is an epimorphism.
350:{\displaystyle g=h}
940:Universal property
578:algebraic topology
446:
414:
385:
347:
321:
279:
213:
187:
1454:
1453:
1412:
1411:
1408:
1407:
1390:monoidal category
1345:
1344:
1217:Enriched category
1169:
1168:
1165:
1164:
1142:Quotient category
1137:Opposite category
1052:
1051:
651:978-0-521-42261-1
614:Opposite category
566:are dual notions.
153:Formal definition
145:and its opposite
141:In the case when
105:concrete category
55:opposite category
16:(Redirected from
1479:
1472:Duality theories
1444:
1443:
1434:
1433:
1424:
1423:
1259:
1258:
1237:Simplex category
1212:Categorification
1203:
1202:
1184:
1183:
1177:
1147:Product category
1132:Kleisli category
1127:Functor category
972:Terminal objects
960:
959:
895:Adjoint functors
848:
847:
837:
836:
822:
815:
808:
799:
798:
793:
766:
739:
721:
703:
680:
679:, p. 53-55.
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668:
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656:
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635:
549:De Morgan's laws
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453:
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423:
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21:
18:Categorical dual
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1467:Category theory
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1017:Initial objects
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949:
842:
831:
829:Category theory
826:
796:
782:
772:Category theory
755:
724:
706:
690:"Dual category"
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684:
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671:
663:
659:
652:
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619:Pulation square
599:Adjoint functor
595:
582:homotopy theory
513:if and only if
509:
495:
429:
426:
425:
424:For a morphism
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39:category theory
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28:
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1249:
1244:
1242:String diagram
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1234:
1232:Model category
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1152:Comma category
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1134:
1129:
1124:
1118:
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1112:
1111:
1109:
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1086:Abelian groups
1083:
1078:
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1068:
1062:
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781:978-0199237180
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224:
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186:
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154:
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41:, a branch of
26:
9:
6:
4:
3:
2:
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1301:
1298:
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1290:Tetracategory
1288:
1286:
1283:
1280:
1279:pseudofunctor
1276:
1273:
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1260:
1257:
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1190:
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1176:
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1153:
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1148:
1145:
1143:
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1133:
1130:
1128:
1125:
1123:
1122:Free category
1120:
1119:
1117:
1113:
1106:
1105:Vector spaces
1102:
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1025:
1023:
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1018:
1015:
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1010:
1006:
1000:
999:Inverse limit
997:
995:
992:
988:
985:
984:
983:
980:
978:
975:
973:
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969:
967:
965:
961:
958:
956:
952:
946:
943:
941:
938:
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933:
931:
928:
926:
925:Kan extension
923:
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918:
916:
913:
911:
908:
906:
903:
901:
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876:
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859:
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838:
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830:
823:
818:
816:
811:
809:
804:
803:
800:
791:
787:
783:
777:
773:
768:
764:
760:
756:
750:
746:
741:
737:
733:
732:
727:
723:
719:
715:
714:
709:
705:
701:
697:
696:
691:
687:
686:
678:
673:
667:, p. 33.
666:
665:Mac Lane 1978
661:
653:
647:
643:
642:
634:
630:
620:
617:
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612:
610:
607:
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602:
600:
597:
596:
587:
583:
579:
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568:
565:
561:
558:
557:
556:
554:
550:
546:
542:
538:
534:
530:
520:
516:
512:
505:
502:
501:
500:
499:
491:
487:
483:
482:partial order
479:
478:
477:
475:
471:
463:
459:
443:
437:
434:
431:
411:
408:
405:
402:
382:
379:
376:
373:
370:
367:
364:
344:
341:
338:
318:
315:
312:
309:
306:
303:
300:
292:
276:
270:
267:
264:
256:
255:
249:
247:
243:
239:
235:
233:
229:
210:
207:
204:
184:
181:
178:
170:
167:
166:
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162:
160:
150:
148:
144:
139:
137:
133:
129:
125:
121:
117:
113:
109:
106:
101:
99:
95:
91:
87:
83:
79:
75:
71:
67:
63:
59:
56:
52:
48:
44:
40:
33:
19:
1370:
1351:Categorified
1255:n-categories
1206:Key concepts
1044:Direct limit
1027:Coequalizers
945:Yoneda lemma
851:Key concepts
841:Key concepts
771:
744:
729:
711:
693:
672:
660:
640:
633:
574:cofibrations
544:
540:
532:
528:
526:
518:
514:
510:
503:
485:
473:
469:
467:
457:
291:monomorphism
245:
241:
237:
236:
232:compositions
225:
163:
156:
146:
142:
140:
131:
127:
123:
119:
115:
111:
107:
102:
97:
93:
89:
85:
81:
61:
57:
50:
46:
36:
1319:-categories
1295:Kan complex
1285:Tricategory
1267:-categories
1157:Subcategory
915:Exponential
883:Preadditive
878:Pre-abelian
677:Awodey 2010
604:Dual object
462:epimorphism
257:A morphism
43:mathematics
1461:Categories
1337:3-category
1327:2-category
1300:∞-groupoid
1275:Bicategory
1022:Coproducts
982:Equalizers
888:Bicategory
754:1441931236
625:References
570:Fibrations
1386:Symmetric
1331:2-functor
1071:Relations
994:Pullbacks
790:740446073
763:851741862
736:EMS Press
726:"Duality"
718:EMS Press
700:EMS Press
460:to be an
441:→
435::
380:∘
368:∘
316:∘
304:∘
274:→
268::
208:∘
182:∘
78:composing
1446:Glossary
1426:Category
1400:n-monoid
1353:concepts
1009:Colimits
977:Products
930:Morphism
873:Concrete
868:Additive
858:Category
593:See also
564:colimits
551:, or of
484:. So if
395:implies
331:implies
252:Examples
103:Given a
74:morphism
72:of each
1436:Outline
1395:n-group
1360:2-group
1315:Strict
1305:∞-topos
1101:Modules
1039:Pushout
987:Kernels
920:Functor
863:Abelian
738:, 2001
720:, 2001
702:, 2001
553:duality
537:lattice
238:Duality
47:duality
1382:Traced
1365:2-ring
1095:Fields
1081:Groups
1076:Magmas
964:Limits
788:
778:
761:
751:
648:
560:Limits
228:arrows
70:target
66:source
1376:-ring
1263:Weak
1247:Topos
1091:Rings
545:joins
541:meets
488:is a
289:is a
197:with
1066:Sets
786:OCLC
776:ISBN
759:OCLC
749:ISBN
646:ISBN
580:and
572:and
562:and
543:and
230:and
134:are
130:and
68:and
910:End
900:CCC
508:new
494:new
490:set
293:if
126:if
37:In
1463::
1388:)
1384:)(
784:.
757:.
734:,
728:,
716:,
710:,
698:,
692:,
517:≤
496:by
248:.
234:.
138:.
100:.
45:,
1380:(
1373:n
1371:E
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1329:(
1317:n
1281:)
1277:(
1265:n
1107:)
1103:(
1097:)
1093:(
821:e
814:t
807:v
792:.
765:.
654:.
588:.
533:B
531:,
529:A
521:.
519:x
515:y
511:y
506:≤
504:x
486:X
474:C
470:C
458:f
444:A
438:B
432:f
412:.
409:h
406:=
403:g
383:f
377:h
374:=
371:f
365:g
345:h
342:=
339:g
319:h
313:f
310:=
307:g
301:f
277:B
271:A
265:f
246:C
242:C
211:g
205:f
185:f
179:g
147:C
143:C
132:C
128:D
124:C
120:D
116:C
112:C
108:C
98:C
94:C
90:C
86:C
82:C
62:C
58:C
51:C
34:.
20:)
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