27:
100:
structures may appear to be unrelated at a superficial or intuitive level, making the notion fairly powerful: it creates the opportunity to "translate" theorems between different kinds of mathematical structures, knowing that the essential meaning of those theorems is preserved under the translation.
760:
are not both full and faithful, then we may view their adjointness relation as expressing a "weaker form of equivalence" of categories. Assuming that the natural transformations for the adjunctions are given, all of these formulations allow for an explicit construction of the necessary data, and no
99:
that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences from many areas of mathematics. Establishing an equivalence involves demonstrating strong similarities between the mathematical structures concerned. In some cases, these
550:
in the underlying set theory), the missing data is not completely specified, and often there are many choices. It is a good idea to specify the missing constructions explicitly whenever possible. Due to this circumstance, a functor with these properties is sometimes called a
2140:. Conversely, for any topology the clopen (i.e. closed and open) subsets yield a Boolean algebra. One obtains a duality between the category of Boolean algebras (with their homomorphisms) and
1672:
to itself. Hence, given the information that the identity functors form an equivalence of categories, in this example one still can choose between two natural isomorphisms for each direction.
1759:
1027:
1073:
2480:. On the other hand, any equivalence between additive categories is necessarily additive. (Note that the latter statement is not true for equivalences between preadditive categories.)
2083:
1466:
1915:
1670:
1621:
653:
621:
1839:
2390:
Dualities "turn all concepts around": they turn initial objects into terminal objects, monomorphisms into epimorphisms, kernels into cokernels, limits into colimits etc.
2540:
1545:
981:
947:
758:
721:
589:
1866:
1592:
1493:
1267:
1180:
913:
886:
839:
2138:
2114:
2056:
2036:
1998:
1970:
1935:
1886:
1807:
1787:
1704:
1641:
1565:
1513:
1421:
1401:
1378:
1358:
1334:
1314:
1290:
1240:
1220:
1200:
1153:
1133:
1113:
1093:
859:
812:
792:
126:
in an algebraic setting, the composite of the functor and its "inverse" is not necessarily the identity mapping. Instead it is sufficient that each object be
2511:
under composition if we consider two auto-equivalences that are naturally isomorphic to be identical. This group captures the essential "symmetries" of
2583:
132:
to its image under this composition. Thus one may describe the functors as being "inverse up to isomorphism". There is indeed a concept of
2145:
48:
2671:
2641:
70:
122:
between the involved categories, which is required to have an "inverse" functor. However, in contrast to the situation common for
503:
2621:
530:
This is a quite useful and commonly applied criterion, because one does not have to explicitly construct the "inverse"
2616:
2094:
1707:
41:
35:
1719:
2204:
986:
2661:
2375:
2345:
1032:
133:
52:
2666:
2611:
2353:
2341:
2317:
2074:
2063:
1426:
128:
104:
96:
2472:
may be turned into a preadditive category (or additive category, or abelian category) in such a way that
2216:
As a rule of thumb, an equivalence of categories preserves all "categorical" concepts and properties. If
2155:
the category of spatial locales is known to be equivalent to the dual of the category of sober spaces.
1891:
1646:
1597:
626:
594:
2457:
1762:
1812:
765:
of an adjunction is an isomorphism if and only if the right adjoint is a full and faithful functor.
136:
where a strict form of inverse functor is required, but this is of much less practical use than the
2508:
2004:
2524:
2159:
2152:
1973:
1941:
1518:
952:
918:
737:
700:
568:
319:) if there exists an equivalence (respectively duality) between them. Furthermore, we say that
2637:
2629:
2579:
2574:
Lutz Schrƶder (2001). "Categories: a free tour". In JĆ¼rgen
Koslowski and Austin Melton (ed.).
2528:
2461:
1766:
2477:
2465:
2365:
2170:
1949:
1676:
563:
431:
1844:
1570:
1471:
1245:
1158:
891:
864:
817:
2598:
2264:
2245:
2015:
556:
547:
84:
542:
and the identity functors. On the other hand, though the above properties guarantee the
303:
One often does not specify all the above data. For instance, we say that the categories
2260:
2253:
2241:
2123:
2117:
2099:
2070:
2041:
2021:
1983:
1955:
1920:
1871:
1792:
1772:
1689:
1626:
1550:
1498:
1406:
1386:
1363:
1343:
1319:
1299:
1275:
1225:
1205:
1185:
1138:
1118:
1098:
1078:
844:
797:
777:
761:
choice principles are needed. The key property that one has to prove here is that the
2655:
2089:
2059:
2012:
1945:
2282:
1713:
359:
2148:
stating a duality between finite partial orders and finite distributive lattices.
2290:
2286:
2268:
2249:
2141:
1977:
1710:
1680:
123:
88:
2545:
2008:
1937:
to the corresponding linear maps is full, faithful and essentially surjective.
1340:
isomorphic in that there are no morphisms between them. Thus any functor from
425:
671:
are equivalent (as defined above in that there are natural isomorphisms from
335:
and the natural isomorphisms: there may be many choices (see example below).
2349:
497:
2357:
2078:
2038:
is associated with the algebra of continuous complex-valued functions on
546:
of a categorical equivalence (given a sufficiently strong version of the
355:
yields an equivalence of categories if and only if it is simultaneously:
327:
and natural isomorphisms as above exist. Note however that knowledge of
2058:, and every commutative C*-algebra is associated with the space of its
119:
1316:
with two objects and only two identity morphisms. The two objects in
2379:
1594:
in place of the required natural isomorphisms between the functor
2602:
2228:
is an equivalence, then the following statements are all true:
2011:
with identity is contravariantly equivalent to the category of
2000:
associates to every affine scheme its ring of global sections.
20:
2144:(with continuous mappings). Another case of Stone duality is
2087:, which is a special instance within the general scheme of
1679:
is equivalent to but not isomorphic with the category of
2081:. Probably the most well-known theorem of this kind is
323:"is" an equivalence of categories if an inverse functor
2519:
is not a small category, then the auto-equivalences of
2077:
that connect certain classes of lattices to classes of
2297:
is a monomorphism (or epimorphism, or isomorphism) in
1567:
is equivalent to itself, which can be shown by taking
555:. (Unfortunately this conflicts with terminology from
2578:. Springer Science & Business Media. p. 10.
2126:
2102:
2044:
2024:
2018:. Under this duality, every compact Hausdorff space
1986:
1958:
1923:
1894:
1874:
1847:
1815:
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1692:
1649:
1629:
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1573:
1553:
1521:
1501:
1474:
1429:
1409:
1389:
1366:
1346:
1322:
1302:
1278:
1248:
1228:
1208:
1188:
1161:
1141:
1121:
1101:
1081:
1035:
989:
955:
921:
894:
867:
847:
820:
800:
780:
740:
703:
629:
597:
571:
1765:(the latter category is explained in the article on
2084:
Stone's representation theorem for
Boolean algebras
288:, assigning each object and morphism to itself. If
2132:
2108:
2050:
2030:
1992:
1964:
1929:
1909:
1880:
1860:
1833:
1801:
1781:
1753:
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1635:
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1586:
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1308:
1284:
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1174:
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1021:
975:
941:
907:
880:
853:
833:
806:
786:
752:
715:
647:
615:
583:
2541:Equivalent definitions of mathematical structures
562:There is also a close relation to the concept of
2116:is mapped to a specific topology on the set of
1754:{\displaystyle D=\mathrm {Mat} (\mathbb {R} )}
1292:with a single object and a single morphism is
8:
2073:, there are a number of dualities, based on
1022:{\displaystyle \alpha \colon d_{1}\to d_{2}}
1623:and itself. However, it is also true that
1068:{\displaystyle \beta \colon d_{2}\to d_{1}}
915:and four morphisms: two identity morphisms
296:are contravariant functors one speaks of a
118:An equivalence of categories consists of a
1115:are equivalent; we can (for example) have
2125:
2101:
2043:
2023:
1985:
1972:associates to every commutative ring its
1957:
1922:
1901:
1897:
1896:
1893:
1873:
1852:
1846:
1814:
1794:
1774:
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1434:
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1388:
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1277:
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1207:
1187:
1166:
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1059:
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988:
965:
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931:
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893:
872:
866:
846:
825:
819:
799:
779:
739:
702:
628:
596:
570:
107:of another category then one speaks of a
71:Learn how and when to remove this message
2636:. New York: Springer. pp. xii+314.
2634:Categories for the working mathematician
2452:is an equivalence of categories, and if
34:This article includes a list of general
2557:
514:is isomorphic to an object of the form
111:, and says that the two categories are
234:denote the respective compositions of
2405:is an equivalence of categories, and
534:and the natural isomorphisms between
331:is usually not enough to reconstruct
7:
1461:{\displaystyle 1_{c},f\colon c\to c}
1380:will not be essentially surjective.
103:If a category is equivalent to the
2203:Any category is equivalent to its
1944:is the duality of the category of
1736:
1733:
1730:
1643:yields a natural isomorphism from
184:, and two natural isomorphisms Īµ:
40:it lacks sufficient corresponding
14:
2386:is cartesian closed (or a topos).
2146:Birkhoff's representation theorem
18:Abstract mathematics relationship
1910:{\displaystyle \mathbb {R} ^{n}}
1665:{\displaystyle \mathbf {I} _{C}}
1652:
1616:{\displaystyle \mathbf {I} _{C}}
1603:
648:{\displaystyle G:D\rightarrow C}
616:{\displaystyle F:C\rightarrow D}
25:
2564:Mac Lane (1998), Theorem IV.4.1
148:Formally, given two categories
2360:, we see that the equivalence
1834:{\displaystyle G\colon D\to C}
1825:
1748:
1740:
1452:
1052:
1006:
639:
607:
553:weak equivalence of categories
504:essentially surjective (dense)
1:
2352:among others. Applying it to
1940:One of the central themes of
339:Alternative characterizations
2007:the category of commutative
1976:, the scheme defined by the
1809:are equivalent: The functor
1495:be the identity morphism on
2617:Encyclopedia of Mathematics
2612:"Equivalence of categories"
2503:. The auto-equivalences of
2324:if and only if the functor
1296:equivalent to the category
434:, i.e. for any two objects
362:, i.e. for any two objects
2690:
2437:are naturally isomorphic.
1980:of the ring. Its adjoint
1683:and point-preserving maps.
1540:{\displaystyle f\circ f=1}
1272:By contrast, the category
95:is a relation between two
2672:Equivalence (mathematics)
2599:equivalence of categories
2376:cartesian closed category
2340:. This can be applied to
1675:The category of sets and
976:{\displaystyle 1_{d_{2}}}
942:{\displaystyle 1_{d_{1}}}
753:{\displaystyle F\dashv G}
716:{\displaystyle F\dashv G}
584:{\displaystyle F\dashv G}
158:equivalence of categories
134:isomorphism of categories
93:equivalence of categories
2576:Categorical Perspectives
659:is the right adjoint of
2336:has limit (or colimit)
2075:representation theorems
794:having a single object
731:are full and faithful.
623:is the left adjoint of
87:, a branch of abstract
55:more precise citations.
2134:
2110:
2064:Gelfand representation
2052:
2032:
1994:
1966:
1931:
1911:
1882:
1862:
1841:which maps the object
1835:
1803:
1783:
1755:
1700:
1686:Consider the category
1666:
1637:
1617:
1588:
1561:
1541:
1509:
1489:
1462:
1417:
1397:
1374:
1354:
1330:
1310:
1286:
1263:
1236:
1216:
1196:
1176:
1149:
1129:
1109:
1089:
1069:
1023:
977:
943:
909:
882:
855:
835:
814:and a single morphism
808:
788:
774:Consider the category
754:
734:When adjoint functors
717:
649:
617:
585:
160:consists of a functor
2135:
2111:
2053:
2033:
1995:
1967:
1932:
1912:
1883:
1863:
1861:{\displaystyle A_{n}}
1836:
1804:
1784:
1756:
1701:
1667:
1638:
1618:
1589:
1587:{\displaystyle 1_{c}}
1562:
1542:
1510:
1490:
1488:{\displaystyle 1_{c}}
1463:
1418:
1398:
1375:
1355:
1331:
1311:
1287:
1264:
1262:{\displaystyle 1_{c}}
1242:and all morphisms to
1237:
1217:
1197:
1177:
1175:{\displaystyle d_{1}}
1150:
1130:
1110:
1090:
1070:
1024:
983:and two isomorphisms
978:
944:
910:
908:{\displaystyle d_{2}}
883:
881:{\displaystyle d_{1}}
856:
836:
834:{\displaystyle 1_{c}}
809:
789:
755:
718:
650:
618:
586:
298:duality of categories
109:duality of categories
2458:preadditive category
2419:are two inverses of
2124:
2100:
2042:
2022:
1984:
1956:
1948:and the category of
1921:
1917:and the matrices in
1892:
1888:to the vector space
1872:
1845:
1813:
1793:
1773:
1720:
1690:
1647:
1627:
1598:
1571:
1551:
1519:
1499:
1472:
1427:
1423:, and two morphisms
1407:
1387:
1383:Consider a category
1364:
1344:
1320:
1300:
1276:
1246:
1226:
1206:
1202:map both objects of
1186:
1159:
1139:
1119:
1099:
1079:
1033:
987:
953:
919:
892:
865:
845:
818:
798:
778:
738:
701:
627:
595:
591:, where we say that
569:
129:naturally isomorphic
2005:functional analysis
1767:additive categories
1716:, and the category
841:, and the category
506:, i.e. each object
2630:Mac Lane, Saunders
2523:may form a proper
2515:. (One caveat: if
2491:is an equivalence
2293:), if and only if
2277:the morphism Ī± in
2188:is equivalent to (
2153:pointless topology
2130:
2106:
2079:topological spaces
2048:
2028:
1990:
1962:
1942:algebraic geometry
1927:
1907:
1878:
1858:
1831:
1799:
1779:
1751:
1696:
1662:
1633:
1613:
1584:
1557:
1537:
1505:
1485:
1458:
1413:
1393:
1370:
1350:
1326:
1306:
1282:
1259:
1232:
1212:
1192:
1172:
1145:
1125:
1105:
1085:
1075:. The categories
1065:
1019:
973:
939:
905:
878:
851:
831:
804:
784:
750:
713:
645:
613:
581:
105:opposite (or dual)
2585:978-0-8176-4186-3
2462:additive category
2382:) if and only if
2133:{\displaystyle B}
2109:{\displaystyle B}
2051:{\displaystyle X}
2031:{\displaystyle X}
1993:{\displaystyle F}
1965:{\displaystyle G}
1950:commutative rings
1930:{\displaystyle D}
1881:{\displaystyle D}
1802:{\displaystyle D}
1782:{\displaystyle C}
1699:{\displaystyle C}
1677:partial functions
1636:{\displaystyle f}
1560:{\displaystyle C}
1508:{\displaystyle c}
1416:{\displaystyle c}
1396:{\displaystyle C}
1373:{\displaystyle E}
1353:{\displaystyle C}
1329:{\displaystyle E}
1309:{\displaystyle E}
1285:{\displaystyle C}
1235:{\displaystyle c}
1215:{\displaystyle D}
1195:{\displaystyle G}
1148:{\displaystyle c}
1128:{\displaystyle F}
1108:{\displaystyle D}
1088:{\displaystyle C}
861:with two objects
854:{\displaystyle D}
807:{\displaystyle c}
787:{\displaystyle C}
697:) if and only if
317:dually equivalent
278:identity functors
113:dually equivalent
81:
80:
73:
2679:
2662:Adjoint functors
2647:
2625:
2590:
2589:
2571:
2565:
2562:
2485:auto-equivalence
2478:additive functor
2466:abelian category
2171:product category
2139:
2137:
2136:
2131:
2115:
2113:
2112:
2107:
2057:
2055:
2054:
2049:
2037:
2035:
2034:
2029:
2016:Hausdorff spaces
1999:
1997:
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1933:
1928:
1916:
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1566:
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1558:
1546:
1544:
1543:
1538:
1514:
1512:
1511:
1506:
1494:
1492:
1491:
1486:
1484:
1483:
1467:
1465:
1464:
1459:
1439:
1438:
1422:
1420:
1419:
1414:
1403:with one object
1402:
1400:
1399:
1394:
1379:
1377:
1376:
1371:
1359:
1357:
1356:
1351:
1335:
1333:
1332:
1327:
1315:
1313:
1312:
1307:
1291:
1289:
1288:
1283:
1268:
1266:
1265:
1260:
1258:
1257:
1241:
1239:
1238:
1233:
1221:
1219:
1218:
1213:
1201:
1199:
1198:
1193:
1181:
1179:
1178:
1173:
1171:
1170:
1154:
1152:
1151:
1146:
1134:
1132:
1131:
1126:
1114:
1112:
1111:
1106:
1094:
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1091:
1086:
1074:
1072:
1071:
1066:
1064:
1063:
1051:
1050:
1028:
1026:
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1020:
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1017:
1005:
1004:
982:
980:
979:
974:
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969:
948:
946:
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911:
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903:
887:
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860:
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852:
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837:
832:
830:
829:
813:
811:
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805:
793:
791:
790:
785:
759:
757:
756:
751:
722:
720:
719:
714:
654:
652:
651:
646:
622:
620:
619:
614:
590:
588:
587:
582:
564:adjoint functors
76:
69:
65:
62:
56:
51:this article by
42:inline citations
29:
28:
21:
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2667:Category theory
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2411:
2265:terminal object
2246:terminal object
2214:
2122:
2121:
2098:
2097:
2095:Boolean algebra
2062:. This is the
2040:
2039:
2020:
2019:
1982:
1981:
1954:
1953:
1952:. The functor
1919:
1918:
1895:
1890:
1889:
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1097:
1096:
1077:
1076:
1055:
1042:
1031:
1030:
1009:
996:
985:
984:
961:
956:
951:
950:
927:
922:
917:
916:
895:
890:
889:
868:
863:
862:
843:
842:
821:
816:
815:
796:
795:
776:
775:
771:
736:
735:
699:
698:
692:
683:
655:, or likewise,
625:
624:
593:
592:
567:
566:
557:homotopy theory
548:axiom of choice
491:
484:
477:
471:
464:
457:
447:
440:
419:
412:
405:
399:
392:
385:
375:
368:
341:
267:
250:
205:
196:
146:
85:category theory
77:
66:
60:
57:
47:Please help to
46:
30:
26:
19:
12:
11:
5:
2687:
2686:
2683:
2675:
2674:
2669:
2664:
2654:
2653:
2649:
2648:
2642:
2626:
2608:
2595:
2592:
2591:
2584:
2566:
2556:
2555:
2553:
2550:
2549:
2548:
2543:
2536:
2533:
2527:rather than a
2487:of a category
2434:
2427:
2416:
2409:
2388:
2387:
2369:
2302:
2275:
2261:initial object
2254:if and only if
2242:initial object
2213:
2210:
2209:
2208:
2201:
2156:
2149:
2129:
2105:
2071:lattice theory
2067:
2060:maximal ideals
2047:
2027:
2001:
1989:
1961:
1946:affine schemes
1938:
1926:
1904:
1899:
1877:
1855:
1851:
1830:
1827:
1824:
1821:
1818:
1798:
1778:
1750:
1746:
1742:
1738:
1735:
1732:
1728:
1725:
1695:
1684:
1673:
1659:
1654:
1632:
1610:
1605:
1581:
1577:
1556:
1547:. Of course,
1536:
1533:
1530:
1527:
1524:
1504:
1482:
1478:
1457:
1454:
1451:
1448:
1445:
1442:
1437:
1433:
1412:
1392:
1381:
1369:
1349:
1325:
1305:
1281:
1270:
1256:
1252:
1231:
1211:
1191:
1169:
1165:
1144:
1124:
1104:
1084:
1062:
1058:
1054:
1049:
1045:
1041:
1038:
1016:
1012:
1008:
1003:
999:
995:
992:
968:
964:
959:
934:
930:
925:
902:
898:
875:
871:
850:
828:
824:
803:
783:
770:
767:
749:
746:
743:
712:
709:
706:
688:
679:
644:
641:
638:
635:
632:
612:
609:
606:
603:
600:
580:
577:
574:
528:
527:
501:
489:
482:
473:
469:
462:
453:
445:
438:
429:
417:
410:
401:
397:
390:
381:
373:
366:
340:
337:
315:(respectively
263:
246:
201:
192:
145:
142:
79:
78:
33:
31:
24:
17:
13:
10:
9:
6:
4:
3:
2:
2685:
2684:
2673:
2670:
2668:
2665:
2663:
2660:
2659:
2657:
2645:
2643:0-387-98403-8
2639:
2635:
2631:
2627:
2623:
2619:
2618:
2613:
2609:
2607:
2605:
2600:
2597:
2596:
2587:
2581:
2577:
2570:
2567:
2561:
2558:
2551:
2547:
2544:
2542:
2539:
2538:
2534:
2532:
2530:
2526:
2522:
2518:
2514:
2510:
2506:
2502:
2498:
2494:
2490:
2486:
2481:
2479:
2475:
2471:
2467:
2463:
2459:
2455:
2451:
2447:
2443:
2438:
2433:
2426:
2422:
2415:
2408:
2404:
2400:
2396:
2391:
2385:
2381:
2377:
2373:
2370:
2367:
2366:exact functor
2363:
2359:
2355:
2351:
2347:
2343:
2339:
2335:
2331:
2327:
2323:
2320:(or colimit)
2319:
2315:
2311:
2307:
2303:
2300:
2296:
2292:
2288:
2284:
2280:
2276:
2274:
2270:
2266:
2262:
2258:
2255:
2251:
2247:
2243:
2239:
2235:
2231:
2230:
2229:
2227:
2223:
2219:
2211:
2206:
2202:
2199:
2195:
2191:
2187:
2183:
2179:
2175:
2172:
2168:
2164:
2161:
2157:
2154:
2150:
2147:
2143:
2127:
2119:
2103:
2096:
2092:
2091:
2090:Stone duality
2086:
2085:
2080:
2076:
2072:
2068:
2065:
2061:
2045:
2025:
2017:
2014:
2010:
2006:
2002:
1987:
1979:
1975:
1959:
1951:
1947:
1943:
1939:
1924:
1902:
1875:
1853:
1849:
1828:
1822:
1819:
1816:
1796:
1776:
1768:
1764:
1726:
1723:
1715:
1714:vector spaces
1712:
1709:
1693:
1685:
1682:
1678:
1674:
1657:
1630:
1608:
1579:
1575:
1554:
1534:
1531:
1528:
1525:
1522:
1502:
1480:
1476:
1455:
1449:
1446:
1443:
1440:
1435:
1431:
1410:
1390:
1382:
1367:
1347:
1339:
1323:
1303:
1295:
1279:
1271:
1254:
1250:
1229:
1209:
1189:
1167:
1163:
1142:
1122:
1102:
1082:
1060:
1056:
1047:
1043:
1039:
1036:
1014:
1010:
1001:
997:
993:
990:
966:
962:
957:
932:
928:
923:
900:
896:
873:
869:
848:
826:
822:
801:
781:
773:
772:
768:
766:
764:
747:
744:
741:
732:
730:
726:
710:
707:
704:
696:
691:
687:
682:
678:
674:
670:
666:
662:
658:
642:
636:
633:
630:
610:
604:
601:
598:
578:
575:
572:
565:
560:
558:
554:
549:
545:
541:
537:
533:
525:
521:
517:
513:
509:
505:
502:
499:
495:
492:) induced by
488:
481:
476:
468:
461:
456:
452:, the map Hom
451:
444:
437:
433:
430:
427:
423:
420:) induced by
416:
409:
404:
396:
389:
384:
380:, the map Hom
379:
372:
365:
361:
358:
357:
356:
354:
350:
346:
338:
336:
334:
330:
326:
322:
318:
314:
310:
306:
301:
299:
295:
291:
287:
283:
279:
275:
271:
266:
262:
258:
254:
249:
245:
241:
237:
233:
229:
225:
221:
217:
213:
209:
204:
200:
197:and Ī· :
195:
191:
187:
183:
179:
175:
171:
167:
163:
159:
155:
151:
143:
141:
139:
135:
131:
130:
125:
121:
116:
114:
110:
106:
101:
98:
94:
90:
86:
75:
72:
64:
54:
50:
44:
43:
37:
32:
23:
22:
16:
2633:
2615:
2603:
2575:
2569:
2560:
2520:
2516:
2512:
2504:
2500:
2496:
2492:
2488:
2484:
2482:
2473:
2469:
2453:
2449:
2445:
2441:
2439:
2431:
2424:
2420:
2413:
2406:
2402:
2398:
2394:
2392:
2389:
2383:
2371:
2361:
2337:
2333:
2329:
2325:
2321:
2313:
2309:
2305:
2304:the functor
2298:
2294:
2283:monomorphism
2278:
2272:
2256:
2237:
2233:
2225:
2221:
2217:
2215:
2197:
2193:
2189:
2185:
2181:
2177:
2173:
2166:
2162:
2142:Stone spaces
2118:ultrafilters
2088:
2082:
1978:prime ideals
1761:of all real
1681:pointed sets
1337:
1293:
762:
733:
728:
724:
694:
689:
685:
680:
676:
672:
668:
664:
660:
656:
561:
552:
543:
539:
535:
531:
529:
523:
519:
515:
511:
507:
493:
486:
479:
474:
466:
459:
454:
449:
442:
435:
421:
414:
407:
402:
394:
387:
382:
377:
370:
363:
352:
348:
344:
342:
332:
328:
324:
320:
316:
312:
308:
304:
302:
297:
293:
289:
285:
281:
277:
273:
269:
264:
260:
256:
252:
247:
243:
239:
235:
231:
227:
223:
219:
215:
211:
207:
202:
198:
193:
189:
185:
181:
177:
173:
172:, a functor
169:
165:
161:
157:
153:
149:
147:
137:
127:
124:isomorphisms
117:
112:
108:
102:
92:
82:
67:
58:
39:
15:
2476:becomes an
2291:isomorphism
2287:epimorphism
2269:zero object
2250:zero object
2232:the object
2009:C*-algebras
1708:dimensional
276:denote the
138:equivalence
89:mathematics
53:introducing
2656:Categories
2552:References
2546:Anafunctor
2350:coproducts
2342:equalizers
2212:Properties
1706:of finite-
426:surjective
343:A functor
313:equivalent
144:Definition
97:categories
36:references
2622:EMS Press
2358:cokernels
1826:→
1820::
1769:). Then
1526:∘
1453:→
1447::
1053:→
1040::
1037:β
1007:→
994::
991:α
745:⊣
723:and both
708:⊣
640:→
608:→
576:⊣
544:existence
498:injective
300:instead.
140:concept.
61:June 2015
2632:(1998).
2535:See also
2495: :
2468:), then
2444: :
2397: :
2346:products
2328: :
2308: :
2220: :
2205:skeleton
2158:For two
2093:. Each
1974:spectrum
1763:matrices
1515:and set
769:Examples
663:. Then
432:faithful
347: :
176: :
164: :
2624:, 2001
2601:at the
2507:form a
2423:, then
2354:kernels
2013:compact
1468:. Let
472:) ā Hom
400:) ā Hom
210:. Here
120:functor
49:improve
2640:
2582:
2378:(or a
2364:is an
2259:is an
2240:is an
2169:, the
763:counit
518:, for
242:, and
38:, but
2525:class
2509:group
2464:, or
2456:is a
2380:topos
2374:is a
2318:limit
2289:, or
2281:is a
2271:) of
2267:, or
2248:, or
2160:rings
500:; and
156:, an
91:, an
2638:ISBN
2580:ISBN
2460:(or
2430:and
2412:and
2356:and
2348:and
2316:has
2285:(or
2263:(or
2244:(or
2165:and
1789:and
1711:real
1336:are
1182:and
1135:map
1095:and
1029:and
727:and
684:and
667:and
441:and
369:and
360:full
311:are
307:and
292:and
284:and
259:and
238:and
222:and
152:and
2606:Lab
2531:.)
2529:set
2483:An
2440:If
2393:If
2252:),
2236:of
2198:Mod
2186:Mod
2178:Mod
2151:In
2120:of
2069:In
2003:In
1868:of
1360:to
1338:not
1294:not
1222:to
1155:to
693:to
675:to
559:.)
522:in
510:in
496:is
448:of
424:is
376:of
280:on
83:In
2658::
2620:,
2614:,
2499:ā
2448:ā
2401:ā
2344:,
2338:Fl
2332:ā
2326:FH
2312:ā
2295:FĪ±
2257:Fc
2224:ā
2196:)-
949:,
888:,
695:GF
673:FG
540:GF
538:,
536:FG
516:Fc
487:Fc
480:Fc
415:Fc
408:Fc
351:ā
268::
251::
226::
224:GF
214::
212:FG
208:GF
186:FG
180:ā
168:ā
115:.
2646:.
2604:n
2588:.
2521:C
2517:C
2513:C
2505:C
2501:C
2497:C
2493:F
2489:C
2474:F
2470:D
2454:C
2450:D
2446:C
2442:F
2435:2
2432:G
2428:1
2425:G
2421:F
2417:2
2414:G
2410:1
2407:G
2403:D
2399:C
2395:F
2384:D
2372:C
2368:.
2362:F
2334:D
2330:I
2322:l
2314:C
2310:I
2306:H
2301:.
2299:D
2279:C
2273:D
2238:C
2234:c
2226:D
2222:C
2218:F
2207:.
2200:.
2194:S
2192:Ć
2190:R
2184:-
2182:S
2180:Ć
2176:-
2174:R
2167:S
2163:R
2128:B
2104:B
2066:.
2046:X
2026:X
1988:F
1960:G
1925:D
1903:n
1898:R
1876:D
1854:n
1850:A
1829:C
1823:D
1817:G
1797:D
1777:C
1749:)
1745:R
1741:(
1737:t
1734:a
1731:M
1727:=
1724:D
1694:C
1658:C
1653:I
1631:f
1609:C
1604:I
1580:c
1576:1
1555:C
1535:1
1532:=
1529:f
1523:f
1503:c
1481:c
1477:1
1456:c
1450:c
1444:f
1441:,
1436:c
1432:1
1411:c
1391:C
1368:E
1348:C
1324:E
1304:E
1280:C
1269:.
1255:c
1251:1
1230:c
1210:D
1190:G
1168:1
1164:d
1143:c
1123:F
1103:D
1083:C
1061:1
1057:d
1048:2
1044:d
1015:2
1011:d
1002:1
998:d
967:2
963:d
958:1
933:1
929:d
924:1
901:2
897:d
874:1
870:d
849:D
827:c
823:1
802:c
782:C
748:G
742:F
729:G
725:F
711:G
705:F
690:C
686:I
681:D
677:I
669:D
665:C
661:F
657:G
643:C
637:D
634::
631:G
611:D
605:C
602::
599:F
579:G
573:F
532:G
526:.
524:C
520:c
512:D
508:d
494:F
490:2
485:,
483:1
478:(
475:D
470:2
467:c
465:,
463:1
460:c
458:(
455:C
450:C
446:2
443:c
439:1
436:c
428:;
422:F
418:2
413:,
411:1
406:(
403:D
398:2
395:c
393:,
391:1
388:c
386:(
383:C
378:C
374:2
371:c
367:1
364:c
353:D
349:C
345:F
333:G
329:F
325:G
321:F
309:D
305:C
294:G
290:F
286:D
282:C
274:D
272:ā
270:D
265:D
261:I
257:C
255:ā
253:C
248:C
244:I
240:G
236:F
232:C
230:ā
228:C
220:D
218:ā
216:D
206:ā
203:C
199:I
194:D
190:I
188:ā
182:C
178:D
174:G
170:D
166:C
162:F
154:D
150:C
74:)
68:(
63:)
59:(
45:.
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