414:
636:
1578:
1091:
342:
1303:
1400:
224:
1224:
936:
845:
803:
403:
761:
721:
1152:
645:
by this relation. It is clear that this results in an additive category if one notes that this is the same as taking the quotient by the subgroup of null-homotopic maps.
1441:
514:
456:
533:
1485:
975:
1639:
1618:
103:
does so only for those that are quasi-isomorphisms for a "good reason", namely actually having an inverse up to homotopy equivalence. Thus,
1234:
236:
49:
is a framework for working with chain homotopies and homotopy equivalences. It lies intermediate between the category of
1679:
1674:
1472:
1243:
1358:
466:. It is clear from the definition that the maps of complexes which are null-homotopic form a group under addition.
1405:
is not distinguished since the cone of the identity map is not isomorphic to the complex 0 (however, the zero map
170:
1161:
897:
808:
766:
1317:
347:
80:
1352:
as well, that category is not triangulated with respect to these distinguished triangles; for example,
734:
694:
1602:
1096:
484:. Its morphisms are "maps of complexes modulo homotopy": that is, we define an equivalence relation
28:
1408:
490:
1653:
1635:
1614:
891:
855:
124:
92:
43:
1451:). Furthermore, the rotation of a distinguished triangle is obviously not distinguished in
947:
939:
76:
61:
1649:
1584:
is the category of complexes whose morphisms do not have to respect the differentials). If
1645:
1610:
887:
883:
50:
631:{\displaystyle \operatorname {Hom} _{K(A)}(A,B)=\operatorname {Hom} _{Kom(A)}(A,B)/\sim }
435:
1634:. Cambridge Studies in Advanced Mathematics. Vol. 38. Cambridge University Press.
1627:
859:
1668:
481:
1573:{\displaystyle \operatorname {Hom} _{Ho(C)}(X,Y)=H^{0}\operatorname {Hom} _{C}(X,Y)}
894:. (The converse is false in general.) This shows that there is a canonical functor
648:
The following variants of the definition are also widely used: if one takes only
32:
17:
1657:
851:
413:
642:
1086:{\displaystyle A:...\to A^{n+1}{\xrightarrow {d_{A}^{n}}}A^{n+2}\to ...}
890:, which are zero in homology. In particular a homotopy equivalence is a
476:
is then defined as follows: its objects are the same as the objects of
1370:
1255:
1027:
1324:, i.e. homotopy equivalent) to the triangles above, for arbitrary
763:, such that the two compositions are homotopic to the identities:
1320:, if one defines distinguished triangles to be isomorphic (in
337:{\displaystyle f^{n}-g^{n}=d_{B}^{n-1}h^{n}+h^{n+1}d_{A}^{n},}
83:, and unlike the latter its formation does not require that
131:
is based on the following definition: if we have complexes
1580:. (This boils down to the homotopy of chain complexes if
672:) complexes instead of unbounded ones, one speaks of the
91:
turns into isomorphisms any maps of complexes that are
1488:
1411:
1361:
1246:
1164:
1099:
978:
900:
811:
769:
737:
697:
536:
493:
438:
350:
239:
173:
1572:
1435:
1394:
1297:
1218:
1146:
1085:
930:
839:
797:
755:
715:
630:
508:
450:
397:
336:
218:
1443:is a homotopy equivalence, so that this triangle
1298:{\displaystyle A{\xrightarrow {f}}B\to C(f)\to A}
1588:has cones and shifts in a suitable sense, then
1395:{\displaystyle X{\xrightarrow {id}}X\to 0\to }
858:induce homotopic (in the above sense) maps of
731:. In detail, this means there is another map
850:The name "homotopy" comes from the fact that
8:
1336:. The same is true for the bounded variants
219:{\displaystyle h^{n}\colon A^{n}\to B^{n-1}}
1455:, but (less obviously) is distinguished in
1546:
1536:
1493:
1487:
1410:
1365:
1360:
1250:
1245:
1204:
1199:
1183:
1169:
1163:
1132:
1119:
1098:
1059:
1046:
1032:
1022:
1010:
977:
899:
878:induce the same maps on homology because
831:
810:
789:
768:
736:
696:
620:
581:
541:
535:
492:
437:
386:
367:
349:
325:
320:
304:
291:
275:
270:
257:
244:
238:
204:
191:
178:
172:
1478:is defined to have the same objects as
1219:{\displaystyle d_{A}^{n}:=-d_{A}^{n+1}}
1632:An introduction to homological algebra
1467:More generally, the homotopy category
7:
1348:. Although triangles make sense in
931:{\displaystyle K(A)\rightarrow D(A)}
471:homotopy category of chain complexes
87:is abelian. Philosophically, while
1459:. See the references for details.
840:{\displaystyle g\circ f\sim Id_{A}}
798:{\displaystyle f\circ g\sim Id_{B}}
398:{\displaystyle f-g=d_{B}h+hd_{A}.}
25:
1592:is a triangulated category, too.
756:{\displaystyle g:B\rightarrow A}
716:{\displaystyle f:A\rightarrow B}
412:
1482:, but morphisms are defined by
1147:{\displaystyle (A)^{n}=A^{n+1}}
674:bounded-below homotopy category
1607:Methods of Homological Algebra
1567:
1555:
1526:
1514:
1506:
1500:
1427:
1424:
1415:
1389:
1383:
1292:
1286:
1280:
1277:
1271:
1265:
1179:
1173:
1116:
1112:
1106:
1100:
1071:
1042:
1036:
1003:
988:
982:
925:
919:
913:
910:
904:
747:
707:
617:
605:
597:
591:
571:
559:
551:
545:
230:a map of complexes) such that
197:
1:
1605:; Gelfand, Sergei I. (2003),
1473:differential graded category
107:is more understandable than
79:; unlike the former it is a
1237:. There are natural maps
1229:For the cone of a morphism
723:which is an isomorphism in
1696:
1436:{\displaystyle C(id)\to 0}
1158:where the differential is
969:is the following complex
954:The triangulated structure
1312:. The homotopy category
1308:This diagram is called a
870:Two chain homotopic maps
676:etc. They are denoted by
509:{\displaystyle f\sim g\ }
408:This can be depicted as:
127:. The homotopy category
42:of chain complexes in an
167:is a collection of maps
1574:
1437:
1396:
1299:
1220:
1148:
1087:
932:
841:
799:
757:
717:
632:
510:
452:
399:
338:
220:
1603:Manin, Yuri Ivanovich
1575:
1438:
1397:
1318:triangulated category
1300:
1221:
1149:
1088:
933:
842:
800:
758:
718:
633:
511:
453:
400:
339:
221:
81:triangulated category
1609:, Berlin, New York:
1486:
1409:
1359:
1244:
1162:
1097:
976:
898:
809:
767:
735:
729:homotopy equivalence
695:
670:A=0 for |n|>>0
534:
491:
436:
348:
237:
171:
1680:Additive categories
1675:Homological algebra
1377:
1259:
1215:
1188:
1052:
1051:
451:{\displaystyle f-g}
330:
286:
29:homological algebra
1628:Weibel, Charles A.
1570:
1433:
1392:
1295:
1216:
1195:
1165:
1144:
1083:
1028:
928:
856:topological spaces
837:
795:
753:
713:
662:A=0 for n>>0
654:A=0 for n<<0
628:
506:
448:
395:
334:
316:
266:
216:
93:quasi-isomorphisms
1641:978-0-521-55987-4
1620:978-3-540-43583-9
1447:distinguished in
1378:
1260:
1053:
892:quasi-isomorphism
505:
420:We also say that
125:additive category
44:additive category
37:homotopy category
16:(Redirected from
1687:
1661:
1623:
1579:
1577:
1576:
1571:
1551:
1550:
1541:
1540:
1510:
1509:
1442:
1440:
1439:
1434:
1401:
1399:
1398:
1393:
1379:
1366:
1304:
1302:
1301:
1296:
1261:
1251:
1225:
1223:
1222:
1217:
1214:
1203:
1187:
1182:
1153:
1151:
1150:
1145:
1143:
1142:
1124:
1123:
1092:
1090:
1089:
1084:
1070:
1069:
1054:
1050:
1045:
1023:
1021:
1020:
940:derived category
937:
935:
934:
929:
846:
844:
843:
838:
836:
835:
804:
802:
801:
796:
794:
793:
762:
760:
759:
754:
722:
720:
719:
714:
688:, respectively.
637:
635:
634:
629:
624:
601:
600:
555:
554:
520:is homotopic to
515:
513:
512:
507:
503:
457:
455:
454:
449:
416:
404:
402:
401:
396:
391:
390:
372:
371:
343:
341:
340:
335:
329:
324:
315:
314:
296:
295:
285:
274:
262:
261:
249:
248:
225:
223:
222:
217:
215:
214:
196:
195:
183:
182:
62:derived category
21:
1695:
1694:
1690:
1689:
1688:
1686:
1685:
1684:
1665:
1664:
1642:
1626:
1621:
1611:Springer-Verlag
1601:
1598:
1542:
1532:
1489:
1484:
1483:
1465:
1407:
1406:
1357:
1356:
1242:
1241:
1160:
1159:
1128:
1115:
1095:
1094:
1055:
1006:
974:
973:
956:
896:
895:
868:
860:singular chains
827:
807:
806:
785:
765:
764:
733:
732:
693:
692:
577:
537:
532:
531:
489:
488:
482:chain complexes
434:
433:
430:chain homotopic
382:
363:
346:
345:
300:
287:
253:
240:
235:
234:
200:
187:
174:
169:
168:
117:
51:chain complexes
23:
22:
15:
12:
11:
5:
1693:
1691:
1683:
1682:
1677:
1667:
1666:
1663:
1662:
1640:
1624:
1619:
1597:
1594:
1569:
1566:
1563:
1560:
1557:
1554:
1549:
1545:
1539:
1535:
1531:
1528:
1525:
1522:
1519:
1516:
1513:
1508:
1505:
1502:
1499:
1496:
1492:
1464:
1463:Generalization
1461:
1432:
1429:
1426:
1423:
1420:
1417:
1414:
1403:
1402:
1391:
1388:
1385:
1382:
1376:
1373:
1369:
1364:
1306:
1305:
1294:
1291:
1288:
1285:
1282:
1279:
1276:
1273:
1270:
1267:
1264:
1258:
1254:
1249:
1213:
1210:
1207:
1202:
1198:
1194:
1191:
1186:
1181:
1178:
1175:
1172:
1168:
1156:
1155:
1141:
1138:
1135:
1131:
1127:
1122:
1118:
1114:
1111:
1108:
1105:
1102:
1082:
1079:
1076:
1073:
1068:
1065:
1062:
1058:
1049:
1044:
1041:
1038:
1035:
1031:
1026:
1019:
1016:
1013:
1009:
1005:
1002:
999:
996:
993:
990:
987:
984:
981:
955:
952:
927:
924:
921:
918:
915:
912:
909:
906:
903:
867:
864:
834:
830:
826:
823:
820:
817:
814:
792:
788:
784:
781:
778:
775:
772:
752:
749:
746:
743:
740:
712:
709:
706:
703:
700:
639:
638:
627:
623:
619:
616:
613:
610:
607:
604:
599:
596:
593:
590:
587:
584:
580:
576:
573:
570:
567:
564:
561:
558:
553:
550:
547:
544:
540:
525:
524:
502:
499:
496:
464:homotopic to 0
460:null-homotopic
447:
444:
441:
418:
417:
406:
405:
394:
389:
385:
381:
378:
375:
370:
366:
362:
359:
356:
353:
333:
328:
323:
319:
313:
310:
307:
303:
299:
294:
290:
284:
281:
278:
273:
269:
265:
260:
256:
252:
247:
243:
213:
210:
207:
203:
199:
194:
190:
186:
181:
177:
157:chain homotopy
116:
113:
24:
18:Chain homotopy
14:
13:
10:
9:
6:
4:
3:
2:
1692:
1681:
1678:
1676:
1673:
1672:
1670:
1659:
1655:
1651:
1647:
1643:
1637:
1633:
1629:
1625:
1622:
1616:
1612:
1608:
1604:
1600:
1599:
1595:
1593:
1591:
1587:
1583:
1564:
1561:
1558:
1552:
1547:
1543:
1537:
1533:
1529:
1523:
1520:
1517:
1511:
1503:
1497:
1494:
1490:
1481:
1477:
1474:
1470:
1462:
1460:
1458:
1454:
1450:
1446:
1430:
1421:
1418:
1412:
1386:
1380:
1374:
1371:
1367:
1362:
1355:
1354:
1353:
1351:
1347:
1343:
1339:
1335:
1331:
1327:
1323:
1319:
1315:
1311:
1289:
1283:
1274:
1268:
1262:
1256:
1252:
1247:
1240:
1239:
1238:
1236:
1232:
1227:
1211:
1208:
1205:
1200:
1196:
1192:
1189:
1184:
1176:
1170:
1166:
1139:
1136:
1133:
1129:
1125:
1120:
1109:
1103:
1080:
1077:
1074:
1066:
1063:
1060:
1056:
1047:
1039:
1033:
1029:
1024:
1017:
1014:
1011:
1007:
1000:
997:
994:
991:
985:
979:
972:
971:
970:
968:
965:of a complex
964:
961:
953:
951:
949:
945:
941:
922:
916:
907:
901:
893:
889:
885:
881:
877:
873:
865:
863:
861:
857:
853:
848:
832:
828:
824:
821:
818:
815:
812:
790:
786:
782:
779:
776:
773:
770:
750:
744:
741:
738:
730:
726:
710:
704:
701:
698:
689:
687:
683:
679:
675:
671:
667:
663:
659:
658:bounded-above
655:
651:
650:bounded-below
646:
644:
625:
621:
614:
611:
608:
602:
594:
588:
585:
582:
578:
574:
568:
565:
562:
556:
548:
542:
538:
530:
529:
528:
523:
519:
500:
497:
494:
487:
486:
485:
483:
479:
475:
472:
467:
465:
461:
445:
442:
439:
431:
427:
423:
415:
411:
410:
409:
392:
387:
383:
379:
376:
373:
368:
364:
360:
357:
354:
351:
331:
326:
321:
317:
311:
308:
305:
301:
297:
292:
288:
282:
279:
276:
271:
267:
263:
258:
254:
250:
245:
241:
233:
232:
231:
229:
211:
208:
205:
201:
192:
188:
184:
179:
175:
166:
162:
158:
154:
150:
146:
142:
138:
134:
130:
126:
122:
114:
112:
110:
106:
102:
98:
94:
90:
86:
82:
78:
74:
70:
66:
63:
59:
55:
52:
48:
45:
41:
38:
34:
30:
19:
1631:
1606:
1589:
1585:
1581:
1479:
1475:
1468:
1466:
1456:
1452:
1448:
1444:
1404:
1349:
1345:
1341:
1337:
1333:
1329:
1325:
1321:
1313:
1309:
1307:
1235:mapping cone
1233:we take the
1230:
1228:
1157:
966:
962:
959:
957:
943:
879:
875:
871:
869:
849:
728:
727:is called a
724:
690:
685:
681:
677:
673:
669:
665:
661:
657:
653:
649:
647:
640:
526:
521:
517:
477:
473:
470:
468:
463:
459:
429:
425:
421:
419:
407:
227:
164:
160:
156:
152:
148:
144:
140:
136:
132:
128:
120:
118:
108:
104:
100:
96:
88:
84:
72:
68:
64:
57:
53:
46:
39:
36:
26:
1093:(note that
691:A morphism
527:and define
115:Definitions
33:mathematics
1669:Categories
1596:References
888:boundaries
641:to be the
432:, or that
344:or simply
1553:
1512:
1428:→
1390:→
1384:→
1281:→
1266:→
1193:−
1072:→
1004:→
914:→
852:homotopic
822:∼
816:∘
780:∼
774:∘
748:→
708:→
626:∼
603:
557:
498:∼
480:, namely
443:−
355:−
280:−
251:−
209:−
198:→
185::
139:and maps
1658:36131259
1630:(1994).
1368:→
1310:triangle
1253:→
1025:→
854:maps of
643:quotient
60:and the
1650:1269324
948:abelian
938:to the
880:(f − g)
866:Remarks
666:bounded
77:abelian
1656:
1648:
1638:
1617:
1453:Kom(A)
1350:Kom(A)
884:cycles
882:sends
664:), or
504:
478:Kom(A)
123:be an
97:Kom(A)
54:Kom(A)
35:, the
1590:Ho(C)
1471:of a
1469:Ho(C)
1316:is a
960:shift
159:from
147:from
71:when
1654:OCLC
1636:ISBN
1615:ISBN
1457:K(A)
1449:K(A)
1346:K(A)
1344:and
1342:K(A)
1338:K(A)
1332:and
1322:K(A)
1314:K(A)
958:The
942:(if
874:and
805:and
725:K(A)
686:K(A)
684:and
682:K(A)
678:K(A)
474:K(A)
469:The
428:are
424:and
155:, a
129:K(A)
119:Let
109:D(A)
105:K(A)
101:K(A)
89:D(A)
65:D(A)
40:K(A)
1544:Hom
1491:Hom
1226:.
950:).
946:is
886:to
656:),
579:Hom
539:Hom
516:if
462:or
458:is
228:not
163:to
151:to
95:in
75:is
67:of
56:of
31:in
27:In
1671::
1652:.
1646:MR
1644:.
1613:,
1445:is
1340:,
1328:,
1190::=
1154:),
862:.
847:.
680:,
143:,
135:,
111:.
99:,
1660:.
1586:C
1582:C
1568:)
1565:Y
1562:,
1559:X
1556:(
1548:C
1538:0
1534:H
1530:=
1527:)
1524:Y
1521:,
1518:X
1515:(
1507:)
1504:C
1501:(
1498:o
1495:H
1480:C
1476:C
1431:0
1425:)
1422:d
1419:i
1416:(
1413:C
1387:0
1381:X
1375:d
1372:i
1363:X
1334:f
1330:B
1326:A
1293:]
1290:1
1287:[
1284:A
1278:)
1275:f
1272:(
1269:C
1263:B
1257:f
1248:A
1231:f
1212:1
1209:+
1206:n
1201:A
1197:d
1185:n
1180:]
1177:1
1174:[
1171:A
1167:d
1140:1
1137:+
1134:n
1130:A
1126:=
1121:n
1117:)
1113:]
1110:1
1107:[
1104:A
1101:(
1081:.
1078:.
1075:.
1067:2
1064:+
1061:n
1057:A
1048:n
1043:]
1040:1
1037:[
1034:A
1030:d
1018:1
1015:+
1012:n
1008:A
1001:.
998:.
995:.
992::
989:]
986:1
983:[
980:A
967:A
963:A
944:A
926:)
923:A
920:(
917:D
911:)
908:A
905:(
902:K
876:g
872:f
833:A
829:d
825:I
819:f
813:g
791:B
787:d
783:I
777:g
771:f
751:A
745:B
742::
739:g
711:B
705:A
702::
699:f
668:(
660:(
652:(
622:/
618:)
615:B
612:,
609:A
606:(
598:)
595:A
592:(
589:m
586:o
583:K
575:=
572:)
569:B
566:,
563:A
560:(
552:)
549:A
546:(
543:K
522:g
518:f
501:g
495:f
446:g
440:f
426:g
422:f
393:.
388:A
384:d
380:h
377:+
374:h
369:B
365:d
361:=
358:g
352:f
332:,
327:n
322:A
318:d
312:1
309:+
306:n
302:h
298:+
293:n
289:h
283:1
277:n
272:B
268:d
264:=
259:n
255:g
246:n
242:f
226:(
212:1
206:n
202:B
193:n
189:A
180:n
176:h
165:g
161:f
153:B
149:A
145:g
141:f
137:B
133:A
121:A
85:A
73:A
69:A
58:A
47:A
20:)
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