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