936:
462:
298:
193:
1310:. dg enhancements of an exact functor between triangulated categories are defined similarly. In general, there need not exist dg enhancements of triangulated categories or functors between them, for example
747:
1115:
1012:
1487:
508:
372:
116:
805:
571:
381:
659:
782:
683:
73:
1160:
334:
1281:
597:
215:
623:
1374:
135:
1551:
1416:
1314:
can be shown not to arise from a dg category in this way. However, various positive results do exist, for example the derived category
688:
1023:
946:
1323:
1546:
1192:
52:
471:
931:{\displaystyle \mathrm {Hom} _{C({\mathcal {A}}),n}(A,B)=\prod _{l\in \mathbb {Z} }\mathrm {Hom} (A_{l},B_{l+n})}
1431:
Alberto
Canonaco; Paolo Stellari (2017), "A tour about existence and uniqueness of dg enhancements and lifts",
1311:
457:{\displaystyle \operatorname {Hom} (A,B)\otimes \operatorname {Hom} (B,C)\rightarrow \operatorname {Hom} (A,C)}
339:
83:
526:
1283:
and a class of distinguished triangles compatible with the suspension, such that its homotopy category Ho(
48:
1288:
631:
1450:
1343:
1338:
1175:
1171:
752:
664:
1188:
32:
56:
1466:
1440:
1399:
1179:
1125:
307:
293:{\displaystyle d\colon \operatorname {Hom} _{n}(A,B)\rightarrow \operatorname {Hom} _{n+1}(A,B)}
1506:
1410:
1391:
520:
1266:
576:
1496:
1458:
1383:
1213:
1518:
1514:
1482:
1348:
1236:
1232:
375:
602:
1454:
1212:
structure such that weak equivalences are those functors that induce an equivalence of
1209:
1205:
1540:
1470:
1403:
1462:
1474:
for a survey of existence and unicity results of dg enhancements dg enhancements.
523:
may be considered to be a DG-category by imposing the trivial grading (i.e. all
28:
207:
127:
1510:
1395:
1531:
1353:
1387:
188:{\displaystyle \bigoplus _{n\in \mathbb {Z} }\operatorname {Hom} _{n}(A,B)}
17:
1306:
is a pretriangulated dg category whose homotopy category is equivalent to
1187:
A DG-category with one object is the same as a DG-ring. A DG-ring over a
1501:
1445:
51:
whose morphism sets are endowed with the additional structure of a
1372:
Tabuada, Gonçalo (2005), "Invariants additifs de DG-catégories",
742:{\displaystyle \operatorname {Hom} _{C({\mathcal {A}}),n}(A,B)}
1170:, respectively. This applies to the category of complexes of
1110:{\displaystyle f_{l+1}\circ d_{A}+(-1)^{n+1}d_{B}\circ f_{l}}
628:
A little bit more sophisticated is the category of complexes
830:
705:
670:
643:
1263:
is called pre-triangulated if it has a suspension functor
464:
is required to be a map of complexes, and for all objects
1243:
is the category of quasi-coherent sheaves on some scheme
1007:{\displaystyle f=(f_{l}\colon A_{l}\rightarrow B_{l+n})}
1269:
1128:
1026:
949:
808:
755:
691:
667:
634:
605:
579:
529:
474:
384:
342:
310:
218:
138:
86:
59:
1227:, there is a notion of smoothness and properness of
1488:
788:need to respect the differentials of the complexes
1275:
1154:
1109:
1006:
930:
776:
741:
677:
653:
617:
591:
565:
502:
456:
366:
328:
292:
187:
110:
67:
8:
503:{\displaystyle d(\operatorname {id} _{A})=0}
378:. Furthermore, the composition of morphisms
1375:International Mathematics Research Notices
1500:
1444:
1268:
1146:
1133:
1127:
1101:
1088:
1072:
1050:
1031:
1025:
989:
976:
963:
948:
913:
900:
882:
876:
875:
868:
829:
828:
821:
810:
807:
754:
704:
703:
696:
690:
669:
668:
666:
642:
641:
633:
604:
578:
542:
531:
528:
485:
473:
383:
367:{\displaystyle \operatorname {Hom} (A,B)}
341:
309:
260:
229:
217:
161:
151:
150:
143:
137:
111:{\displaystyle \operatorname {Hom} (A,B)}
85:
61:
60:
58:
1364:
566:{\displaystyle \mathrm {Hom} _{n}(-,-)}
1408:
1231:that reduces to the usual notions of
202:on this graded group, i.e., for each
7:
1208:dg-categories can be endowed with a
943:The differential of such a morphism
336:. This is equivalent to saying that
1255:Relation to triangulated categories
1485:(1994), "Deriving DG categories",
1415:: CS1 maint: unflagged free DOI (
1270:
889:
886:
883:
817:
814:
811:
538:
535:
532:
25:
654:{\displaystyle C({\mathcal {A}})}
1329:admits a unique dg enhancement.
118:, the morphisms from any object
1433:Journal of Geometry and Physics
1463:10.1016/j.geomphys.2016.11.030
1069:
1059:
1001:
982:
956:
925:
893:
858:
846:
835:
825:
777:{\displaystyle A\rightarrow B}
771:
765:
759:
736:
724:
710:
700:
678:{\displaystyle {\mathcal {A}}}
648:
638:
560:
548:
491:
478:
468:of the category, one requires
451:
439:
430:
427:
415:
403:
391:
361:
349:
287:
275:
253:
250:
238:
182:
170:
105:
93:
1:
1552:Categories in category theory
1324:Grothendieck abelian category
599:) and trivial differential (
198:and there is a differential
68:{\displaystyle \mathbb {Z} }
37:differential graded category
1291:. A triangulated category
1193:differential graded algebra
1155:{\displaystyle d_{A},d_{B}}
80:In detail, this means that
1568:
661:over an additive category
329:{\displaystyle d\circ d=0}
1191:is called DG-algebra, or
1162:are the differentials of
1312:stable homotopy category
1276:{\displaystyle \Sigma }
592:{\displaystyle n\neq 0}
1388:10.1155/IMRN.2005.3309
1277:
1172:quasi-coherent sheaves
1156:
1111:
1008:
932:
778:
743:
679:
655:
619:
593:
567:
504:
458:
368:
330:
294:
189:
112:
69:
1289:triangulated category
1278:
1157:
1112:
1009:
933:
779:
749:is the group of maps
744:
680:
656:
620:
594:
568:
505:
459:
369:
331:
304:which has to satisfy
295:
190:
126:of the category is a
113:
70:
39:, often shortened to
1344:Graded (mathematics)
1339:Differential algebra
1267:
1219:Given a dg-category
1126:
1024:
947:
806:
753:
689:
665:
632:
603:
577:
527:
472:
382:
340:
308:
216:
136:
84:
57:
53:differential graded
1547:Homological algebra
1532:dg-category in nLab
1502:10.24033/asens.1689
1455:2017JGP...122...28C
618:{\displaystyle d=0}
33:homological algebra
1295:is said to have a
1273:
1214:derived categories
1200:Further properties
1152:
1107:
1004:
928:
881:
774:
739:
675:
651:
615:
589:
563:
500:
454:
364:
326:
290:
185:
156:
122:to another object
108:
65:
1382:(53): 3309–3339,
1018:is defined to be
864:
685:. By definition,
521:additive category
139:
16:(Redirected from
1559:
1521:
1504:
1483:Keller, Bernhard
1475:
1473:
1448:
1427:
1421:
1420:
1414:
1406:
1369:
1282:
1280:
1279:
1274:
1237:proper morphisms
1204:The category of
1161:
1159:
1158:
1153:
1151:
1150:
1138:
1137:
1116:
1114:
1113:
1108:
1106:
1105:
1093:
1092:
1083:
1082:
1055:
1054:
1042:
1041:
1013:
1011:
1010:
1005:
1000:
999:
981:
980:
968:
967:
937:
935:
934:
929:
924:
923:
905:
904:
892:
880:
879:
845:
844:
834:
833:
820:
783:
781:
780:
775:
748:
746:
745:
740:
720:
719:
709:
708:
684:
682:
681:
676:
674:
673:
660:
658:
657:
652:
647:
646:
624:
622:
621:
616:
598:
596:
595:
590:
572:
570:
569:
564:
547:
546:
541:
509:
507:
506:
501:
490:
489:
463:
461:
460:
455:
373:
371:
370:
365:
335:
333:
332:
327:
299:
297:
296:
291:
271:
270:
234:
233:
194:
192:
191:
186:
166:
165:
155:
154:
117:
115:
114:
109:
74:
72:
71:
66:
64:
21:
1567:
1566:
1562:
1561:
1560:
1558:
1557:
1556:
1537:
1536:
1528:
1481:
1478:
1430:
1428:
1424:
1407:
1371:
1370:
1366:
1362:
1349:Graded category
1335:
1265:
1264:
1257:
1223:over some ring
1202:
1142:
1129:
1124:
1123:
1097:
1084:
1068:
1046:
1027:
1022:
1021:
985:
972:
959:
945:
944:
909:
896:
809:
804:
803:
751:
750:
692:
687:
686:
663:
662:
630:
629:
601:
600:
575:
574:
530:
525:
524:
516:
481:
470:
469:
380:
379:
376:cochain complex
338:
337:
306:
305:
256:
225:
214:
213:
157:
134:
133:
82:
81:
55:
54:
23:
22:
15:
12:
11:
5:
1565:
1563:
1555:
1554:
1549:
1539:
1538:
1535:
1534:
1527:
1526:External links
1524:
1523:
1522:
1477:
1476:
1422:
1363:
1361:
1358:
1357:
1356:
1351:
1346:
1341:
1334:
1331:
1297:dg enhancement
1272:
1259:A DG category
1256:
1253:
1210:model category
1201:
1198:
1197:
1196:
1184:
1183:
1149:
1145:
1141:
1136:
1132:
1120:
1119:
1118:
1104:
1100:
1096:
1091:
1087:
1081:
1078:
1075:
1071:
1067:
1064:
1061:
1058:
1053:
1049:
1045:
1040:
1037:
1034:
1030:
1003:
998:
995:
992:
988:
984:
979:
975:
971:
966:
962:
958:
955:
952:
941:
940:
939:
927:
922:
919:
916:
912:
908:
903:
899:
895:
891:
888:
885:
878:
874:
871:
867:
863:
860:
857:
854:
851:
848:
843:
840:
837:
832:
827:
824:
819:
816:
813:
798:
797:
773:
770:
767:
764:
761:
758:
738:
735:
732:
729:
726:
723:
718:
715:
712:
707:
702:
699:
695:
672:
650:
645:
640:
637:
626:
614:
611:
608:
588:
585:
582:
562:
559:
556:
553:
550:
545:
540:
537:
534:
515:
512:
499:
496:
493:
488:
484:
480:
477:
453:
450:
447:
444:
441:
438:
435:
432:
429:
426:
423:
420:
417:
414:
411:
408:
405:
402:
399:
396:
393:
390:
387:
363:
360:
357:
354:
351:
348:
345:
325:
322:
319:
316:
313:
302:
301:
289:
286:
283:
280:
277:
274:
269:
266:
263:
259:
255:
252:
249:
246:
243:
240:
237:
232:
228:
224:
221:
196:
195:
184:
181:
178:
175:
172:
169:
164:
160:
153:
149:
146:
142:
107:
104:
101:
98:
95:
92:
89:
63:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1564:
1553:
1550:
1548:
1545:
1544:
1542:
1533:
1530:
1529:
1525:
1520:
1516:
1512:
1508:
1503:
1498:
1495:(1): 63–102,
1494:
1490:
1489:
1484:
1480:
1479:
1472:
1468:
1464:
1460:
1456:
1452:
1447:
1442:
1438:
1434:
1426:
1423:
1418:
1412:
1405:
1401:
1397:
1393:
1389:
1385:
1381:
1377:
1376:
1368:
1365:
1359:
1355:
1352:
1350:
1347:
1345:
1342:
1340:
1337:
1336:
1332:
1330:
1328:
1325:
1321:
1317:
1313:
1309:
1305:
1301:
1298:
1294:
1290:
1286:
1262:
1254:
1252:
1250:
1246:
1242:
1238:
1234:
1230:
1226:
1222:
1217:
1215:
1211:
1207:
1199:
1194:
1190:
1186:
1185:
1181:
1177:
1173:
1169:
1165:
1147:
1143:
1139:
1134:
1130:
1121:
1102:
1098:
1094:
1089:
1085:
1079:
1076:
1073:
1065:
1062:
1056:
1051:
1047:
1043:
1038:
1035:
1032:
1028:
1020:
1019:
1017:
996:
993:
990:
986:
977:
973:
969:
964:
960:
953:
950:
942:
920:
917:
914:
910:
906:
901:
897:
872:
869:
865:
861:
855:
852:
849:
841:
838:
822:
802:
801:
800:
799:
795:
791:
787:
768:
762:
756:
733:
730:
727:
721:
716:
713:
697:
693:
635:
627:
612:
609:
606:
586:
583:
580:
557:
554:
551:
543:
522:
518:
517:
513:
511:
497:
494:
486:
482:
475:
467:
448:
445:
442:
436:
433:
424:
421:
418:
412:
409:
406:
400:
397:
394:
388:
385:
377:
358:
355:
352:
346:
343:
323:
320:
317:
314:
311:
284:
281:
278:
272:
267:
264:
261:
257:
247:
244:
241:
235:
230:
226:
222:
219:
212:
211:
210:
209:
205:
201:
179:
176:
173:
167:
162:
158:
147:
144:
140:
132:
131:
130:
129:
125:
121:
102:
99:
96:
90:
87:
78:
76:
50:
46:
42:
38:
34:
31:, especially
30:
19:
1492:
1486:
1436:
1432:
1425:
1379:
1373:
1367:
1326:
1319:
1315:
1307:
1303:
1299:
1296:
1292:
1284:
1260:
1258:
1248:
1244:
1240:
1228:
1224:
1220:
1218:
1203:
1167:
1163:
1015:
793:
789:
785:
465:
303:
203:
199:
197:
123:
119:
79:
44:
40:
36:
26:
1491:, SĂ©rie 4,
573:vanish for
206:there is a
45:DG category
41:dg-category
29:mathematics
18:Dg category
1541:Categories
1446:1605.00490
1360:References
1014:of degree
208:linear map
128:direct sum
1511:0012-9593
1471:119326832
1439:: 28–52,
1404:119162782
1396:1073-7928
1354:Derivator
1271:Σ
1095:∘
1063:−
1044:∘
983:→
970::
873:∈
866:∏
784:which do
760:→
722:
584:≠
558:−
552:−
437:
431:→
413:
407:⊗
389:
347:
315:∘
273:
254:→
236:
223::
168:
148:∈
141:⨁
91:
1411:citation
1333:See also
1239:in case
514:Examples
49:category
1519:1258406
1451:Bibcode
1322:) of a
1287:) is a
1178:over a
796:, i.e.,
75:-module
47:, is a
1517:
1509:
1469:
1402:
1394:
1233:smooth
1176:scheme
1122:where
1467:S2CID
1441:arXiv
1400:S2CID
1247:over
1206:small
1189:field
1174:on a
374:is a
1507:ISSN
1429:See
1417:link
1392:ISSN
1380:2005
1235:and
1180:ring
1166:and
792:and
519:Any
35:, a
1497:doi
1459:doi
1437:122
1384:doi
1302:if
786:not
694:Hom
434:Hom
410:Hom
386:Hom
344:Hom
258:Hom
227:Hom
159:Hom
88:Hom
43:or
27:In
1543::
1515:MR
1513:,
1505:,
1493:27
1465:,
1457:,
1449:,
1435:,
1413:}}
1409:{{
1398:,
1390:,
1378:,
1251:.
1216:.
625:).
510:.
483:id
77:.
1499::
1461::
1453::
1443::
1419:)
1386::
1327:A
1320:A
1318:(
1316:D
1308:T
1304:C
1300:C
1293:T
1285:C
1261:C
1249:R
1245:X
1241:C
1229:C
1225:R
1221:C
1195:.
1182:.
1168:B
1164:A
1148:B
1144:d
1140:,
1135:A
1131:d
1117:,
1103:l
1099:f
1090:B
1086:d
1080:1
1077:+
1074:n
1070:)
1066:1
1060:(
1057:+
1052:A
1048:d
1039:1
1036:+
1033:l
1029:f
1016:n
1002:)
997:n
994:+
991:l
987:B
978:l
974:A
965:l
961:f
957:(
954:=
951:f
938:.
926:)
921:n
918:+
915:l
911:B
907:,
902:l
898:A
894:(
890:m
887:o
884:H
877:Z
870:l
862:=
859:)
856:B
853:,
850:A
847:(
842:n
839:,
836:)
831:A
826:(
823:C
818:m
815:o
812:H
794:B
790:A
772:]
769:n
766:[
763:B
757:A
737:)
734:B
731:,
728:A
725:(
717:n
714:,
711:)
706:A
701:(
698:C
671:A
649:)
644:A
639:(
636:C
613:0
610:=
607:d
587:0
581:n
561:)
555:,
549:(
544:n
539:m
536:o
533:H
498:0
495:=
492:)
487:A
479:(
476:d
466:A
452:)
449:C
446:,
443:A
440:(
428:)
425:C
422:,
419:B
416:(
404:)
401:B
398:,
395:A
392:(
362:)
359:B
356:,
353:A
350:(
324:0
321:=
318:d
312:d
300:,
288:)
285:B
282:,
279:A
276:(
268:1
265:+
262:n
251:)
248:B
245:,
242:A
239:(
231:n
220:d
204:n
200:d
183:)
180:B
177:,
174:A
171:(
163:n
152:Z
145:n
124:B
120:A
106:)
103:B
100:,
97:A
94:(
62:Z
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
