1064:
as a constant simplicial set; this way, the category of sheaves on the site is included as a subcategory to the homotopy category of simplicial presheaves on the site. The inclusion functor has a left adjoint and that is exactly
59:
from the site to the category of simplicial sets). Equivalently, a simplicial presheaf is a simplicial object in the category of presheaves on a site. The notion was introduced by A. Joyal in the 1970s. Similarly, a
218:
589:
735:
The category of such functors is endowed with (at least) three model structures, namely the projective, the Reedy, and the injective model structure. The weak equivalences / fibrations in the first are maps
300:
474:
933:
829:
1025:
771:
1307:
730:
123:
654:
1222:
1099:
619:
345:
381:
253:
1141:
1062:
161:
166:
479:
1437:
1400:
Axiomatic, enriched and motivic homotopy theory. Proceedings of the NATO Advanced Study
Institute, Cambridge, UK, 9--20 September 2002
939:
1407:
258:
398:
881:
782:
1402:. NATO Science Series II: Mathematics, Physics and Chemistry. Vol. 131. Dordrecht: Kluwer Academic. pp. 29–68.
952:
742:
1234:
1326:
677:
1470:
1465:
90:
137:
624:
1150:
129:, a simplicial object in the site, represents a simplicial presheaf (in fact, often a simplicial sheaf).
1068:
56:
1347:
1384:
851:
69:
1144:
842:. The injective model structure is similar, but with weak equivalences and cofibrations instead.
40:
594:
320:
1475:
1403:
354:
226:
65:
48:
1398:
Jardine, J.F. (2004). "Generalised sheaf cohomology theories". In
Greenlees, J. P. C. (ed.).
1413:
1355:
Introductory
Workshop on Algebraic Stacks, Intersection Theory, and Non-Abelian Hodge Theory
1111:
1417:
1038:
44:
28:
143:
1426:
943:
665:
52:
1459:
865:
76:
1312:
where the left denotes a sheaf cohomology and the right the homotopy class of maps.
1321:
17:
1143:
by doing classifying space construction levelwise (the notion comes from the
255:
is a local weak equivalence of simplicial presheaves, then the induced map
36:
671:
Some of them are obtained by viewing simplicial presheaves as functors
213:{\displaystyle B\operatorname {GL} =\varinjlim B\operatorname {GL_{n}} }
1450:
664:
The category of simplicial presheaves on a site admits many different
584:{\displaystyle (\pi _{i}^{\text{pr}}(F,s))(f)=\pi _{i}(F(Y),f^{*}(s))}
1108:
is a sheaf of abelian group (on the same site), then we define
834:
is a weak equivalence / fibration of simplicial sets, for all
807:
788:
758:
748:
295:{\displaystyle \mathbb {Z} f:\mathbb {Z} X\to \mathbb {Z} Y}
469:{\displaystyle (\pi _{0}^{\text{pr}}F)(X)=\pi _{0}(F(X))}
928:{\displaystyle F(X)\to \operatorname {holim} F(H_{n})}
824:{\displaystyle {\mathcal {F}}(U)\to {\mathcal {G}}(U)}
1237:
1153:
1114:
1071:
1041:
955:
884:
785:
745:
680:
627:
597:
482:
401:
357:
323:
261:
229:
169:
146:
93:
1438:
Simplicial presheaves and derived algebraic geometry
1035:
on the site can be considered as a stack by viewing
1301:
1216:
1135:
1093:
1056:
1019:
927:
823:
765:
724:
648:
613:
583:
468:
375:
339:
294:
247:
212:
155:
117:
1020:{\displaystyle =\{0,1,\dots ,n\}\mapsto F(H_{n})}
766:{\displaystyle {\mathcal {F}}\to {\mathcal {G}}}
140:section-wise, one obtains a simplicial presheaf
621:to be the sheaf associated with the pre-sheaf
220:. These types of examples appear in K-theory.
8:
1302:{\displaystyle \operatorname {H} ^{i}(X;A)=}
992:
968:
942:as simplicial sets, where the right is the
725:{\displaystyle S^{op}\to \Delta ^{op}Sets}
1385:Model structures on simplicial presheaves
1242:
1236:
1152:
1113:
1082:
1070:
1040:
1008:
954:
916:
883:
806:
805:
787:
786:
784:
757:
756:
747:
746:
744:
701:
685:
679:
637:
632:
626:
602:
596:
563:
535:
495:
490:
481:
442:
414:
409:
400:
356:
328:
322:
306:Homotopy sheaves of a simplicial presheaf
285:
284:
274:
273:
263:
262:
260:
228:
203:
195:
179:
168:
145:
118:{\displaystyle \operatorname {Hom} (-,U)}
92:
860:on a site is called a stack if, for any
314:be a simplicial presheaf on a site. The
136:be a presheaf of groupoids. Then taking
1367:
1338:
1224:. One can show (by induction): for any
649:{\displaystyle \pi _{i}^{\text{pr}}F}
27:In mathematics, more specifically in
7:
1217:{\displaystyle K(A,i)=K(K(A,i-1),1)}
87:in the site represents the presheaf
1348:"Stacks and Non-abelian cohomology"
1239:
1094:{\displaystyle F\mapsto \pi _{0}F}
698:
302:is also a local weak equivalence.
204:
200:
196:
25:
351:is defined as follows. For any
1296:
1293:
1281:
1269:
1263:
1251:
1211:
1202:
1184:
1178:
1169:
1157:
1130:
1118:
1075:
1051:
1045:
1014:
1001:
995:
962:
956:
922:
909:
897:
894:
888:
818:
812:
802:
799:
793:
753:
694:
578:
575:
569:
553:
547:
541:
525:
519:
516:
513:
501:
483:
463:
460:
454:
448:
432:
426:
423:
402:
367:
281:
239:
112:
100:
1:
163:. For example, one might set
383:in the site and a 0-simplex
1492:
849:
1327:N-group (category theory)
614:{\displaystyle \pi _{i}F}
340:{\displaystyle \pi _{*}F}
376:{\displaystyle f:X\to Y}
248:{\displaystyle f:X\to Y}
1451:J.F. Jardine's homepage
1427:"Simplicial presheaves"
1346:Toën, Bertrand (2002),
1425:Jardine, J.F. (2007).
1303:
1218:
1137:
1136:{\displaystyle K(A,1)}
1095:
1058:
1021:
929:
856:A simplicial presheaf
825:
767:
726:
650:
615:
585:
470:
377:
341:
296:
249:
214:
157:
119:
75:Example: Consider the
1304:
1219:
1138:
1096:
1059:
1022:
930:
826:
768:
727:
651:
616:
586:
471:
378:
342:
297:
250:
215:
158:
120:
57:contravariant functor
1235:
1151:
1112:
1069:
1057:{\displaystyle F(X)}
1039:
953:
882:
875:, the canonical map
783:
743:
678:
625:
595:
480:
399:
355:
321:
259:
227:
167:
144:
91:
852:Stack (mathematics)
642:
500:
419:
68:in the category of
51:) taking values in
33:simplicial presheaf
1299:
1214:
1145:obstruction theory
1133:
1091:
1054:
1017:
925:
821:
763:
722:
646:
628:
611:
581:
486:
466:
405:
373:
337:
292:
245:
210:
187:
156:{\displaystyle BG}
153:
115:
49:topological spaces
640:
498:
417:
180:
127:simplicial scheme
66:simplicial object
16:(Redirected from
1483:
1433:
1431:
1421:
1383:Konrad Voelkel,
1371:
1365:
1359:
1358:
1352:
1343:
1308:
1306:
1305:
1300:
1247:
1246:
1223:
1221:
1220:
1215:
1142:
1140:
1139:
1134:
1100:
1098:
1097:
1092:
1087:
1086:
1063:
1061:
1060:
1055:
1026:
1024:
1023:
1018:
1013:
1012:
940:weak equivalence
934:
932:
931:
926:
921:
920:
830:
828:
827:
822:
811:
810:
792:
791:
772:
770:
769:
764:
762:
761:
752:
751:
731:
729:
728:
723:
709:
708:
693:
692:
666:model structures
660:Model structures
655:
653:
652:
647:
641:
638:
636:
620:
618:
617:
612:
607:
606:
590:
588:
587:
582:
568:
567:
540:
539:
499:
496:
494:
475:
473:
472:
467:
447:
446:
418:
415:
413:
382:
380:
379:
374:
346:
344:
343:
338:
333:
332:
316:homotopy sheaves
301:
299:
298:
293:
288:
277:
266:
254:
252:
251:
246:
219:
217:
216:
211:
209:
208:
207:
188:
162:
160:
159:
154:
124:
122:
121:
116:
62:simplicial sheaf
21:
1491:
1490:
1486:
1485:
1484:
1482:
1481:
1480:
1471:Simplicial sets
1466:Homotopy theory
1456:
1455:
1447:
1442:
1429:
1424:
1410:
1397:
1393:
1380:
1378:Further reading
1375:
1374:
1366:
1362:
1350:
1345:
1344:
1340:
1335:
1318:
1238:
1233:
1232:
1149:
1148:
1110:
1109:
1078:
1067:
1066:
1037:
1036:
1004:
951:
950:
912:
880:
879:
854:
848:
781:
780:
741:
740:
697:
681:
676:
675:
662:
623:
622:
598:
593:
592:
559:
531:
478:
477:
438:
397:
396:
353:
352:
324:
319:
318:
308:
257:
256:
225:
224:
199:
165:
164:
142:
141:
89:
88:
64:on a site is a
53:simplicial sets
29:homotopy theory
23:
22:
15:
12:
11:
5:
1489:
1487:
1479:
1478:
1473:
1468:
1458:
1457:
1454:
1453:
1446:
1445:External links
1443:
1441:
1440:
1434:
1422:
1408:
1394:
1392:
1389:
1388:
1387:
1379:
1376:
1373:
1372:
1360:
1337:
1336:
1334:
1331:
1330:
1329:
1324:
1317:
1314:
1310:
1309:
1298:
1295:
1292:
1289:
1286:
1283:
1280:
1277:
1274:
1271:
1268:
1265:
1262:
1259:
1256:
1253:
1250:
1245:
1241:
1213:
1210:
1207:
1204:
1201:
1198:
1195:
1192:
1189:
1186:
1183:
1180:
1177:
1174:
1171:
1168:
1165:
1162:
1159:
1156:
1132:
1129:
1126:
1123:
1120:
1117:
1090:
1085:
1081:
1077:
1074:
1053:
1050:
1047:
1044:
1029:
1028:
1016:
1011:
1007:
1003:
1000:
997:
994:
991:
988:
985:
982:
979:
976:
973:
970:
967:
964:
961:
958:
944:homotopy limit
936:
935:
924:
919:
915:
911:
908:
905:
902:
899:
896:
893:
890:
887:
850:Main article:
847:
844:
832:
831:
820:
817:
814:
809:
804:
801:
798:
795:
790:
774:
773:
760:
755:
750:
733:
732:
721:
718:
715:
712:
707:
704:
700:
696:
691:
688:
684:
661:
658:
645:
635:
631:
610:
605:
601:
591:. We then set
580:
577:
574:
571:
566:
562:
558:
555:
552:
549:
546:
543:
538:
534:
530:
527:
524:
521:
518:
515:
512:
509:
506:
503:
493:
489:
485:
465:
462:
459:
456:
453:
450:
445:
441:
437:
434:
431:
428:
425:
422:
412:
408:
404:
372:
369:
366:
363:
360:
336:
331:
327:
307:
304:
291:
287:
283:
280:
276:
272:
269:
265:
244:
241:
238:
235:
232:
206:
202:
198:
194:
191:
186:
183:
178:
175:
172:
152:
149:
114:
111:
108:
105:
102:
99:
96:
24:
18:Homotopy sheaf
14:
13:
10:
9:
6:
4:
3:
2:
1488:
1477:
1474:
1472:
1469:
1467:
1464:
1463:
1461:
1452:
1449:
1448:
1444:
1439:
1435:
1428:
1423:
1419:
1415:
1411:
1409:1-4020-1833-9
1405:
1401:
1396:
1395:
1390:
1386:
1382:
1381:
1377:
1369:
1364:
1361:
1356:
1349:
1342:
1339:
1332:
1328:
1325:
1323:
1320:
1319:
1315:
1313:
1290:
1287:
1284:
1278:
1275:
1272:
1266:
1260:
1257:
1254:
1248:
1243:
1231:
1230:
1229:
1228:in the site,
1227:
1208:
1205:
1199:
1196:
1193:
1190:
1187:
1181:
1175:
1172:
1166:
1163:
1160:
1154:
1146:
1127:
1124:
1121:
1115:
1107:
1102:
1088:
1083:
1079:
1072:
1048:
1042:
1034:
1009:
1005:
998:
989:
986:
983:
980:
977:
974:
971:
965:
959:
949:
948:
947:
945:
941:
917:
913:
906:
903:
900:
891:
885:
878:
877:
876:
874:
870:
867:
866:hypercovering
863:
859:
853:
845:
843:
841:
837:
815:
796:
779:
778:
777:
739:
738:
737:
719:
716:
713:
710:
705:
702:
689:
686:
682:
674:
673:
672:
669:
667:
659:
657:
643:
633:
629:
608:
603:
599:
572:
564:
560:
556:
550:
544:
536:
532:
528:
522:
510:
507:
504:
491:
487:
457:
451:
443:
439:
435:
429:
420:
410:
406:
394:
390:
386:
370:
364:
361:
358:
350:
334:
329:
325:
317:
313:
305:
303:
289:
278:
270:
267:
242:
236:
233:
230:
221:
192:
189:
184:
181:
176:
173:
170:
150:
147:
139:
135:
132:Example: Let
130:
128:
109:
106:
103:
97:
94:
86:
82:
78:
73:
72:on the site.
71:
67:
63:
58:
54:
50:
46:
42:
38:
34:
30:
19:
1399:
1368:Jardine 2007
1363:
1354:
1341:
1311:
1225:
1105:
1103:
1032:
1030:
937:
872:
868:
861:
857:
855:
839:
838:in the site
835:
833:
775:
734:
670:
663:
392:
388:
384:
348:
315:
311:
309:
222:
133:
131:
126:
84:
80:
79:of a scheme
74:
61:
32:
26:
1322:cubical set
1147:) and set
776:such that
43:(e.g., the
1460:Categories
1418:1063.55004
1391:References
1031:Any sheaf
125:. Thus, a
77:étale site
1436:B. Toën,
1249:
1197:−
1080:π
1076:↦
996:↦
984:…
904:
898:→
803:→
754:→
699:Δ
695:→
630:π
600:π
565:∗
533:π
488:π
440:π
407:π
368:→
330:∗
326:π
282:→
240:→
190:
185:→
104:−
98:
55:(i.e., a
1476:Functors
1316:See also
864:and any
45:category
37:presheaf
395:), set
83:. Each
70:sheaves
1416:
1406:
1357:, MSRI
138:nerves
1430:(PDF)
1351:(PDF)
1333:Notes
938:is a
901:holim
846:Stack
39:on a
35:is a
1404:ISBN
1370:, §1
476:and
310:Let
41:site
31:, a
1414:Zbl
1104:If
946:of
387:in
347:of
223:If
182:lim
95:Hom
47:of
1462::
1412:.
1353:,
1101:.
668:.
656:.
639:pr
497:pr
416:pr
174:GL
1432:.
1420:.
1297:]
1294:)
1291:i
1288:,
1285:A
1282:(
1279:K
1276:,
1273:X
1270:[
1267:=
1264:)
1261:A
1258:;
1255:X
1252:(
1244:i
1240:H
1226:X
1212:)
1209:1
1206:,
1203:)
1200:1
1194:i
1191:,
1188:A
1185:(
1182:K
1179:(
1176:K
1173:=
1170:)
1167:i
1164:,
1161:A
1158:(
1155:K
1131:)
1128:1
1125:,
1122:A
1119:(
1116:K
1106:A
1089:F
1084:0
1073:F
1052:)
1049:X
1046:(
1043:F
1033:F
1027:.
1015:)
1010:n
1006:H
1002:(
999:F
993:}
990:n
987:,
981:,
978:1
975:,
972:0
969:{
966:=
963:]
960:n
957:[
923:)
918:n
914:H
910:(
907:F
895:)
892:X
889:(
886:F
873:X
871:→
869:H
862:X
858:F
840:S
836:U
819:)
816:U
813:(
808:G
800:)
797:U
794:(
789:F
759:G
749:F
720:s
717:t
714:e
711:S
706:p
703:o
690:p
687:o
683:S
644:F
634:i
609:F
604:i
579:)
576:)
573:s
570:(
561:f
557:,
554:)
551:Y
548:(
545:F
542:(
537:i
529:=
526:)
523:f
520:(
517:)
514:)
511:s
508:,
505:F
502:(
492:i
484:(
464:)
461:)
458:X
455:(
452:F
449:(
444:0
436:=
433:)
430:X
427:(
424:)
421:F
411:0
403:(
393:X
391:(
389:F
385:s
371:Y
365:X
362::
359:f
349:F
335:F
312:F
290:Y
286:Z
279:X
275:Z
271::
268:f
264:Z
243:Y
237:X
234::
231:f
205:n
201:L
197:G
193:B
177:=
171:B
151:G
148:B
134:G
113:)
110:U
107:,
101:(
85:U
81:S
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.