338:
218:
590:
783:
1326:
484:
402:
974:
682:
1433:
1021:
631:
1260:
1108:
1063:
1503:
1199:
1173:
853:
827:
86:
1400:
229:
97:
491:
687:
1293:
1540:
410:
1117:
However, unlike in the previous example with Lie groups, a groupoid object in the category of manifolds is not necessarily a
860:
1438:
1509:
1502:
Behrend, Kai; Conrad, Brian; Edidin, Dan; Fulton, William; Fantechi, Barbara; Göttsche, Lothar; Kresch, Andrew (2006),
1577:
56:
1442:
1110:. When the term "groupoid" can naturally refer to a groupoid object in some particular category in mind, the term
343:
1587:
1122:
910:
60:
1403:
636:
1406:
1287:
1582:
49:
1344:
887:
in the usual sense: a category in which every morphism is an isomorphism. Indeed, given such a category
868:
1142:
979:
597:
1367:
1356:
1223:
1233:
1068:
1026:
1527:
1454:
856:
1549:
880:
1561:
1557:
1217:
1210:
17:
1178:
1152:
832:
806:
65:
1121:, since the maps s and t fail to satisfy further requirements (they are not necessarily
1360:
1373:
1571:
1553:
333:{\displaystyle s\circ e=t\circ e=1_{U},\,s\circ m=s\circ p_{1},t\circ m=t\circ p_{2}}
1523:
1202:
1118:
800:
33:
1333:
21:
864:
213:{\displaystyle s,t:R\to U,\ e:U\to R,\ m:R\times _{U,t,s}R\to R,\ i:R\to R}
1351:
to the category of groupoids. This way, each groupoid object determines a
1352:
884:
89:
32:
which is built on richer structures than sets, and a generalization of a
29:
585:{\displaystyle m\circ (e\circ s,1_{R})=m\circ (1_{R},e\circ t)=1_{R},}
778:{\displaystyle m\circ (1_{R},i)=e\circ s,\,m\circ (i,1_{R})=e\circ t}
1321:{\displaystyle R{\overset {s}{\underset {t}{\rightrightarrows }}}U}
1114:
is used to refer to a groupoid object in the category of sets.
1336:
of the same diagram, if any, is the quotient of the groupoid.
479:{\displaystyle m\circ (1_{R}\times m)=m\circ (m\times 1_{R}),}
1409:
1376:
1296:
1270:
the given action. This determines a groupoid scheme.
1236:
1181:
1155:
1071:
1029:
982:
913:
835:
809:
690:
639:
600:
494:
413:
346:
232:
100:
68:
1534:, Proceedings of the Luminy conference on algebraic
1201:
are necessarily the structure map) is the same as a
36:when the multiplication is only partially defined.
1427:
1394:
1366:The main use of the notion is that it provides an
1320:
1254:
1193:
1167:
1102:
1057:
1015:
968:
847:
821:
777:
676:
625:
584:
478:
396:
332:
212:
80:
1209:, to convey the idea it is a generalization of
803:is a special case of a groupoid object, where
1528:"Intersection theory on algebraic stacks and
8:
1473:
1445:. Conversely, any DM stack is of this form.
397:{\displaystyle p_{i}:R\times _{U,t,s}R\to R}
1408:
1375:
1300:
1295:
1235:
1180:
1154:
1091:
1070:
1049:
1028:
981:
969:{\displaystyle s(x\to y)=x,\,t(x\to y)=y}
941:
912:
834:
808:
754:
734:
704:
689:
658:
638:
617:
599:
573:
545:
520:
493:
464:
427:
412:
367:
351:
345:
324:
293:
270:
261:
231:
223:satisfying the following groupoid axioms
162:
99:
67:
1141:is a groupoid object in the category of
1466:
1328:, if any, is a group object called the
677:{\displaystyle s\circ i=t,\,t\circ i=s}
1485:
1428:{\displaystyle (R\rightrightarrows U)}
1205:. A groupoid scheme is also called an
1355:in groupoids. This prestack is not a
7:
1370:for a stack. More specifically, let
1541:Journal of Pure and Applied Algebra
1339:Each groupoid object in a category
14:
1343:(if any) may be thought of as a
1175:, then a groupoid scheme (where
895:to be the set of all objects in
1441:; in fact, (in a nice case), a
1016:{\displaystyle m(f,g)=g\circ f}
1422:
1416:
1410:
1389:
1383:
1377:
1303:
1081:
1075:
1039:
1033:
998:
986:
957:
951:
945:
929:
923:
917:
907:, the five morphisms given by
861:category of topological spaces
760:
741:
716:
697:
626:{\displaystyle i\circ i=1_{R}}
563:
538:
526:
501:
470:
451:
439:
420:
388:
204:
183:
137:
116:
28:is both a generalization of a
1:
1439:category fibered in groupoids
1554:10.1016/0022-4049(84)90036-7
1145:over some fixed base scheme
1255:{\displaystyle R=U\times G}
1226:from the right on a scheme
1103:{\displaystyle i(f)=f^{-1}}
1604:
1058:{\displaystyle e(x)=1_{x}}
1278:Given a groupoid object (
903:the set of all arrows in
879:A groupoid object in the
855:. One recovers therefore
1538:-theory (Luminy, 1983),
1216:For example, suppose an
404:are the two projections,
1429:
1396:
1322:
1256:
1195:
1169:
1104:
1059:
1017:
970:
849:
823:
779:
678:
627:
586:
480:
398:
334:
214:
82:
59:consists of a pair of
1443:Deligne–Mumford stack
1430:
1397:
1345:contravariant functor
1332:of the groupoid. The
1323:
1257:
1196:
1170:
1105:
1060:
1018:
971:
869:category of manifolds
850:
824:
780:
679:
628:
587:
481:
399:
335:
215:
83:
1407:
1374:
1294:
1234:
1179:
1153:
1069:
1027:
980:
911:
833:
807:
688:
637:
598:
492:
411:
344:
230:
98:
66:
1402:be the category of
1213:and their actions.
1194:{\displaystyle s=t}
1168:{\displaystyle U=S}
848:{\displaystyle s=t}
822:{\displaystyle R=U}
88:together with five
81:{\displaystyle R,U}
1578:Algebraic geometry
1425:
1392:
1363:to yield a stack.
1318:
1309:
1252:
1207:algebraic groupoid
1191:
1165:
1100:
1055:
1013:
966:
857:topological groups
845:
819:
775:
674:
623:
582:
476:
394:
330:
210:
78:
1455:Simplicial scheme
1313:
1302:
194:
148:
127:
55:admitting finite
1595:
1564:
1548:(2–3): 193–240,
1537:
1531:
1519:
1518:
1517:
1508:, archived from
1505:Algebraic stacks
1489:
1483:
1477:
1474:Algebraic stacks
1471:
1434:
1432:
1431:
1426:
1401:
1399:
1398:
1395:{\displaystyle }
1393:
1327:
1325:
1324:
1319:
1314:
1301:
1266:the projection,
1261:
1259:
1258:
1253:
1211:algebraic groups
1200:
1198:
1197:
1192:
1174:
1172:
1171:
1166:
1129:Groupoid schemes
1109:
1107:
1106:
1101:
1099:
1098:
1064:
1062:
1061:
1056:
1054:
1053:
1022:
1020:
1019:
1014:
975:
973:
972:
967:
881:category of sets
854:
852:
851:
846:
828:
826:
825:
820:
784:
782:
781:
776:
759:
758:
709:
708:
683:
681:
680:
675:
632:
630:
629:
624:
622:
621:
591:
589:
588:
583:
578:
577:
550:
549:
525:
524:
485:
483:
482:
477:
469:
468:
432:
431:
407:(associativity)
403:
401:
400:
395:
384:
383:
356:
355:
339:
337:
336:
331:
329:
328:
298:
297:
266:
265:
219:
217:
216:
211:
192:
179:
178:
146:
125:
87:
85:
84:
79:
1603:
1602:
1598:
1597:
1596:
1594:
1593:
1592:
1588:Category theory
1568:
1567:
1535:
1529:
1522:
1515:
1513:
1501:
1498:
1493:
1492:
1484:
1480:
1472:
1468:
1463:
1451:
1437:. Then it is a
1405:
1404:
1372:
1371:
1292:
1291:
1276:
1232:
1231:
1218:algebraic group
1177:
1176:
1151:
1150:
1131:
1087:
1067:
1066:
1045:
1025:
1024:
978:
977:
909:
908:
883:is precisely a
877:
831:
830:
805:
804:
797:
792:
750:
700:
686:
685:
635:
634:
613:
596:
595:
569:
541:
516:
490:
489:
460:
423:
409:
408:
363:
347:
342:
341:
320:
289:
257:
228:
227:
158:
96:
95:
64:
63:
46:groupoid object
42:
26:groupoid object
18:category theory
12:
11:
5:
1601:
1599:
1591:
1590:
1585:
1580:
1570:
1569:
1566:
1565:
1520:
1497:
1494:
1491:
1490:
1478:
1465:
1464:
1462:
1459:
1458:
1457:
1450:
1447:
1424:
1421:
1418:
1415:
1412:
1391:
1388:
1385:
1382:
1379:
1359:but it can be
1317:
1312:
1308:
1305:
1299:
1275:
1272:
1251:
1248:
1245:
1242:
1239:
1190:
1187:
1184:
1164:
1161:
1158:
1130:
1127:
1097:
1094:
1090:
1086:
1083:
1080:
1077:
1074:
1052:
1048:
1044:
1041:
1038:
1035:
1032:
1012:
1009:
1006:
1003:
1000:
997:
994:
991:
988:
985:
965:
962:
959:
956:
953:
950:
947:
944:
940:
937:
934:
931:
928:
925:
922:
919:
916:
876:
873:
867:by taking the
859:by taking the
844:
841:
838:
818:
815:
812:
796:
793:
791:
788:
787:
786:
774:
771:
768:
765:
762:
757:
753:
749:
746:
743:
740:
737:
733:
730:
727:
724:
721:
718:
715:
712:
707:
703:
699:
696:
693:
673:
670:
667:
664:
661:
657:
654:
651:
648:
645:
642:
620:
616:
612:
609:
606:
603:
592:
581:
576:
572:
568:
565:
562:
559:
556:
553:
548:
544:
540:
537:
534:
531:
528:
523:
519:
515:
512:
509:
506:
503:
500:
497:
486:
475:
472:
467:
463:
459:
456:
453:
450:
447:
444:
441:
438:
435:
430:
426:
422:
419:
416:
405:
393:
390:
387:
382:
379:
376:
373:
370:
366:
362:
359:
354:
350:
327:
323:
319:
316:
313:
310:
307:
304:
301:
296:
292:
288:
285:
282:
279:
276:
273:
269:
264:
260:
256:
253:
250:
247:
244:
241:
238:
235:
221:
220:
209:
206:
203:
200:
197:
191:
188:
185:
182:
177:
174:
171:
168:
165:
161:
157:
154:
151:
145:
142:
139:
136:
133:
130:
124:
121:
118:
115:
112:
109:
106:
103:
77:
74:
71:
57:fiber products
41:
38:
20:, a branch of
13:
10:
9:
6:
4:
3:
2:
1600:
1589:
1586:
1584:
1583:Scheme theory
1581:
1579:
1576:
1575:
1573:
1563:
1559:
1555:
1551:
1547:
1543:
1542:
1533:
1525:
1524:Gillet, Henri
1521:
1512:on 2008-05-05
1511:
1507:
1506:
1500:
1499:
1495:
1487:
1482:
1479:
1475:
1470:
1467:
1460:
1456:
1453:
1452:
1448:
1446:
1444:
1440:
1436:
1419:
1413:
1386:
1380:
1369:
1364:
1362:
1358:
1354:
1350:
1346:
1342:
1337:
1335:
1331:
1330:inertia group
1315:
1310:
1306:
1297:
1289:
1285:
1281:
1274:Constructions
1273:
1271:
1269:
1265:
1249:
1246:
1243:
1240:
1237:
1229:
1225:
1222:
1219:
1214:
1212:
1208:
1204:
1188:
1185:
1182:
1162:
1159:
1156:
1148:
1144:
1140:
1138:
1128:
1126:
1124:
1120:
1115:
1113:
1095:
1092:
1088:
1084:
1078:
1072:
1050:
1046:
1042:
1036:
1030:
1010:
1007:
1004:
1001:
995:
992:
989:
983:
963:
960:
954:
948:
942:
938:
935:
932:
926:
920:
914:
906:
902:
898:
894:
890:
886:
882:
874:
872:
870:
866:
862:
858:
842:
839:
836:
816:
813:
810:
802:
795:Group objects
794:
789:
772:
769:
766:
763:
755:
751:
747:
744:
738:
735:
731:
728:
725:
722:
719:
713:
710:
705:
701:
694:
691:
671:
668:
665:
662:
659:
655:
652:
649:
646:
643:
640:
618:
614:
610:
607:
604:
601:
593:
579:
574:
570:
566:
560:
557:
554:
551:
546:
542:
535:
532:
529:
521:
517:
513:
510:
507:
504:
498:
495:
487:
473:
465:
461:
457:
454:
448:
445:
442:
436:
433:
428:
424:
417:
414:
406:
391:
385:
380:
377:
374:
371:
368:
364:
360:
357:
352:
348:
325:
321:
317:
314:
311:
308:
305:
302:
299:
294:
290:
286:
283:
280:
277:
274:
271:
267:
262:
258:
254:
251:
248:
245:
242:
239:
236:
233:
226:
225:
224:
207:
201:
198:
195:
189:
186:
180:
175:
172:
169:
166:
163:
159:
155:
152:
149:
143:
140:
134:
131:
128:
122:
119:
113:
110:
107:
104:
101:
94:
93:
92:
91:
75:
72:
69:
62:
58:
54:
51:
47:
39:
37:
35:
34:group objects
31:
27:
23:
19:
1545:
1539:
1514:, retrieved
1510:the original
1504:
1481:
1476:, Ch 3. § 1.
1469:
1365:
1348:
1340:
1338:
1329:
1283:
1279:
1277:
1267:
1263:
1230:. Then take
1227:
1220:
1215:
1206:
1203:group scheme
1146:
1136:
1134:
1132:
1119:Lie groupoid
1116:
1112:groupoid set
1111:
904:
900:
896:
892:
888:
878:
801:group object
798:
222:
52:
45:
43:
25:
15:
1532:-varieties"
1486:Gillet 1984
1334:coequalizer
1123:submersions
22:mathematics
1572:Categories
1516:2014-02-11
1496:References
1361:stackified
865:Lie groups
594:(inverse)
340:where the
40:Definition
1417:⇉
1384:⇉
1304:⇉
1288:equalizer
1247:×
1135:groupoid
1093:−
1008:∘
952:→
924:→
875:Groupoids
770:∘
739:∘
726:∘
695:∘
663:∘
644:∘
605:∘
558:∘
536:∘
508:∘
499:∘
458:×
449:∘
434:×
418:∘
389:→
365:×
318:∘
306:∘
287:∘
275:∘
249:∘
237:∘
205:→
184:→
160:×
138:→
117:→
90:morphisms
1526:(1984),
1449:See also
1435:-torsors
1353:prestack
885:groupoid
790:Examples
50:category
30:groupoid
1562:0772058
1286:), the
1143:schemes
1139:-scheme
891:, take
871:, etc.
488:(unit)
61:objects
1560:
193:
147:
126:
1461:Notes
1368:atlas
1357:stack
1347:from
1149:. If
863:, or
48:in a
1224:acts
1065:and
829:and
24:, a
1550:doi
1290:of
1125:).
16:In
1574::
1558:MR
1556:,
1546:34
1544:,
1282:,
1262:,
1133:A
1023:,
976:,
899:,
799:A
684:,
633:,
44:A
1552::
1536:K
1530:Q
1488:.
1423:)
1420:U
1414:R
1411:(
1390:]
1387:U
1381:R
1378:[
1349:C
1341:C
1316:U
1311:s
1307:t
1298:R
1284:U
1280:R
1268:t
1264:s
1250:G
1244:U
1241:=
1238:R
1228:U
1221:G
1189:t
1186:=
1183:s
1163:S
1160:=
1157:U
1147:S
1137:S
1096:1
1089:f
1085:=
1082:)
1079:f
1076:(
1073:i
1051:x
1047:1
1043:=
1040:)
1037:x
1034:(
1031:e
1011:f
1005:g
1002:=
999:)
996:g
993:,
990:f
987:(
984:m
964:y
961:=
958:)
955:y
949:x
946:(
943:t
939:,
936:x
933:=
930:)
927:y
921:x
918:(
915:s
905:C
901:R
897:C
893:U
889:C
843:t
840:=
837:s
817:U
814:=
811:R
785:.
773:t
767:e
764:=
761:)
756:R
752:1
748:,
745:i
742:(
736:m
732:,
729:s
723:e
720:=
717:)
714:i
711:,
706:R
702:1
698:(
692:m
672:s
669:=
666:i
660:t
656:,
653:t
650:=
647:i
641:s
619:R
615:1
611:=
608:i
602:i
580:,
575:R
571:1
567:=
564:)
561:t
555:e
552:,
547:R
543:1
539:(
533:m
530:=
527:)
522:R
518:1
514:,
511:s
505:e
502:(
496:m
474:,
471:)
466:R
462:1
455:m
452:(
446:m
443:=
440:)
437:m
429:R
425:1
421:(
415:m
392:R
386:R
381:s
378:,
375:t
372:,
369:U
361:R
358::
353:i
349:p
326:2
322:p
315:t
312:=
309:m
303:t
300:,
295:1
291:p
284:s
281:=
278:m
272:s
268:,
263:U
259:1
255:=
252:e
246:t
243:=
240:e
234:s
208:R
202:R
199::
196:i
190:,
187:R
181:R
176:s
173:,
170:t
167:,
164:U
156:R
153::
150:m
144:,
141:R
135:U
132::
129:e
123:,
120:U
114:R
111::
108:t
105:,
102:s
76:U
73:,
70:R
53:C
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