349:
229:
601:
794:
1337:
495:
413:
985:
693:
1444:
1032:
642:
1271:
1119:
1074:
1514:
1210:
1184:
864:
838:
97:
1411:
240:
108:
502:
698:
1304:
1551:
421:
1128:
However, unlike in the previous example with Lie groups, a groupoid object in the category of manifolds is not necessarily a
871:
1449:
1520:
1513:
Behrend, Kai; Conrad, Brian; Edidin, Dan; Fulton, William; Fantechi, Barbara; Göttsche, Lothar; Kresch, Andrew (2006),
1588:
67:
1453:
1121:. When the term "groupoid" can naturally refer to a groupoid object in some particular category in mind, the term
354:
1598:
1133:
921:
71:
1414:
647:
1417:
1298:
1593:
60:
1355:
898:
in the usual sense: a category in which every morphism is an isomorphism. Indeed, given such a category
879:
1153:
990:
608:
1378:
1367:
1234:
1244:
1079:
1037:
1538:
1465:
867:
1560:
891:
1572:
1568:
1228:
1221:
28:
1189:
1163:
843:
817:
76:
1132:, since the maps s and t fail to satisfy further requirements (they are not necessarily
1371:
1384:
1582:
1564:
344:{\displaystyle s\circ e=t\circ e=1_{U},\,s\circ m=s\circ p_{1},t\circ m=t\circ p_{2}}
17:
1534:
1213:
1129:
811:
44:
1344:
32:
875:
224:{\displaystyle s,t:R\to U,\ e:U\to R,\ m:R\times _{U,t,s}R\to R,\ i:R\to R}
1362:
to the category of groupoids. This way, each groupoid object determines a
1363:
895:
100:
43:
which is built on richer structures than sets, and a generalization of a
40:
596:{\displaystyle m\circ (e\circ s,1_{R})=m\circ (1_{R},e\circ t)=1_{R},}
789:{\displaystyle m\circ (1_{R},i)=e\circ s,\,m\circ (i,1_{R})=e\circ t}
1332:{\displaystyle R{\overset {s}{\underset {t}{\rightrightarrows }}}U}
1125:
is used to refer to a groupoid object in the category of sets.
1347:
of the same diagram, if any, is the quotient of the groupoid.
490:{\displaystyle m\circ (1_{R}\times m)=m\circ (m\times 1_{R}),}
1420:
1387:
1307:
1281:
the given action. This determines a groupoid scheme.
1247:
1192:
1166:
1082:
1040:
993:
924:
846:
820:
701:
650:
611:
505:
424:
357:
243:
111:
79:
1545:, Proceedings of the Luminy conference on algebraic
1212:
are necessarily the structure map) is the same as a
47:when the multiplication is only partially defined.
1438:
1405:
1377:The main use of the notion is that it provides an
1331:
1265:
1204:
1178:
1113:
1068:
1026:
979:
858:
832:
788:
687:
636:
595:
489:
407:
343:
223:
91:
1220:, to convey the idea it is a generalization of
814:is a special case of a groupoid object, where
1539:"Intersection theory on algebraic stacks and
8:
1484:
1456:. Conversely, any DM stack is of this form.
408:{\displaystyle p_{i}:R\times _{U,t,s}R\to R}
1419:
1386:
1311:
1306:
1246:
1191:
1165:
1102:
1081:
1060:
1039:
992:
980:{\displaystyle s(x\to y)=x,\,t(x\to y)=y}
952:
923:
845:
819:
765:
745:
715:
700:
669:
649:
628:
610:
584:
556:
531:
504:
475:
438:
423:
378:
362:
356:
335:
304:
281:
272:
242:
234:satisfying the following groupoid axioms
173:
110:
78:
1152:is a groupoid object in the category of
1477:
1339:, if any, is a group object called the
688:{\displaystyle s\circ i=t,\,t\circ i=s}
1496:
1439:{\displaystyle (R\rightrightarrows U)}
1216:. A groupoid scheme is also called an
1366:in groupoids. This prestack is not a
7:
1381:for a stack. More specifically, let
1552:Journal of Pure and Applied Algebra
1350:Each groupoid object in a category
25:
1354:(if any) may be thought of as a
1186:, then a groupoid scheme (where
906:to be the set of all objects in
1452:; in fact, (in a nice case), a
1027:{\displaystyle m(f,g)=g\circ f}
1433:
1427:
1421:
1400:
1394:
1388:
1314:
1092:
1086:
1050:
1044:
1009:
997:
968:
962:
956:
940:
934:
928:
918:, the five morphisms given by
872:category of topological spaces
771:
752:
727:
708:
637:{\displaystyle i\circ i=1_{R}}
574:
549:
537:
512:
481:
462:
450:
431:
399:
215:
194:
148:
127:
39:is both a generalization of a
1:
1450:category fibered in groupoids
1565:10.1016/0022-4049(84)90036-7
1156:over some fixed base scheme
1266:{\displaystyle R=U\times G}
1237:from the right on a scheme
1114:{\displaystyle i(f)=f^{-1}}
1615:
1069:{\displaystyle e(x)=1_{x}}
1289:Given a groupoid object (
914:the set of all arrows in
890:A groupoid object in the
866:. One recovers therefore
1549:-theory (Luminy, 1983),
1227:For example, suppose an
415:are the two projections,
1440:
1407:
1333:
1267:
1206:
1180:
1115:
1070:
1028:
981:
860:
834:
790:
689:
638:
597:
491:
409:
345:
225:
93:
70:consists of a pair of
1454:Deligne–Mumford stack
1441:
1408:
1356:contravariant functor
1343:of the groupoid. The
1334:
1268:
1207:
1181:
1116:
1071:
1029:
982:
880:category of manifolds
861:
835:
791:
690:
639:
598:
492:
410:
346:
226:
94:
1418:
1385:
1305:
1245:
1190:
1164:
1080:
1038:
991:
922:
844:
818:
699:
648:
609:
503:
422:
355:
241:
109:
77:
18:Topological groupoid
1413:be the category of
1224:and their actions.
1205:{\displaystyle s=t}
1179:{\displaystyle U=S}
859:{\displaystyle s=t}
833:{\displaystyle R=U}
99:together with five
92:{\displaystyle R,U}
1589:Algebraic geometry
1436:
1403:
1374:to yield a stack.
1329:
1320:
1263:
1218:algebraic groupoid
1202:
1176:
1111:
1066:
1024:
977:
868:topological groups
856:
830:
786:
685:
634:
593:
487:
405:
341:
221:
89:
1466:Simplicial scheme
1324:
1313:
205:
159:
138:
66:admitting finite
16:(Redirected from
1606:
1575:
1559:(2–3): 193–240,
1548:
1542:
1530:
1529:
1528:
1519:, archived from
1516:Algebraic stacks
1500:
1494:
1488:
1485:Algebraic stacks
1482:
1445:
1443:
1442:
1437:
1412:
1410:
1409:
1406:{\displaystyle }
1404:
1338:
1336:
1335:
1330:
1325:
1312:
1277:the projection,
1272:
1270:
1269:
1264:
1222:algebraic groups
1211:
1209:
1208:
1203:
1185:
1183:
1182:
1177:
1140:Groupoid schemes
1120:
1118:
1117:
1112:
1110:
1109:
1075:
1073:
1072:
1067:
1065:
1064:
1033:
1031:
1030:
1025:
986:
984:
983:
978:
892:category of sets
865:
863:
862:
857:
839:
837:
836:
831:
795:
793:
792:
787:
770:
769:
720:
719:
694:
692:
691:
686:
643:
641:
640:
635:
633:
632:
602:
600:
599:
594:
589:
588:
561:
560:
536:
535:
496:
494:
493:
488:
480:
479:
443:
442:
418:(associativity)
414:
412:
411:
406:
395:
394:
367:
366:
350:
348:
347:
342:
340:
339:
309:
308:
277:
276:
230:
228:
227:
222:
203:
190:
189:
157:
136:
98:
96:
95:
90:
21:
1614:
1613:
1609:
1608:
1607:
1605:
1604:
1603:
1599:Category theory
1579:
1578:
1546:
1540:
1533:
1526:
1524:
1512:
1509:
1504:
1503:
1495:
1491:
1483:
1479:
1474:
1462:
1448:. Then it is a
1416:
1415:
1383:
1382:
1303:
1302:
1287:
1243:
1242:
1229:algebraic group
1188:
1187:
1162:
1161:
1142:
1098:
1078:
1077:
1056:
1036:
1035:
989:
988:
920:
919:
894:is precisely a
888:
842:
841:
816:
815:
808:
803:
761:
711:
697:
696:
646:
645:
624:
607:
606:
580:
552:
527:
501:
500:
471:
434:
420:
419:
374:
358:
353:
352:
331:
300:
268:
239:
238:
169:
107:
106:
75:
74:
57:groupoid object
53:
37:groupoid object
29:category theory
23:
22:
15:
12:
11:
5:
1612:
1610:
1602:
1601:
1596:
1591:
1581:
1580:
1577:
1576:
1531:
1508:
1505:
1502:
1501:
1489:
1476:
1475:
1473:
1470:
1469:
1468:
1461:
1458:
1435:
1432:
1429:
1426:
1423:
1402:
1399:
1396:
1393:
1390:
1370:but it can be
1328:
1323:
1319:
1316:
1310:
1286:
1283:
1262:
1259:
1256:
1253:
1250:
1201:
1198:
1195:
1175:
1172:
1169:
1141:
1138:
1108:
1105:
1101:
1097:
1094:
1091:
1088:
1085:
1063:
1059:
1055:
1052:
1049:
1046:
1043:
1023:
1020:
1017:
1014:
1011:
1008:
1005:
1002:
999:
996:
976:
973:
970:
967:
964:
961:
958:
955:
951:
948:
945:
942:
939:
936:
933:
930:
927:
887:
884:
878:by taking the
870:by taking the
855:
852:
849:
829:
826:
823:
807:
804:
802:
799:
798:
797:
785:
782:
779:
776:
773:
768:
764:
760:
757:
754:
751:
748:
744:
741:
738:
735:
732:
729:
726:
723:
718:
714:
710:
707:
704:
684:
681:
678:
675:
672:
668:
665:
662:
659:
656:
653:
631:
627:
623:
620:
617:
614:
603:
592:
587:
583:
579:
576:
573:
570:
567:
564:
559:
555:
551:
548:
545:
542:
539:
534:
530:
526:
523:
520:
517:
514:
511:
508:
497:
486:
483:
478:
474:
470:
467:
464:
461:
458:
455:
452:
449:
446:
441:
437:
433:
430:
427:
416:
404:
401:
398:
393:
390:
387:
384:
381:
377:
373:
370:
365:
361:
338:
334:
330:
327:
324:
321:
318:
315:
312:
307:
303:
299:
296:
293:
290:
287:
284:
280:
275:
271:
267:
264:
261:
258:
255:
252:
249:
246:
232:
231:
220:
217:
214:
211:
208:
202:
199:
196:
193:
188:
185:
182:
179:
176:
172:
168:
165:
162:
156:
153:
150:
147:
144:
141:
135:
132:
129:
126:
123:
120:
117:
114:
88:
85:
82:
68:fiber products
52:
49:
31:, a branch of
24:
14:
13:
10:
9:
6:
4:
3:
2:
1611:
1600:
1597:
1595:
1594:Scheme theory
1592:
1590:
1587:
1586:
1584:
1574:
1570:
1566:
1562:
1558:
1554:
1553:
1544:
1536:
1535:Gillet, Henri
1532:
1523:on 2008-05-05
1522:
1518:
1517:
1511:
1510:
1506:
1498:
1493:
1490:
1486:
1481:
1478:
1471:
1467:
1464:
1463:
1459:
1457:
1455:
1451:
1447:
1430:
1424:
1397:
1391:
1380:
1375:
1373:
1369:
1365:
1361:
1357:
1353:
1348:
1346:
1342:
1341:inertia group
1326:
1321:
1317:
1308:
1300:
1296:
1292:
1285:Constructions
1284:
1282:
1280:
1276:
1260:
1257:
1254:
1251:
1248:
1240:
1236:
1233:
1230:
1225:
1223:
1219:
1215:
1199:
1196:
1193:
1173:
1170:
1167:
1159:
1155:
1151:
1149:
1139:
1137:
1135:
1131:
1126:
1124:
1106:
1103:
1099:
1095:
1089:
1083:
1061:
1057:
1053:
1047:
1041:
1021:
1018:
1015:
1012:
1006:
1003:
1000:
994:
974:
971:
965:
959:
953:
949:
946:
943:
937:
931:
925:
917:
913:
909:
905:
901:
897:
893:
885:
883:
881:
877:
873:
869:
853:
850:
847:
827:
824:
821:
813:
806:Group objects
805:
800:
783:
780:
777:
774:
766:
762:
758:
755:
749:
746:
742:
739:
736:
733:
730:
724:
721:
716:
712:
705:
702:
682:
679:
676:
673:
670:
666:
663:
660:
657:
654:
651:
629:
625:
621:
618:
615:
612:
604:
590:
585:
581:
577:
571:
568:
565:
562:
557:
553:
546:
543:
540:
532:
528:
524:
521:
518:
515:
509:
506:
498:
484:
476:
472:
468:
465:
459:
456:
453:
447:
444:
439:
435:
428:
425:
417:
402:
396:
391:
388:
385:
382:
379:
375:
371:
368:
363:
359:
336:
332:
328:
325:
322:
319:
316:
313:
310:
305:
301:
297:
294:
291:
288:
285:
282:
278:
273:
269:
265:
262:
259:
256:
253:
250:
247:
244:
237:
236:
235:
218:
212:
209:
206:
200:
197:
191:
186:
183:
180:
177:
174:
170:
166:
163:
160:
154:
151:
145:
142:
139:
133:
130:
124:
121:
118:
115:
112:
105:
104:
103:
102:
86:
83:
80:
73:
69:
65:
62:
58:
50:
48:
46:
45:group objects
42:
38:
34:
30:
19:
1556:
1550:
1525:, retrieved
1521:the original
1515:
1492:
1487:, Ch 3. § 1.
1480:
1376:
1359:
1351:
1349:
1340:
1294:
1290:
1288:
1278:
1274:
1241:. Then take
1238:
1231:
1226:
1217:
1214:group scheme
1157:
1147:
1145:
1143:
1130:Lie groupoid
1127:
1123:groupoid set
1122:
915:
911:
907:
903:
899:
889:
812:group object
809:
233:
63:
56:
54:
36:
26:
1543:-varieties"
1497:Gillet 1984
1345:coequalizer
1134:submersions
33:mathematics
1583:Categories
1527:2014-02-11
1507:References
1372:stackified
876:Lie groups
605:(inverse)
351:where the
51:Definition
1428:⇉
1395:⇉
1315:⇉
1299:equalizer
1258:×
1146:groupoid
1104:−
1019:∘
963:→
935:→
886:Groupoids
781:∘
750:∘
737:∘
706:∘
674:∘
655:∘
616:∘
569:∘
547:∘
519:∘
510:∘
469:×
460:∘
445:×
429:∘
400:→
376:×
329:∘
317:∘
298:∘
286:∘
260:∘
248:∘
216:→
195:→
171:×
149:→
128:→
101:morphisms
1537:(1984),
1460:See also
1446:-torsors
1364:prestack
896:groupoid
801:Examples
61:category
41:groupoid
1573:0772058
1297:), the
1154:schemes
1150:-scheme
902:, take
882:, etc.
499:(unit)
72:objects
1571:
204:
158:
137:
1472:Notes
1379:atlas
1368:stack
1358:from
1160:. If
874:, or
59:in a
1235:acts
1076:and
840:and
35:, a
1561:doi
1301:of
1136:).
27:In
1585::
1569:MR
1567:,
1557:34
1555:,
1293:,
1273:,
1144:A
1034:,
987:,
910:,
810:A
695:,
644:,
55:A
1563::
1547:K
1541:Q
1499:.
1434:)
1431:U
1425:R
1422:(
1401:]
1398:U
1392:R
1389:[
1360:C
1352:C
1327:U
1322:s
1318:t
1309:R
1295:U
1291:R
1279:t
1275:s
1261:G
1255:U
1252:=
1249:R
1239:U
1232:G
1200:t
1197:=
1194:s
1174:S
1171:=
1168:U
1158:S
1148:S
1107:1
1100:f
1096:=
1093:)
1090:f
1087:(
1084:i
1062:x
1058:1
1054:=
1051:)
1048:x
1045:(
1042:e
1022:f
1016:g
1013:=
1010:)
1007:g
1004:,
1001:f
998:(
995:m
975:y
972:=
969:)
966:y
960:x
957:(
954:t
950:,
947:x
944:=
941:)
938:y
932:x
929:(
926:s
916:C
912:R
908:C
904:U
900:C
854:t
851:=
848:s
828:U
825:=
822:R
796:.
784:t
778:e
775:=
772:)
767:R
763:1
759:,
756:i
753:(
747:m
743:,
740:s
734:e
731:=
728:)
725:i
722:,
717:R
713:1
709:(
703:m
683:s
680:=
677:i
671:t
667:,
664:t
661:=
658:i
652:s
630:R
626:1
622:=
619:i
613:i
591:,
586:R
582:1
578:=
575:)
572:t
566:e
563:,
558:R
554:1
550:(
544:m
541:=
538:)
533:R
529:1
525:,
522:s
516:e
513:(
507:m
485:,
482:)
477:R
473:1
466:m
463:(
457:m
454:=
451:)
448:m
440:R
436:1
432:(
426:m
403:R
397:R
392:s
389:,
386:t
383:,
380:U
372:R
369::
364:i
360:p
337:2
333:p
326:t
323:=
320:m
314:t
311:,
306:1
302:p
295:s
292:=
289:m
283:s
279:,
274:U
270:1
266:=
263:e
257:t
254:=
251:e
245:s
219:R
213:R
210::
207:i
201:,
198:R
192:R
187:s
184:,
181:t
178:,
175:U
167:R
164::
161:m
155:,
152:R
146:U
143::
140:e
134:,
131:U
125:R
122::
119:t
116:,
113:s
87:U
84:,
81:R
64:C
20:)
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