1269:
There is a variant of "filtered category" known as a "Îș-filtered category", defined as follows. This begins with the following observation: the three conditions in the definition of filtered category above say respectively that there exists a
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39:
understood as a category (hence called a directed category; while some use directed category as a synonym for a filtered category). There is a dual notion of
1644:
1657:
1381:
969:
1544:
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1138:
879:
1297:
1271:
52:
1416:. The existence of cocones for these three shapes of diagrams turns out to imply that cocones exist for
510:
1096:
975:
1623:
731:
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270:
205:
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is filtered (according to the above definition) if and only if there is a cocone over any
28:
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17:
1547:
is of cardinality Îș if the morphism set of its domain is of cardinality Îș.)
1635:). Lecture Notes in Mathematics 269, Springer Verlag, 1972. Exposé I, 2.7.
1014:
that is a small filtered colimit of representable presheaves, is called an
1231:
is the opposite of the category of ind-objects in the opposite category
1551:
883:
422:
418:
1503:
is defined to be Îș-filtered if there is a cocone over every diagram
539:
is filtered. In detail, a category is cofiltered when
1591:
1559:
1529:
1509:
1489:
1450:
1426:
1409:{\displaystyle \{i\rightrightarrows j\}\rightarrow J}
1384:
1335:
1300:
1280:
1237:
1217:
1141:
1099:
1064:
1044:
1024:
978:
954:
923:
891:
844:
812:
792:
772:
734:
711:
669:
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572:
552:
513:
490:
462:
430:
383:
351:
331:
311:
273:
250:
208:
176:
156:
136:
111:
91:
60:
1597:
1577:
1535:
1515:
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1468:
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1370:
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1286:
1253:
1223:
1203:
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862:
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655:
623:
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531:
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194:
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142:
122:
97:
66:
1628:Séminaire de Géométrie Algébrique du Bois Marie
8:
1397:
1385:
1359:
1336:
1310:
1301:
1420:finite diagram; in other words, a category
1371:{\displaystyle \{j\ \ \ j'\}\rightarrow J}
1590:
1558:
1528:
1508:
1488:
1449:
1425:
1383:
1334:
1299:
1279:
1242:
1236:
1216:
1192:
1179:
1140:
1104:
1098:
1093:in the category of functors (presheaves)
1063:
1043:
1023:
983:
977:
953:
922:
890:
843:
811:
791:
771:
733:
710:
668:
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571:
551:
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461:
429:
382:
350:
330:
310:
272:
249:
207:
175:
155:
135:
110:
90:
59:
43:category, which will be recalled below.
1645:Categories for the Working Mathematician
1543:of cardinality smaller than Îș. (A small
1550:A Îș-filtered colimit is a colimit of a
1204:{\displaystyle Pro(C)=Ind(C^{op})^{op}}
1322:{\displaystyle \{\ \ \}\rightarrow J}
7:
1648:(2nd ed.), Berlin, New York:
523:
520:
25:
532:{\displaystyle J^{\mathrm {op} }}
1569:
1460:
1400:
1391:
1362:
1313:
1189:
1172:
1157:
1151:
1113:
1080:
1074:
992:
901:
822:
750:
728:for every two parallel arrows
684:
647:
440:
361:
289:
267:for every two parallel arrows
228:
186:
1:
1128:{\displaystyle C^{op}\to Set}
1007:{\displaystyle C^{op}\to Set}
1038:. Ind-objects of a category
941:Ind-objects and pro-objects
1694:
1605:is a Îș-filtered category.
937:is a cofiltered category.
759:{\displaystyle u,v:j\to i}
698:{\displaystyle f':k\to j'}
298:{\displaystyle u,v:i\to j}
237:{\displaystyle f':j'\to k}
786:, there exists an object
476:is a filtered category.
325:, there exists an object
35:generalize the notion of
1578:{\displaystyle F:J\to C}
1479:Extending this, given a
1469:{\displaystyle d:D\to J}
1058:form a full subcategory
910:{\displaystyle F:J\to C}
831:{\displaystyle w:k\to j}
656:{\displaystyle f:k\to j}
449:{\displaystyle F:J\to C}
370:{\displaystyle w:j\to k}
195:{\displaystyle f:j\to k}
611:there exists an object
150:there exists an object
1599:
1579:
1537:
1517:
1497:
1470:
1434:
1410:
1372:
1323:
1288:
1255:
1254:{\displaystyle C^{op}}
1225:
1205:
1129:
1087:
1086:{\displaystyle Ind(C)}
1052:
1032:
1008:
962:
931:
911:
864:
832:
800:
780:
760:
719:
699:
657:
625:
605:
585:
560:
546:for every two objects
533:
498:
470:
450:
403:
371:
339:
319:
299:
258:
238:
196:
164:
144:
124:
99:
85:for every two objects
68:
1600:
1580:
1538:
1518:
1498:
1471:
1435:
1411:
1373:
1324:
1289:
1265:Îș-filtered categories
1256:
1226:
1206:
1130:
1088:
1053:
1033:
1009:
963:
932:
912:
865:
863:{\displaystyle uw=vw}
833:
801:
781:
761:
720:
700:
658:
626:
606:
586:
561:
534:
504:is cofiltered if the
499:
480:Cofiltered categories
471:
451:
404:
402:{\displaystyle wu=wv}
372:
340:
320:
300:
259:
239:
197:
165:
145:
125:
100:
69:
1589:
1557:
1527:
1507:
1487:
1448:
1424:
1382:
1333:
1298:
1278:
1274:over any diagram in
1235:
1215:
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1097:
1062:
1042:
1022:
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921:
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842:
810:
790:
770:
732:
709:
667:
635:
615:
595:
570:
550:
511:
488:
460:
428:
381:
349:
329:
309:
271:
248:
206:
174:
154:
134:
109:
89:
58:
47:Filtered categories
33:filtered categories
1640:Mac Lane, Saunders
1595:
1575:
1533:
1513:
1493:
1466:
1430:
1406:
1368:
1319:
1284:
1251:
1221:
1211:of pro-objects in
1201:
1125:
1083:
1048:
1028:
1004:
958:
927:
907:
860:
828:
796:
776:
756:
715:
695:
653:
621:
601:
584:{\displaystyle j'}
581:
556:
529:
494:
466:
446:
399:
367:
335:
315:
295:
254:
234:
192:
160:
140:
123:{\displaystyle j'}
120:
95:
64:
1659:978-0-387-98403-2
1598:{\displaystyle J}
1536:{\displaystyle J}
1516:{\displaystyle d}
1496:{\displaystyle J}
1433:{\displaystyle J}
1350:
1347:
1344:
1309:
1306:
1287:{\displaystyle J}
1224:{\displaystyle C}
1051:{\displaystyle C}
1031:{\displaystyle C}
961:{\displaystyle C}
930:{\displaystyle J}
799:{\displaystyle k}
779:{\displaystyle J}
718:{\displaystyle J}
624:{\displaystyle k}
604:{\displaystyle J}
559:{\displaystyle j}
506:opposite category
497:{\displaystyle J}
469:{\displaystyle J}
338:{\displaystyle k}
318:{\displaystyle J}
257:{\displaystyle J}
163:{\displaystyle k}
143:{\displaystyle J}
98:{\displaystyle j}
67:{\displaystyle J}
16:(Redirected from
1685:
1662:
1620:Grothendieck, A.
1604:
1602:
1601:
1596:
1584:
1582:
1581:
1576:
1542:
1540:
1539:
1534:
1522:
1520:
1519:
1514:
1502:
1500:
1499:
1494:
1481:regular cardinal
1475:
1473:
1472:
1467:
1439:
1437:
1436:
1431:
1415:
1413:
1412:
1407:
1377:
1375:
1374:
1369:
1358:
1348:
1345:
1342:
1328:
1326:
1325:
1320:
1307:
1304:
1293:
1291:
1290:
1285:
1260:
1258:
1257:
1252:
1250:
1249:
1230:
1228:
1227:
1222:
1210:
1208:
1207:
1202:
1200:
1199:
1187:
1186:
1134:
1132:
1131:
1126:
1112:
1111:
1092:
1090:
1089:
1084:
1057:
1055:
1054:
1049:
1037:
1035:
1034:
1029:
1018:of the category
1013:
1011:
1010:
1005:
991:
990:
967:
965:
964:
959:
936:
934:
933:
928:
916:
914:
913:
908:
876:cofiltered limit
869:
867:
866:
861:
837:
835:
834:
829:
805:
803:
802:
797:
785:
783:
782:
777:
765:
763:
762:
757:
724:
722:
721:
716:
704:
702:
701:
696:
694:
677:
662:
660:
659:
654:
630:
628:
627:
622:
610:
608:
607:
602:
590:
588:
587:
582:
580:
565:
563:
562:
557:
543:it is not empty,
538:
536:
535:
530:
528:
527:
526:
503:
501:
500:
495:
475:
473:
472:
467:
455:
453:
452:
447:
415:filtered colimit
408:
406:
405:
400:
376:
374:
373:
368:
344:
342:
341:
336:
324:
322:
321:
316:
304:
302:
301:
296:
263:
261:
260:
255:
243:
241:
240:
235:
227:
216:
201:
199:
198:
193:
169:
167:
166:
161:
149:
147:
146:
141:
129:
127:
126:
121:
119:
104:
102:
101:
96:
82:it is not empty,
73:
71:
70:
65:
21:
1693:
1692:
1688:
1687:
1686:
1684:
1683:
1682:
1678:Category theory
1668:
1667:
1666:
1663:, section IX.1.
1660:
1650:Springer-Verlag
1638:
1611:
1587:
1586:
1555:
1554:
1525:
1524:
1505:
1504:
1485:
1484:
1446:
1445:
1422:
1421:
1380:
1379:
1351:
1331:
1330:
1296:
1295:
1276:
1275:
1267:
1238:
1233:
1232:
1213:
1212:
1188:
1175:
1137:
1136:
1135:. The category
1100:
1095:
1094:
1060:
1059:
1040:
1039:
1020:
1019:
979:
974:
973:
950:
949:
943:
919:
918:
887:
886:
840:
839:
808:
807:
788:
787:
768:
767:
730:
729:
707:
706:
687:
670:
665:
664:
633:
632:
631:and two arrows
613:
612:
593:
592:
573:
568:
567:
548:
547:
514:
509:
508:
486:
485:
482:
458:
457:
426:
425:
379:
378:
347:
346:
327:
326:
307:
306:
269:
268:
246:
245:
220:
209:
204:
203:
172:
171:
170:and two arrows
152:
151:
132:
131:
112:
107:
106:
87:
86:
56:
55:
49:
29:category theory
23:
22:
15:
12:
11:
5:
1691:
1689:
1681:
1680:
1670:
1669:
1665:
1664:
1658:
1636:
1624:Verdier, J.-L.
1612:
1610:
1607:
1594:
1574:
1571:
1568:
1565:
1562:
1532:
1512:
1492:
1483:Îș, a category
1465:
1462:
1459:
1456:
1453:
1429:
1405:
1402:
1399:
1396:
1393:
1390:
1387:
1367:
1364:
1361:
1357:
1354:
1341:
1338:
1318:
1315:
1312:
1303:
1283:
1266:
1263:
1248:
1245:
1241:
1220:
1198:
1195:
1191:
1185:
1182:
1178:
1174:
1171:
1168:
1165:
1162:
1159:
1156:
1153:
1150:
1147:
1144:
1124:
1121:
1118:
1115:
1110:
1107:
1103:
1082:
1079:
1076:
1073:
1070:
1067:
1047:
1027:
1003:
1000:
997:
994:
989:
986:
982:
957:
947:small category
942:
939:
926:
906:
903:
900:
897:
894:
872:
871:
859:
856:
853:
850:
847:
827:
824:
821:
818:
815:
795:
775:
755:
752:
749:
746:
743:
740:
737:
726:
714:
693:
690:
686:
683:
680:
676:
673:
652:
649:
646:
643:
640:
620:
600:
579:
576:
555:
544:
525:
522:
517:
493:
481:
478:
465:
445:
442:
439:
436:
433:
411:
410:
398:
395:
392:
389:
386:
366:
363:
360:
357:
354:
334:
314:
294:
291:
288:
285:
282:
279:
276:
265:
253:
233:
230:
226:
223:
219:
215:
212:
191:
188:
185:
182:
179:
159:
139:
118:
115:
94:
83:
63:
48:
45:
24:
18:Filtered limit
14:
13:
10:
9:
6:
4:
3:
2:
1690:
1679:
1676:
1675:
1673:
1661:
1655:
1651:
1647:
1646:
1641:
1637:
1634:
1630:
1629:
1625:
1621:
1617:
1614:
1613:
1608:
1606:
1592:
1572:
1566:
1563:
1560:
1553:
1548:
1546:
1530:
1510:
1490:
1482:
1477:
1463:
1457:
1454:
1451:
1443:
1427:
1419:
1403:
1394:
1388:
1365:
1355:
1352:
1339:
1316:
1281:
1273:
1264:
1262:
1246:
1243:
1239:
1218:
1196:
1193:
1183:
1180:
1176:
1169:
1166:
1163:
1160:
1154:
1148:
1145:
1142:
1122:
1119:
1116:
1108:
1105:
1101:
1077:
1071:
1068:
1065:
1045:
1025:
1017:
1001:
998:
995:
987:
984:
980:
971:
955:
948:
940:
938:
924:
904:
898:
895:
892:
885:
881:
877:
857:
854:
851:
848:
845:
825:
819:
816:
813:
806:and an arrow
793:
773:
753:
747:
744:
741:
738:
735:
727:
712:
691:
688:
681:
678:
674:
671:
650:
644:
641:
638:
618:
598:
577:
574:
553:
545:
542:
541:
540:
515:
507:
491:
479:
477:
463:
443:
437:
434:
431:
424:
420:
416:
396:
393:
390:
387:
384:
364:
358:
355:
352:
345:and an arrow
332:
312:
292:
286:
283:
280:
277:
274:
266:
251:
231:
224:
221:
217:
213:
210:
189:
183:
180:
177:
157:
137:
116:
113:
92:
84:
81:
80:
79:
77:
61:
54:
46:
44:
42:
38:
34:
30:
19:
1643:
1632:
1626:
1549:
1478:
1441:
1417:
1294:of the form
1268:
1015:
944:
875:
873:
483:
414:
412:
75:
50:
40:
37:directed set
32:
26:
484:A category
1609:References
1016:ind-object
838:such that
377:such that
41:cofiltered
1616:Artin, M.
1570:→
1461:→
1401:→
1392:⇉
1363:→
1314:→
1114:→
993:→
902:→
823:→
751:→
685:→
648:→
441:→
362:→
290:→
229:→
187:→
1672:Category
1642:(1998),
1444:diagram
1356:′
972:of sets
970:presheaf
945:Given a
692:′
675:′
578:′
225:′
214:′
117:′
76:filtered
53:category
1552:functor
1545:diagram
884:functor
423:functor
419:colimit
1656:
1585:where
1442:finite
1349:
1346:
1343:
1308:
1305:
1272:cocone
917:where
456:where
1633:SGA 4
1378:, or
882:of a
880:limit
878:is a
421:of a
417:is a
78:when
1654:ISBN
1622:and
968:, a
663:and
566:and
202:and
105:and
1523:in
1418:any
766:in
705:in
591:in
305:in
244:in
130:in
74:is
27:In
1674::
1652:,
1618:,
1476:.
1329:,
1261:.
874:A
413:A
51:A
31:,
1631:(
1593:J
1573:C
1567:J
1564::
1561:F
1531:J
1511:d
1491:J
1464:J
1458:D
1455::
1452:d
1428:J
1404:J
1398:}
1395:j
1389:i
1386:{
1366:J
1360:}
1353:j
1340:j
1337:{
1317:J
1311:}
1302:{
1282:J
1247:p
1244:o
1240:C
1219:C
1197:p
1194:o
1190:)
1184:p
1181:o
1177:C
1173:(
1170:d
1167:n
1164:I
1161:=
1158:)
1155:C
1152:(
1149:o
1146:r
1143:P
1123:t
1120:e
1117:S
1109:p
1106:o
1102:C
1081:)
1078:C
1075:(
1072:d
1069:n
1066:I
1046:C
1026:C
1002:t
999:e
996:S
988:p
985:o
981:C
956:C
925:J
905:C
899:J
896::
893:F
870:.
858:w
855:v
852:=
849:w
846:u
826:j
820:k
817::
814:w
794:k
774:J
754:i
748:j
745::
742:v
739:,
736:u
725:,
713:J
689:j
682:k
679::
672:f
651:j
645:k
642::
639:f
619:k
599:J
575:j
554:j
524:p
521:o
516:J
492:J
464:J
444:C
438:J
435::
432:F
409:.
397:v
394:w
391:=
388:u
385:w
365:k
359:j
356::
353:w
333:k
313:J
293:j
287:i
284::
281:v
278:,
275:u
264:,
252:J
232:k
222:j
218::
211:f
190:k
184:j
181::
178:f
158:k
138:J
114:j
93:j
62:J
20:)
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