286:
1917:
1727:
171:
594:
1186:
1806:
182:
2264:
403:
1616:
1834:
1644:
1537:
658:
2423:
2467:
660:
always commutes with finite colimits since these are limits. Then, one uses a presentation of filtered colimits as a coequalizer (which is a finite colimit) of an infinite coproduct.
2339:
2054:
71:
1956:
500:
1288:
1084:
1003:
2620:, which are in particular compact. In the context of ∞-categories, dualizable and compact objects tend to be more closely linked, for example in the ∞-category of complexes of
2122:
778:
435:
The terminology is motivated by an example arising from topology mentioned below. Several authors also use a terminology which is more closely related to algebraic categories:
2293:
946:
855:
2146:
2082:
2011:
1472:
1421:
1342:
353:
2378:
1033:
1826:
1257:
430:
316:
2561:
2535:
2507:
2487:
2209:
2189:
2166:
1976:
1636:
1557:
1441:
1390:
1366:
1311:
1230:
1210:
1053:
969:
915:
895:
875:
824:
1092:
2567:
is the filtered colimit of its finite-dimensional (i.e., compact) subspaces. Hence the category of vector spaces (over a fixed field) is compactly generated.
1735:
2850:
2810:
701:
Similar results hold for any category of algebraic structures given by operations on a set obeying equational laws. Such categories, called
2624:-modules, compact and dualizable objects agree. This and more general example where dualizable and compact objects agree are discussed in
2692:
Hall, Jack; Neeman, Amnon; Rydh, David (2015-12-03). "One positive and two negative results for derived categories of algebraic stacks".
806:. The link between compactness in topology and the above categorical notion of compactness is as follows: for a fixed topological space
2124:, there are no compact objects other than the zero object. This observation can be generalized to the following theorem: if the stack
281:{\displaystyle \operatorname {colim} \operatorname {Hom} _{C}(X,Y_{i})\to \operatorname {Hom} _{C}(X,\operatorname {colim} _{i}Y_{i})}
2730:
2216:
615:
is compact in Neeman's sense if and only if it is compact in the ∞-categorical sense. The reason is that in a stable ∞-category,
471:(and the filtered colimits are understood in the ∞-categorical sense, sometimes also referred to as filtered homotopy colimits).
2747:
Ben-Zvi, David; Francis, John; Nadler, David (2010), "Integral transforms and
Drinfeld centers in derived algebraic geometry",
799:
362:
1912:{\displaystyle \psi \in {\underset {i\in I}{\text{colim}}}{\text{ Hom}}({\mathcal {F}}^{\bullet },\mathbb {Z} _{U_{i}})}
1958:, which is not guaranteed. Proving this requires showing that any compact object has support in some compact subset of
1722:{\displaystyle \phi \in {\text{Hom}}({\mathcal {F}}^{\bullet },{\underset {i\in I}{\text{colim}}}\mathbb {Z} _{U_{i}})}
1565:
702:
2517:
In most categories, the condition of being compact is quite strong, so that most objects are not compact. A category
2872:
2832:
1494:
618:
791:
688:
2383:
2889:
2089:
2428:
166:{\displaystyle \operatorname {Hom} _{C}(X,\cdot ):C\to \mathrm {Sets} ,Y\mapsto \operatorname {Hom} _{C}(X,Y)}
2301:
2016:
1559:, it is generally not a compactly generated category. Some evidence for this can be found by considering an
611:
admitting all filtered colimits. (This condition is widely satisfied, but not automatic.) Then an object in
589:{\displaystyle \operatorname {Hom} _{C}(X,\cdot ):C\to \mathrm {Ab} ,Y\mapsto \operatorname {Hom} _{C}(X,Y)}
2057:
1925:
32:
608:
1262:
1058:
977:
480:
2594:), there is another condition imposing some kind of finiteness, namely the condition that an object is
2095:
746:
2269:
920:
829:
2127:
2063:
1992:
1446:
1395:
1316:
2571:
680:
2836:
2782:
2756:
2693:
325:
2846:
2806:
2726:
2617:
2596:
2591:
2347:
803:
604:
2798:
2766:
2718:
1560:
1488:
1087:
1008:
741:
706:
669:
52:
2860:
2820:
2778:
2740:
1811:
1235:
599:
commutes with coproducts. The relation of this notion and the above is as follows: suppose
408:
294:
2856:
2816:
2774:
2736:
1987:
785:
1181:{\displaystyle h_{(-)}:C\to {\text{PreShv}}(C),X\mapsto h_{X}:=\operatorname {Hom} (-,X)}
464:
2546:
2520:
2492:
2472:
2194:
2174:
2151:
1961:
1801:{\displaystyle {\mathcal {F}}^{\bullet }\in {\text{Ob}}(D({\text{Sh}}(X;{\text{Ab}})))}
1621:
1542:
1426:
1375:
1351:
1296:
1215:
1195:
1038:
954:
900:
880:
860:
809:
2883:
2085:
2786:
56:
2770:
467:, provided that the above set of morphisms gets replaced by the mapping space in
2570:
Categories which are compactly generated and also admit all colimits are called
1618:(which can never be refined to a finite subcover using the non-compactness of
1345:
2722:
322:. Since elements in the filtered colimit at the left are represented by maps
2341:
is the zero object. In particular, the category is not compactly generated.
487:
2543:
if any object can be expressed as a filtered colimit of compact objects in
2802:
972:
2841:
2612:-module, so this observation can be applied. Indeed, in the category of
2604:
is compact, then any dualizable object is compact as well. For example,
698:
is a field, then compact objects are finite-dimensional vector spaces.
725:) are precisely the finitely presented models. For example: suppose
359:, the surjectivity of the above map amounts to requiring that a map
2761:
2698:
2670:
1483:
Derived category of sheaves of
Abelian groups on a noncompact X
802:. Instead these are precisely the finite sets endowed with the
2013:
over positive characteristic, the unbounded derived category
2259:{\displaystyle {\overline {G}}=G\otimes _{k}{\overline {k}}}
1982:
Derived category of quasi-coherent sheaves on an Artin stack
1873:
1742:
1665:
1571:
733:) is the category of groups, and the compact objects in Mod(
2616:-modules the dualizable objects are the finitely presented
176:
commutes with filtered colimits, i.e., if the natural map
1313:
can be regarded as a full subcategory of the category
2549:
2523:
2495:
2475:
2431:
2386:
2350:
2304:
2272:
2219:
2197:
2177:
2154:
2130:
2098:
2066:
2019:
1995:
1964:
1928:
1837:
1814:
1738:
1647:
1624:
1568:
1545:
1497:
1449:
1429:
1398:
1378:
1354:
1319:
1299:
1265:
1238:
1218:
1198:
1095:
1061:
1041:
1011:
980:
957:
923:
903:
883:
863:
832:
812:
749:
621:
503:
411:
365:
328:
297:
185:
74:
2625:
398:{\displaystyle X\to \operatorname {colim} _{i}Y_{i}}
2586:with a well-behaved tensor product (more formally,
2671:"On the derived category of sheaves on a manifold"
2555:
2529:
2501:
2481:
2461:
2417:
2372:
2333:
2287:
2258:
2203:
2183:
2160:
2140:
2116:
2076:
2048:
2005:
1970:
1950:
1911:
1820:
1800:
1721:
1630:
1610:
1551:
1531:
1466:
1435:
1415:
1384:
1360:
1336:
1305:
1282:
1251:
1224:
1204:
1180:
1078:
1047:
1027:
997:
963:
940:
909:
889:
869:
849:
818:
772:
652:
588:
424:
397:
347:
310:
291:is a bijection for any filtered system of objects
280:
165:
1611:{\displaystyle {\mathcal {U}}=\{U_{i}\}_{i\in I}}
1368:. Regarded as an object of this larger category,
1035:to sets) has all colimits. The original category
444:
2871:, Annals of Mathematics Studies, vol. 148,
2831:, Annals of Mathematics Studies, vol. 170,
2060:is in general not compactly generated, even if
1978:, and then showing this subset must be empty.
897:is a compact topological space if and only if
2715:Locally presentable and accessible categories
2656:
1532:{\displaystyle D({\text{Sh}}(X;{\text{Ab}}))}
653:{\displaystyle \operatorname {Hom} _{C}(X,-)}
436:
8:
2793:Kashiwara, Masaki; Schapira, Pierre (2006),
2749:Journal of the American Mathematical Society
1593:
1579:
1392:is compact. In fact, the compact objects of
35:satisfying a certain finiteness condition.
2418:{\displaystyle \mathbb {G} _{a}\to GL_{n}}
2840:
2760:
2697:
2548:
2522:
2494:
2474:
2444:
2440:
2439:
2430:
2409:
2393:
2389:
2388:
2385:
2364:
2349:
2322:
2321:
2309:
2303:
2279:
2275:
2274:
2271:
2246:
2240:
2220:
2218:
2196:
2176:
2153:
2132:
2131:
2129:
2108:
2104:
2103:
2097:
2068:
2067:
2065:
2037:
2036:
2024:
2018:
1997:
1996:
1994:
1963:
1940:
1935:
1931:
1930:
1927:
1898:
1893:
1889:
1888:
1878:
1872:
1871:
1862:
1844:
1836:
1813:
1784:
1770:
1756:
1747:
1741:
1740:
1737:
1708:
1703:
1699:
1698:
1679:
1670:
1664:
1663:
1654:
1646:
1623:
1596:
1586:
1570:
1569:
1567:
1544:
1518:
1504:
1496:
1450:
1448:
1428:
1399:
1397:
1377:
1353:
1320:
1318:
1298:
1266:
1264:
1243:
1237:
1217:
1197:
1148:
1121:
1100:
1094:
1062:
1060:
1040:
1016:
1010:
981:
979:
956:
924:
922:
902:
882:
862:
833:
831:
811:
762:
748:
626:
620:
562:
541:
508:
502:
416:
410:
389:
376:
364:
339:
327:
302:
296:
269:
256:
234:
218:
196:
184:
139:
112:
79:
73:
2462:{\displaystyle x\in \mathbb {G} _{a}(S)}
2637:
2334:{\displaystyle D_{qc}({\mathfrak {X}})}
2049:{\displaystyle D_{qc}({\mathfrak {X}})}
2344:This theorem applies, for example, to
857:whose objects are the open subsets of
705:, can be studied systematically using
491:
475:Compactness in triangulated categories
2644:
1922:it would have to factor through some
1443:(or, more precisely, their images in
1005:(i.e., the category of functors from
877:(and inclusions as morphisms). Then,
737:) are the finitely presented groups.
7:
2713:Adámek, Jiří; Rosický, Jiří (1994),
2626:Ben-Zvi, Francis & Nadler (2010)
1951:{\displaystyle \mathbb {Z} _{U_{i}}}
1539:for a non-compact topological space
459:The same definition also applies if
2323:
2133:
2092:. In fact, for the algebraic stack
2069:
2038:
1998:
494:defines an object to be compact if
1283:{\displaystyle {\text{PreShv}}(C)}
1079:{\displaystyle {\text{PreShv}}(C)}
998:{\displaystyle {\text{PreShv}}(C)}
729:is the theory of groups. Then Mod(
545:
542:
122:
119:
116:
113:
14:
2117:{\displaystyle B\mathbb {G} _{a}}
971:is any category, the category of
773:{\displaystyle D(R-{\text{Mod}})}
721:, and the compact objects in Mod(
2563:. For example, any vector space
2298:then the only compact object in
2288:{\displaystyle \mathbb {G} _{a}}
1293:In a similar vein, any category
941:{\displaystyle {\text{Open}}(X)}
850:{\displaystyle {\text{Open}}(X)}
2141:{\displaystyle {\mathfrak {X}}}
2077:{\displaystyle {\mathfrak {X}}}
2006:{\displaystyle {\mathfrak {X}}}
1467:{\displaystyle {\text{Ind}}(C)}
1416:{\displaystyle {\text{Ind}}(C)}
1337:{\displaystyle {\text{Ind}}(C)}
672:are precisely the finite sets.
445:Kashiwara & Schapira (2006)
2717:, Cambridge University Press,
2578:Relation to dualizable objects
2513:Compactly generated categories
2456:
2450:
2399:
2328:
2318:
2043:
2033:
1906:
1867:
1795:
1792:
1789:
1775:
1767:
1761:
1716:
1659:
1526:
1523:
1509:
1501:
1461:
1455:
1410:
1404:
1331:
1325:
1277:
1271:
1175:
1163:
1141:
1132:
1126:
1118:
1107:
1101:
1073:
1067:
992:
986:
935:
929:
844:
838:
800:category of topological spaces
767:
753:
647:
635:
583:
571:
555:
538:
529:
517:
449:objects of finite presentation
369:
332:
275:
243:
227:
224:
205:
160:
148:
132:
109:
100:
88:
29:objects of finite presentation
1:
2771:10.1090/S0894-0347-10-00669-7
2509:-th column in the first row.
2266:has a subgroup isomorphic to
1491:of sheaves of Abelian groups
1423:are precisely the objects of
679:, the compact objects in the
2469:to the identity matrix plus
2251:
2225:
694:-modules. In particular, if
917:is compact as an object in
798:the compact objects in the
784:-modules are precisely the
740:The compact objects in the
668:The compact objects in the
455:Compactness in ∞-categories
443:instead of compact object.
437:Adámek & Rosický (1994)
2906:
2873:Princeton University Press
2833:Princeton University Press
2657:Adámek & Rosický (1994
2600:. If the monoidal unit in
2380:by means of the embedding
2211:of positive characteristic
792:Compact topological spaces
713:, there is a category Mod(
348:{\displaystyle X\to Y_{i}}
25:finitely presented objects
709:. For any Lawvere theory
441:finitely presented object
2723:10.1017/CBO9780511600579
2373:{\displaystyle G=GL_{n}}
2191:is defined over a field
1259:is a compact object (of
826:, there is the category
2869:Triangulated Categories
2148:has a stabilizer group
2867:Neeman, Amnon (2001),
2795:Categories and sheaves
2557:
2531:
2503:
2483:
2463:
2419:
2374:
2335:
2289:
2260:
2205:
2185:
2162:
2142:
2118:
2078:
2058:quasi-coherent sheaves
2050:
2007:
1972:
1952:
1920:
1913:
1822:
1802:
1730:
1723:
1632:
1612:
1553:
1533:
1468:
1437:
1417:
1386:
1362:
1338:
1307:
1284:
1253:
1226:
1206:
1182:
1080:
1049:
1029:
1028:{\displaystyle C^{op}}
999:
965:
942:
911:
891:
871:
851:
820:
774:
654:
590:
426:
399:
349:
312:
282:
167:
23:, also referred to as
2827:Lurie, Jacob (2009),
2803:10.1007/3-540-27950-4
2675:Documenta Mathematica
2572:accessible categories
2558:
2532:
2504:
2484:
2464:
2420:
2375:
2336:
2290:
2261:
2206:
2186:
2163:
2143:
2119:
2079:
2051:
2008:
1973:
1953:
1914:
1830:
1828:to lift to an element
1823:
1821:{\displaystyle \phi }
1808:. Then, for this map
1803:
1724:
1640:
1633:
1613:
1554:
1534:
1469:
1438:
1418:
1387:
1363:
1339:
1308:
1285:
1254:
1252:{\displaystyle h_{X}}
1227:
1207:
1183:
1081:
1050:
1030:
1000:
966:
943:
912:
892:
872:
852:
821:
775:
655:
591:
481:triangulated category
427:
425:{\displaystyle Y_{i}}
400:
350:
313:
311:{\displaystyle Y_{i}}
283:
168:
2590:is required to be a
2547:
2521:
2493:
2473:
2429:
2384:
2348:
2302:
2270:
2217:
2195:
2175:
2152:
2128:
2096:
2064:
2017:
1993:
1962:
1926:
1835:
1812:
1736:
1645:
1622:
1566:
1543:
1495:
1447:
1427:
1396:
1376:
1352:
1317:
1297:
1263:
1236:
1216:
1196:
1093:
1059:
1039:
1009:
978:
955:
921:
901:
881:
861:
830:
810:
747:
619:
501:
439:use the terminology
409:
363:
326:
295:
183:
72:
16:Mathematical concept
2829:Higher topos theory
2797:, Springer Verlag,
2540:compactly generated
31:, are objects in a
2618:projective modules
2553:
2527:
2499:
2479:
2459:
2415:
2370:
2331:
2285:
2256:
2201:
2181:
2158:
2138:
2114:
2074:
2046:
2003:
1968:
1948:
1909:
1860:
1818:
1798:
1719:
1695:
1638:) and taking a map
1628:
1608:
1549:
1529:
1464:
1433:
1413:
1382:
1358:
1334:
1303:
1280:
1249:
1222:
1202:
1178:
1076:
1045:
1025:
995:
961:
938:
907:
887:
867:
847:
816:
770:
689:finitely presented
687:are precisely the
650:
586:
422:
405:factors over some
395:
345:
308:
278:
163:
2852:978-0-691-14049-0
2812:978-3-540-27949-5
2608:is compact as an
2592:monoidal category
2556:{\displaystyle C}
2530:{\displaystyle C}
2502:{\displaystyle n}
2482:{\displaystyle x}
2254:
2228:
2204:{\displaystyle k}
2184:{\displaystyle G}
2161:{\displaystyle G}
1971:{\displaystyle X}
1865:
1848:
1845:
1787:
1773:
1759:
1683:
1680:
1657:
1631:{\displaystyle X}
1552:{\displaystyle X}
1521:
1507:
1487:In the unbounded
1453:
1436:{\displaystyle C}
1402:
1385:{\displaystyle C}
1361:{\displaystyle C}
1323:
1306:{\displaystyle C}
1269:
1225:{\displaystyle C}
1205:{\displaystyle X}
1124:
1065:
1048:{\displaystyle C}
984:
964:{\displaystyle C}
927:
910:{\displaystyle X}
890:{\displaystyle X}
870:{\displaystyle X}
836:
819:{\displaystyle X}
804:discrete topology
786:perfect complexes
765:
609:stable ∞-category
605:homotopy category
486:which admits all
53:filtered colimits
51:which admits all
2897:
2875:
2863:
2844:
2823:
2789:
2764:
2743:
2704:
2703:
2701:
2689:
2683:
2682:
2666:
2660:
2654:
2648:
2642:
2562:
2560:
2559:
2554:
2536:
2534:
2533:
2528:
2508:
2506:
2505:
2500:
2488:
2486:
2485:
2480:
2468:
2466:
2465:
2460:
2449:
2448:
2443:
2425:sending a point
2424:
2422:
2421:
2416:
2414:
2413:
2398:
2397:
2392:
2379:
2377:
2376:
2371:
2369:
2368:
2340:
2338:
2337:
2332:
2327:
2326:
2317:
2316:
2294:
2292:
2291:
2286:
2284:
2283:
2278:
2265:
2263:
2262:
2257:
2255:
2247:
2245:
2244:
2229:
2221:
2210:
2208:
2207:
2202:
2190:
2188:
2187:
2182:
2167:
2165:
2164:
2159:
2147:
2145:
2144:
2139:
2137:
2136:
2123:
2121:
2120:
2115:
2113:
2112:
2107:
2083:
2081:
2080:
2075:
2073:
2072:
2055:
2053:
2052:
2047:
2042:
2041:
2032:
2031:
2012:
2010:
2009:
2004:
2002:
2001:
1988:algebraic stacks
1977:
1975:
1974:
1969:
1957:
1955:
1954:
1949:
1947:
1946:
1945:
1944:
1934:
1918:
1916:
1915:
1910:
1905:
1904:
1903:
1902:
1892:
1883:
1882:
1877:
1876:
1866:
1863:
1861:
1859:
1846:
1827:
1825:
1824:
1819:
1807:
1805:
1804:
1799:
1788:
1785:
1774:
1771:
1760:
1757:
1752:
1751:
1746:
1745:
1728:
1726:
1725:
1720:
1715:
1714:
1713:
1712:
1702:
1696:
1694:
1681:
1675:
1674:
1669:
1668:
1658:
1655:
1637:
1635:
1634:
1629:
1617:
1615:
1614:
1609:
1607:
1606:
1591:
1590:
1575:
1574:
1558:
1556:
1555:
1550:
1538:
1536:
1535:
1530:
1522:
1519:
1508:
1505:
1489:derived category
1473:
1471:
1470:
1465:
1454:
1451:
1442:
1440:
1439:
1434:
1422:
1420:
1419:
1414:
1403:
1400:
1391:
1389:
1388:
1383:
1367:
1365:
1364:
1359:
1343:
1341:
1340:
1335:
1324:
1321:
1312:
1310:
1309:
1304:
1289:
1287:
1286:
1281:
1270:
1267:
1258:
1256:
1255:
1250:
1248:
1247:
1231:
1229:
1228:
1223:
1211:
1209:
1208:
1203:
1187:
1185:
1184:
1179:
1153:
1152:
1125:
1122:
1111:
1110:
1088:Yoneda embedding
1085:
1083:
1082:
1077:
1066:
1063:
1055:is connected to
1054:
1052:
1051:
1046:
1034:
1032:
1031:
1026:
1024:
1023:
1004:
1002:
1001:
996:
985:
982:
970:
968:
967:
962:
947:
945:
944:
939:
928:
925:
916:
914:
913:
908:
896:
894:
893:
888:
876:
874:
873:
868:
856:
854:
853:
848:
837:
834:
825:
823:
822:
817:
779:
777:
776:
771:
766:
763:
742:derived category
707:Lawvere theories
670:category of sets
659:
657:
656:
651:
631:
630:
595:
593:
592:
587:
567:
566:
548:
513:
512:
431:
429:
428:
423:
421:
420:
404:
402:
401:
396:
394:
393:
381:
380:
354:
352:
351:
346:
344:
343:
317:
315:
314:
309:
307:
306:
287:
285:
284:
279:
274:
273:
261:
260:
239:
238:
223:
222:
201:
200:
172:
170:
169:
164:
144:
143:
125:
84:
83:
19:In mathematics,
2905:
2904:
2900:
2899:
2898:
2896:
2895:
2894:
2890:Category theory
2880:
2879:
2878:
2866:
2853:
2842:math.CT/0608040
2826:
2813:
2792:
2746:
2733:
2712:
2708:
2707:
2691:
2690:
2686:
2669:Neeman, Amnon.
2668:
2667:
2663:
2655:
2651:
2643:
2639:
2634:
2582:For categories
2580:
2545:
2544:
2519:
2518:
2515:
2491:
2490:
2471:
2470:
2438:
2427:
2426:
2405:
2387:
2382:
2381:
2360:
2346:
2345:
2305:
2300:
2299:
2273:
2268:
2267:
2236:
2215:
2214:
2193:
2192:
2173:
2172:
2150:
2149:
2126:
2125:
2102:
2094:
2093:
2090:quasi-separated
2062:
2061:
2020:
2015:
2014:
1991:
1990:
1984:
1960:
1959:
1936:
1929:
1924:
1923:
1894:
1887:
1870:
1849:
1833:
1832:
1810:
1809:
1739:
1734:
1733:
1704:
1697:
1684:
1662:
1643:
1642:
1620:
1619:
1592:
1582:
1564:
1563:
1541:
1540:
1493:
1492:
1485:
1480:
1445:
1444:
1425:
1424:
1394:
1393:
1374:
1373:
1350:
1349:
1315:
1314:
1295:
1294:
1261:
1260:
1239:
1234:
1233:
1214:
1213:
1194:
1193:
1144:
1096:
1091:
1090:
1057:
1056:
1037:
1036:
1012:
1007:
1006:
976:
975:
953:
952:
919:
918:
899:
898:
879:
878:
859:
858:
828:
827:
808:
807:
745:
744:
717:) of models of
666:
622:
617:
616:
558:
504:
499:
498:
477:
457:
447:call these the
412:
407:
406:
385:
372:
361:
360:
335:
324:
323:
298:
293:
292:
265:
252:
230:
214:
192:
181:
180:
135:
75:
70:
69:
65:if the functor
55:(also known as
41:
21:compact objects
17:
12:
11:
5:
2903:
2901:
2893:
2892:
2882:
2881:
2877:
2876:
2864:
2851:
2824:
2811:
2790:
2755:(4): 909–966,
2744:
2731:
2709:
2706:
2705:
2684:
2661:
2659:, Chapter 1.A)
2649:
2636:
2635:
2633:
2630:
2579:
2576:
2552:
2526:
2514:
2511:
2498:
2478:
2458:
2455:
2452:
2447:
2442:
2437:
2434:
2412:
2408:
2404:
2401:
2396:
2391:
2367:
2363:
2359:
2356:
2353:
2330:
2325:
2320:
2315:
2312:
2308:
2296:
2295:
2282:
2277:
2253:
2250:
2243:
2239:
2235:
2232:
2227:
2224:
2212:
2200:
2180:
2157:
2135:
2111:
2106:
2101:
2071:
2045:
2040:
2035:
2030:
2027:
2023:
2000:
1983:
1980:
1967:
1943:
1939:
1933:
1908:
1901:
1897:
1891:
1886:
1881:
1875:
1869:
1858:
1855:
1852:
1843:
1840:
1817:
1797:
1794:
1791:
1783:
1780:
1777:
1769:
1766:
1763:
1755:
1750:
1744:
1718:
1711:
1707:
1701:
1693:
1690:
1687:
1678:
1673:
1667:
1661:
1653:
1650:
1627:
1605:
1602:
1599:
1595:
1589:
1585:
1581:
1578:
1573:
1548:
1528:
1525:
1517:
1514:
1511:
1503:
1500:
1484:
1481:
1479:
1476:
1463:
1460:
1457:
1432:
1412:
1409:
1406:
1381:
1357:
1333:
1330:
1327:
1302:
1279:
1276:
1273:
1246:
1242:
1221:
1201:
1177:
1174:
1171:
1168:
1165:
1162:
1159:
1156:
1151:
1147:
1143:
1140:
1137:
1134:
1131:
1128:
1120:
1117:
1114:
1109:
1106:
1103:
1099:
1075:
1072:
1069:
1044:
1022:
1019:
1015:
994:
991:
988:
960:
937:
934:
931:
906:
886:
866:
846:
843:
840:
815:
769:
761:
758:
755:
752:
665:
662:
649:
646:
643:
640:
637:
634:
629:
625:
603:arises as the
597:
596:
585:
582:
579:
576:
573:
570:
565:
561:
557:
554:
551:
547:
544:
540:
537:
534:
531:
528:
525:
522:
519:
516:
511:
507:
476:
473:
456:
453:
419:
415:
392:
388:
384:
379:
375:
371:
368:
342:
338:
334:
331:
305:
301:
289:
288:
277:
272:
268:
264:
259:
255:
251:
248:
245:
242:
237:
233:
229:
226:
221:
217:
213:
210:
207:
204:
199:
195:
191:
188:
174:
173:
162:
159:
156:
153:
150:
147:
142:
138:
134:
131:
128:
124:
121:
118:
115:
111:
108:
105:
102:
99:
96:
93:
90:
87:
82:
78:
47:in a category
40:
37:
15:
13:
10:
9:
6:
4:
3:
2:
2902:
2891:
2888:
2887:
2885:
2874:
2870:
2865:
2862:
2858:
2854:
2848:
2843:
2838:
2834:
2830:
2825:
2822:
2818:
2814:
2808:
2804:
2800:
2796:
2791:
2788:
2784:
2780:
2776:
2772:
2768:
2763:
2758:
2754:
2750:
2745:
2742:
2738:
2734:
2732:0-521-42261-2
2728:
2724:
2720:
2716:
2711:
2710:
2700:
2695:
2688:
2685:
2680:
2676:
2672:
2665:
2662:
2658:
2653:
2650:
2646:
2641:
2638:
2631:
2629:
2627:
2623:
2619:
2615:
2611:
2607:
2603:
2599:
2598:
2593:
2589:
2585:
2577:
2575:
2573:
2568:
2566:
2550:
2542:
2541:
2524:
2512:
2510:
2496:
2476:
2453:
2445:
2435:
2432:
2410:
2406:
2402:
2394:
2365:
2361:
2357:
2354:
2351:
2342:
2313:
2310:
2306:
2280:
2248:
2241:
2237:
2233:
2230:
2222:
2213:
2198:
2178:
2171:
2170:
2169:
2155:
2109:
2099:
2091:
2087:
2086:quasi-compact
2059:
2028:
2025:
2021:
1989:
1981:
1979:
1965:
1941:
1937:
1919:
1899:
1895:
1884:
1879:
1856:
1853:
1850:
1841:
1838:
1829:
1815:
1781:
1778:
1764:
1753:
1748:
1729:
1709:
1705:
1691:
1688:
1685:
1676:
1671:
1651:
1648:
1639:
1625:
1603:
1600:
1597:
1587:
1583:
1576:
1562:
1546:
1515:
1512:
1498:
1490:
1482:
1477:
1475:
1458:
1430:
1407:
1379:
1371:
1355:
1347:
1328:
1300:
1291:
1274:
1244:
1240:
1219:
1199:
1191:
1172:
1169:
1166:
1160:
1157:
1154:
1149:
1145:
1138:
1135:
1129:
1115:
1112:
1104:
1097:
1089:
1070:
1042:
1020:
1017:
1013:
989:
974:
958:
949:
932:
904:
884:
864:
841:
813:
805:
801:
797:
793:
789:
787:
783:
759:
756:
750:
743:
738:
736:
732:
728:
724:
720:
716:
712:
708:
704:
699:
697:
693:
690:
686:
684:
678:
673:
671:
663:
661:
644:
641:
638:
632:
627:
623:
614:
610:
606:
602:
580:
577:
574:
568:
563:
559:
552:
549:
535:
532:
526:
523:
520:
514:
509:
505:
497:
496:
495:
493:
492:Neeman (2001)
489:
485:
482:
474:
472:
470:
466:
462:
454:
452:
450:
446:
442:
438:
433:
417:
413:
390:
386:
382:
377:
373:
366:
358:
340:
336:
329:
321:
303:
299:
270:
266:
262:
257:
253:
249:
246:
240:
235:
231:
219:
215:
211:
208:
202:
197:
193:
189:
186:
179:
178:
177:
157:
154:
151:
145:
140:
136:
129:
126:
106:
103:
97:
94:
91:
85:
80:
76:
68:
67:
66:
64:
63:
58:
57:direct limits
54:
50:
46:
38:
36:
34:
30:
26:
22:
2868:
2828:
2794:
2752:
2748:
2714:
2687:
2678:
2674:
2664:
2652:
2640:
2621:
2613:
2609:
2605:
2601:
2595:
2587:
2583:
2581:
2569:
2564:
2539:
2538:
2516:
2343:
2297:
1985:
1921:
1831:
1731:
1641:
1486:
1478:Non-examples
1369:
1292:
1189:
950:
795:
790:
781:
739:
734:
730:
726:
722:
718:
714:
710:
700:
695:
691:
682:
681:category of
676:
674:
667:
612:
600:
598:
483:
478:
468:
460:
458:
448:
440:
434:
356:
319:
290:
175:
61:
60:
59:) is called
48:
44:
42:
28:
24:
20:
18:
2645:Lurie (2009
1346:ind-objects
675:For a ring
355:, for some
2681:: 483–488.
2632:References
2597:dualizable
2168:such that
1561:open cover
1372:object of
973:presheaves
488:coproducts
465:∞-category
43:An object
39:Definition
2762:0805.0157
2699:1405.1888
2647:, §5.3.4)
2436:∈
2400:→
2252:¯
2238:⊗
2226:¯
1880:∙
1864: Hom
1854:∈
1842:∈
1839:ψ
1816:ϕ
1754:∈
1749:∙
1732:for some
1689:∈
1672:∙
1652:∈
1649:ϕ
1601:∈
1167:−
1161:
1142:↦
1119:→
1105:−
760:−
703:varieties
645:−
633:
569:
556:↦
539:→
527:⋅
515:
383:
370:→
333:→
263:
241:
228:→
203:
190:
146:
133:↦
110:→
98:⋅
86:
2884:Category
685:-modules
664:Examples
33:category
2861:2522659
2821:2182076
2787:2202294
2779:2669705
2741:1294136
2489:at the
1192:object
1086:by the
62:compact
2859:
2849:
2819:
2809:
2785:
2777:
2739:
2729:
1268:PreShv
1188:. For
1123:PreShv
1064:PreShv
983:PreShv
479:For a
463:is an
2837:arXiv
2783:S2CID
2757:arXiv
2694:arXiv
1847:colim
1682:colim
607:of a
374:colim
254:colim
187:colim
27:, or
2847:ISBN
2807:ISBN
2727:ISBN
2088:and
1986:For
926:Open
835:Open
794:are
2799:doi
2767:doi
2719:doi
2537:is
2084:is
2056:of
1656:Hom
1474:).
1452:Ind
1401:Ind
1370:any
1348:in
1344:of
1322:Ind
1290:).
1212:of
1190:any
1158:Hom
951:If
796:not
780:of
764:Mod
624:Hom
560:Hom
506:Hom
318:in
232:Hom
194:Hom
137:Hom
77:Hom
2886::
2857:MR
2855:,
2845:,
2835:,
2817:MR
2815:,
2805:,
2781:,
2775:MR
2773:,
2765:,
2753:23
2751:,
2737:MR
2735:,
2725:,
2677:.
2673:.
2628:.
2574:.
1786:Ab
1772:Sh
1758:Ob
1520:Ab
1506:Sh
1232:,
1155::=
948:.
788:.
490:,
451:.
432:.
2839::
2801::
2769::
2759::
2721::
2702:.
2696::
2679:6
2622:R
2614:R
2610:R
2606:R
2602:C
2588:C
2584:C
2565:V
2551:C
2525:C
2497:n
2477:x
2457:)
2454:S
2451:(
2446:a
2441:G
2433:x
2411:n
2407:L
2403:G
2395:a
2390:G
2366:n
2362:L
2358:G
2355:=
2352:G
2329:)
2324:X
2319:(
2314:c
2311:q
2307:D
2281:a
2276:G
2249:k
2242:k
2234:G
2231:=
2223:G
2199:k
2179:G
2156:G
2134:X
2110:a
2105:G
2100:B
2070:X
2044:)
2039:X
2034:(
2029:c
2026:q
2022:D
1999:X
1966:X
1942:i
1938:U
1932:Z
1907:)
1900:i
1896:U
1890:Z
1885:,
1874:F
1868:(
1857:I
1851:i
1796:)
1793:)
1790:)
1782:;
1779:X
1776:(
1768:(
1765:D
1762:(
1743:F
1717:)
1710:i
1706:U
1700:Z
1692:I
1686:i
1677:,
1666:F
1660:(
1626:X
1604:I
1598:i
1594:}
1588:i
1584:U
1580:{
1577:=
1572:U
1547:X
1527:)
1524:)
1516:;
1513:X
1510:(
1502:(
1499:D
1462:)
1459:C
1456:(
1431:C
1411:)
1408:C
1405:(
1380:C
1356:C
1332:)
1329:C
1326:(
1301:C
1278:)
1275:C
1272:(
1245:X
1241:h
1220:C
1200:X
1176:)
1173:X
1170:,
1164:(
1150:X
1146:h
1139:X
1136:,
1133:)
1130:C
1127:(
1116:C
1113::
1108:)
1102:(
1098:h
1074:)
1071:C
1068:(
1043:C
1021:p
1018:o
1014:C
993:)
990:C
987:(
959:C
936:)
933:X
930:(
905:X
885:X
865:X
845:)
842:X
839:(
814:X
782:R
768:)
757:R
754:(
751:D
735:T
731:T
727:T
723:T
719:T
715:T
711:T
696:R
692:R
683:R
677:R
648:)
642:,
639:X
636:(
628:C
613:C
601:C
584:)
581:Y
578:,
575:X
572:(
564:C
553:Y
550:,
546:b
543:A
536:C
533::
530:)
524:,
521:X
518:(
510:C
484:C
469:C
461:C
418:i
414:Y
391:i
387:Y
378:i
367:X
357:i
341:i
337:Y
330:X
320:C
304:i
300:Y
276:)
271:i
267:Y
258:i
250:,
247:X
244:(
236:C
225:)
220:i
216:Y
212:,
209:X
206:(
198:C
161:)
158:Y
155:,
152:X
149:(
141:C
130:Y
127:,
123:s
120:t
117:e
114:S
107:C
104::
101:)
95:,
92:X
89:(
81:C
49:C
45:X
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