855:
521:
1604:
Another common overcategory considered in the literature are overcategories of spaces, such as schemes, smooth manifolds, or topological spaces. These categories encode objects relative to a fixed object, such as the category of schemes over
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1524:
850:{\displaystyle {\begin{matrix}X&=&X\\\psi \downarrow {\text{ }}&{\text{ }}&{\text{ }}\downarrow \psi '\\B&\xrightarrow {g} &B'\end{matrix}}}
516:{\displaystyle {\begin{matrix}A&\xrightarrow {f} &A'\\\pi \downarrow {\text{ }}&{\text{ }}&{\text{ }}\downarrow \pi '\\X&=&X\end{matrix}}}
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49:
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in these categories can be considered intersections, given the objects are subobjects of the fixed object.
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31:. They were introduced as a mechanism for keeping track of data surrounding a fixed object
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34:
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1526:. If we consider the opposite category, it is an overcategory of affine schemes,
1132:. One of the canonical examples comes directly from topology, where the category
1203:, and the morphisms are given by inclusion maps. Then, for a fixed open subset
861:
1040:, and through universal properties, there exists a unique morphism either to
1709:
1016:
have these properties since the product and coproduct can be constructed in
949:
1460:
for the category of commutative rings. This is because the structure of an
1128:
is a categorical generalization of a topological space first introduced by
1085:
256:
900:
are inherited by the associated over and undercategories for an object
75:. There is a dual notion of undercategory, which is defined similarly.
1739:
1710:"Section 4.32 (02XG): Categories over categories—The Stacks project"
827:
436:
1694:
23:, an overcategory (and undercategory) is a distinguished class of
868:, with definitions either analogous or essentially the same.
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Leinster, Tom (2016-12-29). "Basic
Category Theory".
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1733:Lurie, Jacob (2008-07-31). "Higher Topos Theory".
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191:is an associated category whose objects are pairs
183:
143:
119:
99:
67:
43:
1560:{\displaystyle {\text{Aff}}/{\text{Spec}}(A)}
8:
1262:is canonically equivalent to the category
860:These two notions have generalizations in
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36:
1402:Category of algebras as an undercategory
747:such that the following diagram commutes
342:{\displaystyle f:(A,\pi )\to (A',\pi ')}
27:used in multiple contexts, such as with
1680:
1500:is directly encoded by a ring morphism
7:
1430:is equivalent to the undercategory
283:. Then, a morphism between objects
1351:{\displaystyle {\text{Open}}(X)/U}
1319:. This is because every object in
1255:{\displaystyle {\text{Open}}(X)/U}
526:There is a dual notion called the
14:
952:, it is immediate the categories
1589:{\displaystyle {\text{Aff}}_{A}}
1453:{\displaystyle A/{\text{CRing}}}
1286:{\displaystyle {\text{Open}}(U)}
1156:{\displaystyle {\text{Open}}(X)}
1009:{\displaystyle X/{\mathcal {C}}}
977:{\displaystyle {\mathcal {C}}/X}
679:{\displaystyle X/{\mathcal {C}}}
559:{\displaystyle X/{\mathcal {C}}}
410:such that the following diagram
184:{\displaystyle {\mathcal {C}}/X}
1480:-algebra on a commutative ring
1163:whose objects are open subsets
1080:. In addition, this applies to
876:Many categorical properties of
1648:{\displaystyle {\text{Sch}}/S}
1554:
1548:
1510:
1337:
1331:
1280:
1274:
1241:
1235:
1150:
1144:
1121:{\displaystyle {\mathcal {C}}}
1033:{\displaystyle {\mathcal {C}}}
937:{\displaystyle {\mathcal {C}}}
893:{\displaystyle {\mathcal {C}}}
803:
781:
740:{\displaystyle {\mathcal {C}}}
702:
647:{\displaystyle {\mathcal {C}}}
614:
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459:
403:{\displaystyle {\mathcal {C}}}
365:
336:
314:
311:
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296:
276:{\displaystyle {\mathcal {C}}}
239:
210:
198:
144:{\displaystyle {\mathcal {C}}}
100:{\displaystyle {\mathcal {C}}}
68:{\displaystyle {\mathcal {C}}}
29:covering spaces (espace etale)
1:
19:In mathematics, specifically
1406:The category of commutative
1312:{\displaystyle U\subseteq X}
1293:for the induced topology on
623:{\displaystyle \psi :X\to B}
248:{\displaystyle \pi :A\to X}
1777:
1183:of some topological space
716:{\displaystyle g:B\to B'}
591:{\displaystyle (B,\psi )}
379:{\displaystyle f:A\to A'}
1714:stacks.math.columbia.edu
1600:Overcategories of spaces
1097:Overcategories on a site
566:whose objects are pairs
216:{\displaystyle (A,\pi )}
686:are given by morphisms
349:is given by a morphism
16:Category theory concept
1649:
1619:
1590:
1561:
1520:
1519:{\displaystyle A\to B}
1494:
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866:higher category theory
858:
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749:
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681:
654:. Then, morphisms in
649:
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122:
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111:
87:
55:
35:
1223:, the overcategory
831:
440:
1645:
1615:
1586:
1557:
1516:
1490:
1470:
1450:
1416:
1388:
1368:
1358:is an open subset
1348:
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1252:
1213:
1193:
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1006:
974:
934:
920:. For example, if
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511:
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376:
339:
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181:
141:
127:a fixed object of
117:
107:be a category and
97:
65:
41:
1635:
1618:{\displaystyle S}
1578:
1546:
1536:
1493:{\displaystyle B}
1473:{\displaystyle A}
1448:
1419:{\displaystyle A}
1391:{\displaystyle U}
1371:{\displaystyle V}
1329:
1272:
1233:
1216:{\displaystyle U}
1196:{\displaystyle X}
1176:{\displaystyle U}
1142:
1073:{\displaystyle X}
1053:{\displaystyle X}
913:{\displaystyle X}
862:2-category theory
832:
801:
794:
787:
630:is a morphism in
479:
472:
465:
441:
120:{\displaystyle X}
51:in some category
44:{\displaystyle X}
1768:
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621:
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548:
532:coslice category
522:
520:
519:
514:
512:
491:
480:
477:
473:
470:
466:
463:
451:
432:
409:
407:
406:
401:
399:
398:
386:in the category
385:
383:
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377:
375:
348:
346:
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126:
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118:
106:
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103:
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63:
50:
48:
47:
42:
1776:
1775:
1771:
1770:
1769:
1767:
1766:
1765:
1761:Category theory
1751:
1750:
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1727:
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1703:
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724:
705:
688:
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631:
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568:
567:
536:
535:
530:(also called a
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493:
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351:
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328:
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261:
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193:
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161:
160:
155:(also called a
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128:
109:
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85:
84:
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53:
52:
33:
32:
21:category theory
17:
12:
11:
5:
1774:
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1752:
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1679:
1678:
1676:
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1671:
1669:Comma category
1664:
1661:
1657:Fiber products
1644:
1640:
1614:
1601:
1598:
1583:
1556:
1553:
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1489:
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1101:Recall that a
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157:slice category
138:
116:
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62:
40:
15:
13:
10:
9:
6:
4:
3:
2:
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1429:
1413:
1401:
1399:
1385:
1378:contained in
1365:
1345:
1341:
1334:
1306:
1303:
1300:
1277:
1249:
1245:
1238:
1210:
1190:
1170:
1147:
1131:
1104:
1096:
1091:
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1083:
1067:
1047:
995:
991:
971:
967:
951:
947:
907:
871:
869:
867:
863:
857:
839:
836:
828:
824:
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778:
771:
766:
761:
748:
709:
706:
699:
696:
693:
665:
661:
617:
611:
608:
605:
582:
579:
576:
545:
541:
533:
529:
528:undercategory
523:
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258:
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204:
201:
178:
174:
158:
154:
114:
78:
76:
38:
30:
26:
22:
1740:math/0608040
1728:
1717:. Retrieved
1713:
1704:
1683:
1603:
1405:
1130:Grothendieck
1100:
875:
859:
750:
531:
527:
525:
416:
156:
153:overcategory
152:
82:
18:
944:has finite
1719:2020-10-16
1695:1612.09375
1675:References
1567:, or just
950:coproducts
872:Properties
79:Definition
25:categories
1511:→
1304:⊆
1088:as well.
808:ψ
804:↓
782:↓
779:ψ
703:→
615:→
606:ψ
583:ψ
486:π
482:↓
460:↓
457:π
366:→
330:π
312:→
306:π
240:→
231:π
208:π
1755:Category
1663:See also
1428:algebras
1092:Examples
1086:colimits
1060:or from
946:products
840:′
825:→
811:′
710:′
489:′
449:′
434:→
412:commutes
373:′
333:′
322:′
257:morphism
1082:limits
800:
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598:where
478:
471:
464:
223:where
151:. The
1735:arXiv
1690:arXiv
1447:CRing
255:is a
1545:Spec
1328:Open
1271:Open
1232:Open
1141:Open
1103:site
1084:and
984:and
948:and
864:and
83:Let
1634:Sch
1577:Aff
1535:Aff
723:in
259:in
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1596:.
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1026:C
1002:C
996:/
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837:B
829:g
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767:=
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700:B
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666:/
662:X
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618:B
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574:(
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319:A
315:(
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300:A
297:(
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291:f
269:C
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237:A
234::
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199:(
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175:/
169:C
137:C
115:X
93:C
61:C
39:X
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