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of the diagonal functor. Similarly, the colimit functor (which exists if the category is cocomplete) is the left-adjoint of the diagonal functor. For example, the diagonal functor
237:
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1049:. These cones and their factorizations correspond precisely to the objects and morphisms of the
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can be employed to give a succinct alternate description of the product of objects
1698:
21:
543:{\displaystyle \Delta :{\mathcal {C}}\rightarrow {\mathcal {C}}^{\mathcal {J}}}
1452:{\displaystyle {\mathcal {C}}\rightarrow {\mathcal {C}}\times {\mathcal {C}}}
875:{\displaystyle {\mathcal {C}}\rightarrow {\mathcal {C}}\times {\mathcal {C}}}
64:{\displaystyle {\mathcal {C}}\rightarrow {\mathcal {C}}\times {\mathcal {C}}}
1464:
123:
1601:
1182:
933:{\displaystyle {\mathcal {F}}:{\mathcal {J}}\rightarrow {\mathcal {C}}}
890:
127:
1529:
Sheaves in geometry and logic a first introduction to topos theory
1570:
1354:), then the operation of taking limits is itself a functor from
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399:{\displaystyle \Delta _{a}:{\mathcal {J}}\to {\mathcal {C}}}
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1459:described above is the left-adjoint of the binary
1451:
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1342:
1318:
1294:
1267:
1238:{\displaystyle ({\mathcal {F}}\downarrow \Delta )}
1237:
1201:
1173:
1141:{\displaystyle (\Delta \downarrow {\mathcal {F}})}
1140:
1104:
1081:{\displaystyle (\Delta \downarrow {\mathcal {F}})}
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110:
63:
1500:Awodey, Steve (2006). "Functors and Naturality".
1268:{\displaystyle {\mathcal {F}}\rightarrow \Delta }
1174:{\displaystyle \Delta \rightarrow {\mathcal {F}}}
111:{\displaystyle \Delta (a)=\langle a,a\rangle }
1734:
1586:
1531:. New York: Springer-Verlag. pp. 20–23.
1380:{\displaystyle {\mathcal {C}}^{\mathcal {J}}}
970:{\displaystyle \Delta _{a}\to {\mathcal {F}}}
726:{\displaystyle {\mathcal {C}}^{\mathcal {J}}}
301:{\displaystyle {\mathcal {C}}^{\mathcal {J}}}
8:
239:. The arrow comprises the projection maps.
226:
214:
105:
93:
1555:. University of Chicago Press. p. 16.
1527:Mac Lane, Saunders; Moerdijk, Ieke (1992).
1209:is an initial object in the comma category
1741:
1727:
1593:
1579:
1571:
885:Diagonal functors provide a way to define
1510:10.1093/acprof:oso/9780198568612.003.0007
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810:). Thus, for example, in the case that
1326:has a limit (which will be the case if
838:with two objects, the diagonal functor
1549:A Concise Course in Algebraic Topology
7:
1695:
1693:
1463:and the right-adjoint of the binary
232:{\displaystyle \langle a,b\rangle }
1713:. You can help Knowledge (XXG) by
1262:
1229:
1158:
1122:
1062:
948:
602:
508:
367:
308:, the objects of which are called
194:
78:
14:
1697:
646:{\displaystyle f:a\rightarrow b}
1429:
1404:{\displaystyle {\mathcal {C}}}
1343:{\displaystyle {\mathcal {C}}}
1319:{\displaystyle {\mathcal {C}}}
1295:{\displaystyle {\mathcal {J}}}
1259:
1232:
1226:
1216:
1202:{\displaystyle {\mathcal {F}}}
1161:
1135:
1125:
1119:
1105:{\displaystyle {\mathcal {F}}}
1075:
1065:
1059:
1042:{\displaystyle {\mathcal {F}}}
1014:{\displaystyle {\mathcal {C}}}
957:
920:
852:
827:{\displaystyle {\mathcal {J}}}
770:{\displaystyle {\mathcal {J}}}
670:{\displaystyle {\mathcal {C}}}
637:
587:{\displaystyle {\mathcal {C}}}
521:
467:{\displaystyle {\mathcal {J}}}
423:{\displaystyle {\mathcal {J}}}
386:
349:{\displaystyle {\mathcal {C}}}
265:{\displaystyle {\mathcal {J}}}
154:{\displaystyle {\mathcal {C}}}
87:
81:
41:
1:
1411:. The limit functor is the
940:, a natural transformation
803:{\displaystyle \eta _{j}=f}
614:{\displaystyle \Delta _{a}}
1781:
1692:
1245:, i.e., a universal arrow
406:that maps every object in
187:is a universal arrow from
1609:
1477:Diagram (category theory)
242:More generally, given a
180:{\displaystyle a\times b}
1112:is a terminal object in
733:(given for every object
272:, one may construct the
621:, and to each morphism
550:assigns to each object
501:. The diagonal functor
200:{\displaystyle \Delta }
1709:-related article is a
1644:Essentially surjective
1482:Cone (category theory)
1453:
1405:
1381:
1344:
1320:
1296:
1278:If every functor from
1269:
1239:
1203:
1175:
1142:
1106:
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1043:
1015:
991:
971:
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679:natural transformation
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468:
450:and every morphism in
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65:
1765:Category theory stubs
1454:
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1204:
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1016:
992:
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893:of diagrams. Given a
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693:{\displaystyle \eta }
672:
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496:
494:{\displaystyle 1_{a}}
469:
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1504:. pp. 125–158.
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1546:May, J. P. (1999).
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312:. For each object
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1659:Full and faithful
1519:978-0-19-856861-2
1487:Diagonal morphism
1465:coproduct functor
1088:, and a limit of
990:{\displaystyle a}
977:(for some object
836:discrete category
746:{\displaystyle j}
563:{\displaystyle a}
443:{\displaystyle a}
325:{\displaystyle a}
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358:constant diagram
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274:functor category
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26:diagonal functor
1780:
1779:
1775:
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1771:
1770:
1769:
1760:Category theory
1750:
1749:
1748:
1747:
1707:category theory
1690:
1688:
1683:
1605:
1599:
1569:
1563:
1552:
1545:
1539:
1526:
1520:
1502:Category Theory
1499:
1495:
1473:
1461:product functor
1417:
1416:
1389:
1388:
1361:
1356:
1355:
1328:
1327:
1304:
1303:
1280:
1279:
1247:
1246:
1211:
1210:
1187:
1186:
1153:
1152:
1150:universal arrow
1114:
1113:
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1053:
1027:
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998:
979:
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811:
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475:
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29:
28:
18:category theory
12:
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5:
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1099:
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1064:
1061:
1051:comma category
1036:
1021:) is called a
1008:
986:
964:
959:
954:
950:
927:
922:
917:
912:
907:
882:is recovered.
869:
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796:
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787:
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259:
247:index category
228:
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196:
176:
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148:
107:
104:
101:
98:
95:
92:
89:
86:
83:
80:
58:
53:
48:
43:
38:
20:, a branch of
13:
10:
9:
6:
4:
3:
2:
1777:
1766:
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1724:
1718:
1716:
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1708:
1703:
1700:
1696:
1691:
1680:
1677:
1675:
1674:Representable
1672:
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1584:
1582:
1577:
1576:
1573:
1564:
1562:0-226-51183-9
1558:
1551:
1550:
1544:
1540:
1538:9780387977102
1534:
1530:
1525:
1521:
1515:
1511:
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1498:
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1483:
1480:
1478:
1475:
1474:
1470:
1468:
1466:
1462:
1439:
1414:
1413:right-adjoint
1353:
1276:
1184:
1151:
1052:
1024:
984:
952:
910:
896:
892:
888:
883:
862:
837:
797:
794:
789:
785:
740:
687:
680:
640:
634:
631:
628:
606:
557:
511:
486:
482:
437:
376:
371:
359:
356:, there is a
319:
311:
275:
248:
245:
240:
223:
220:
217:
174:
171:
168:
137:
133:
129:
125:
121:
118:, which maps
102:
99:
96:
90:
84:
51:
27:
23:
19:
1715:expanding it
1704:
1689:
1633:
1624:Conservative
1548:
1528:
1501:
1277:
1181:. Dually, a
884:
594:the diagram
241:
161:: a product
131:
71:is given by
25:
15:
122:as well as
22:mathematics
1754:Categories
1493:References
1148:, i.e., a
1654:Forgetful
1440:×
1430:→
1263:Δ
1260:→
1230:Δ
1227:↓
1162:→
1159:Δ
1126:↓
1123:Δ
1066:↓
1063:Δ
958:→
949:Δ
921:→
863:×
853:→
786:η
688:η
638:→
603:Δ
522:→
509:Δ
387:→
368:Δ
227:⟩
215:⟨
195:Δ
172:×
124:morphisms
106:⟩
94:⟨
79:Δ
52:×
42:→
1669:Monoidal
1639:Enriched
1634:Diagonal
1614:Additive
1471:See also
1352:complete
891:colimits
310:diagrams
136:category
126:. This
1664:Logical
1629:Derived
1619:Adjoint
1602:Functor
1183:colimit
895:diagram
128:functor
120:objects
1679:Smooth
1559:
1535:
1516:
887:limits
132:within
24:, the
1705:This
1649:Exact
1604:types
1553:(PDF)
834:is a
244:small
1711:stub
1557:ISBN
1533:ISBN
1514:ISBN
1467:.
1025:for
1023:cone
889:and
677:the
134:the
1506:doi
1387:to
1350:is
1302:to
1185:of
997:of
777:by
753:of
700:in
653:in
570:of
474:to
430:to
332:in
207:to
16:In
1756::
1512:.
1275:.
1742:e
1735:t
1728:v
1717:.
1594:e
1587:t
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1565:.
1541:.
1522:.
1508::
1445:C
1435:C
1425:C
1397:C
1372:J
1365:C
1336:C
1312:C
1288:J
1255:F
1233:)
1222:F
1217:(
1195:F
1167:F
1136:)
1131:F
1120:(
1098:F
1076:)
1071:F
1060:(
1035:F
1007:C
985:a
963:F
953:a
926:C
916:J
911::
906:F
868:C
858:C
848:C
820:J
798:f
795:=
790:j
763:J
741:j
718:J
711:C
663:C
641:b
635:a
632::
629:f
607:a
580:C
558:a
535:J
528:C
517:C
512::
487:a
483:1
460:J
438:a
416:J
392:C
382:J
377::
372:a
342:C
320:a
293:J
286:C
258:J
224:b
221:,
218:a
175:b
169:a
147:C
103:a
100:,
97:a
91:=
88:)
85:a
82:(
57:C
47:C
37:C
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