465:
977:
1074:
417:
Note that fixing the first argument of Hom naturally gives rise to a covariant functor and fixing the second argument naturally gives a contravariant functor. This is an artifact of the way in which one must compose the morphisms.
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which is contravariant in the first argument and covariant in the second. Equivalently, we may say that Hom(–, –) is a bifunctor
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1885:
1605:
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559:(–, –) is sometimes used for Hom(–, –) in order to emphasize the category forming the domain.
1854:
660:
natural transformation between Hom functors is of this form. In other words, the Hom functors give rise to a
1265:
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683:
Some categories may possess a functor that behaves like a Hom functor, but takes values in the category
1308:
661:
347:
221:
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1178:
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1130:
1533:
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992:
36:
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972:{\displaystyle \operatorname {Hom} (I,\operatorname {hom} (-,-))\simeq \operatorname {Hom} (-,-)}
1069:{\displaystyle \operatorname {Hom} (X,Y\Rightarrow Z)\simeq \operatorname {Hom} (X\otimes Y,Z)}
592:
430:
187:
68:
64:
1725:
1581:, into limits. In a certain sense, this can be taken as the definition of a limit or colimit.
1160:
1082:
1781:
1351:
1183:
885:
270:
1354:(or representable copresheaf); likewise, a contravariant functor equivalent to Hom(–,
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1233:
8:
1335:
1112:
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and have numerous applications in category theory and other branches of mathematics.
1776:
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95:
44:
1900:
1124:
893:
24:
1123:. In other words, in a closed monoidal category, the internal Hom functor is an
1659:
506:
The commutativity of the above diagram implies that Hom(–, –) is a
1957:
1941:
1715:
1655:
1643:
573:
Referring to the above commutative diagram, one observes that every morphism
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988:
815:
to emphasize its functorial nature, or sometimes merely in lower-case:
20:
1374:
1100:
877:{\displaystyle \operatorname {hom} (-,-):C^{\text{op}}\times C\to C.}
675:(covariant or contravariant depending on which Hom functor is used).
507:
996:
892:
Categories that possess an internal Hom functor are referred to as
32:
1523:{\displaystyle \operatorname {hom} (-,X)\colon C^{\text{op}}\to C}
1260:
Internal Homs, when chained together, form a language, called the
805:{\displaystyle \mathop {\Rightarrow } :C^{\text{op}}\times C\to C}
1815:, and this encodes the arrow-reversing behaviour of Hom(–,
1558:
40:
28:
464:
1409:{\displaystyle \operatorname {id} _{C}\colon C\nrightarrow C}
1945:
1933:
1466:{\displaystyle \operatorname {hom} (X,-)\colon C\to C}
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16:
Functor mapping hom objects to an underlying category
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1955:
752:{\displaystyle \left:C^{\text{op}}\times C\to C}
1264:of the category. The most famous of these are
1127:to the internal product functor. The object
762:to emphasize its product-like nature, or as
991:of the closed category. For the case of a
1863:
1899:
1847:Categories for the Working Mathematician
1841:
1867:Topoi, the Categorial Analysis of Logic
1361:Note that Hom(–, –) :
691:. Such a functor is referred to as the
678:
633:gives rise to a natural transformation
1956:
1907:. Vol. 2 (2nd ed.). Dover.
1278:closed symmetric monoidal categories
1276:, which is the internal language of
1268:, which is the internal language of
54:
1828:Jacobson (2009), p. 149, Prop. 3.9.
1611:
1419:The internal hom functor preserves
1358:) might be called corepresentable.
71:for which hom-classes are actually
13:
463:
14:
1975:
1923:
562:
369:{\displaystyle g\mapsto g\circ h}
243:{\displaystyle g\mapsto f\circ g}
1600:can be given the structure of a
1288:Note that a functor of the form
995:, this extends to the notion of
1822:
1793:
1514:
1498:
1486:
1473:sends limits to limits, while
1457:
1448:
1436:
1219:{\displaystyle Y\Rightarrow Z}
1210:
1146:{\displaystyle Y\Rightarrow Z}
1137:
1063:
1045:
1033:
1027:
1015:
966:
954:
942:
939:
927:
912:
865:
843:
831:
796:
773:
743:
354:
228:
174:, –) maps each morphism
94:we define two functors to the
1:
1849:(Second ed.). Springer.
1834:
1606:environment (or reader) monad
1550:{\displaystyle C^{\text{op}}}
1283:
526:Hom(–, –) :
288:to the set of morphisms, Hom(
159:to the set of morphisms, Hom(
1266:simply typed lambda calculus
433:. For any pair of morphisms
425:, –) and Hom(–,
47:. These functors are called
7:
1864:Goldblatt, Robert (2006) .
1765:
1604:; this monad is called the
1270:Cartesian closed categories
10:
1980:
1642:, –) is a covariant
1230:, and is often written as
695:, and is often written as
664:embedding of the category
566:
421:The pair of functors Hom(
400:The functor Hom(–,
39:) give rise to important
1787:
1735:{\displaystyle \otimes }
1170:{\displaystyle \otimes }
1105:internal product functor
1092:{\displaystyle \otimes }
993:closed monoidal category
1193:{\displaystyle \times }
645:) → Hom(–,
606:, –) → Hom(
1799:Also commonly denoted
1736:
1575:
1551:
1524:
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1410:
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1171:
1147:
1111:. The isomorphism is
1093:
1070:
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806:
753:
641:) : Hom(–,
602:, –) : Hom(
593:natural transformation
468:
457:the following diagram
370:
244:
65:locally small category
1782:Representable functor
1737:
1576:
1552:
1525:
1468:
1411:
1352:representable functor
1252:
1250:{\displaystyle Z^{Y}}
1221:
1195:
1172:
1148:
1094:
1071:
974:
886:Category of relations
879:
807:
754:
467:
404:) is also called the
371:
303:) maps each morphism
271:contravariant functor
245:
1870:(Revised ed.).
1726:
1565:
1534:
1477:
1427:
1381:
1342:, –) for some
1336:naturally isomorphic
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1204:
1184:
1161:
1131:
1083:
1006:
903:
822:
769:
702:
693:internal Hom functor
687:itself, rather than
679:Internal Hom functor
348:
222:
614:and every morphism
553:. The notation Hom
429:) are related in a
280:) maps each object
1872:Dover Publications
1845:(September 1998).
1843:Mac Lane, Saunders
1732:
1592:, –) :
1571:
1547:
1520:
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1406:
1274:linear type system
1247:
1228:exponential object
1216:
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1167:
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1066:
969:
884:For examples, see
874:
802:
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240:
110:, –) :
23:, specifically in
1914:978-0-486-47187-7
1881:978-0-486-45026-1
1813:opposite category
1692:. The functor Hom
1574:{\displaystyle C}
1544:
1511:
1262:internal language
1179:Cartesian product
1109:monoidal category
894:closed categories
856:
787:
734:
715:
662:full and faithful
610:′, –)
547:opposite category
407:functor of points
396:
395:
138:covariant functor
55:Formal definition
1971:
1918:
1901:Jacobson, Nathan
1896:
1894:
1893:
1884:. Archived from
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1797:
1777:Functor category
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1722:functor –
1650:to the category
1628:is an object of
1622:abelian category
1612:Other properties
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1554:
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584:′ →
471:Both paths send
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96:category of sets
82:For all objects
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1658:. It is exact
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1888:on 2020-03-21
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1942:Internal Hom
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1890:. Retrieved
1886:the original
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1811:denotes the
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49:hom-functors
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1930:Hom functor
1772:Ext functor
1586:endofunctor
1423:; that is,
1107:defining a
989:unit object
147:, –)
21:mathematics
1892:2009-11-25
1835:References
1667:projective
1644:left-exact
1632:, then Hom
1557:, that is
1375:profunctor
1322:A functor
1317:copresheaf
1284:Properties
1272:, and the
453:′ →
342:) given by
273:given by:
269:This is a
216:) given by
140:given by:
136:This is a
1730:⊗
1515:→
1502::
1490:−
1484:
1458:→
1452::
1446:−
1434:
1401:↛
1395::
1296:) :
1211:⇒
1188:×
1165:⊗
1138:⇒
1101:bifunctor
1087:⊗
1052:⊗
1043:
1037:≃
1028:⇒
1013:
964:−
958:−
952:
946:≃
937:−
931:−
925:
910:
866:→
860:×
841:−
835:−
829:
797:→
791:×
774:⇒
744:→
738:×
717:−
711:−
668:into the
508:bifunctor
376:for each
361:∘
355:↦
250:for each
235:∘
229:↦
124:) :
33:morphisms
1964:Functors
1958:Category
1903:(2009).
1807:, where
1803:→
1766:See also
1596:→
1559:colimits
1369:→
1334:that is
1330:→
1326: :
1309:presheaf
1300:→
1157:. When
1115:in both
997:currying
649:′)
625:→
621: :
580: :
534:→
495: :
475: :
459:commutes
449: :
437: :
334:) → Hom(
307: :
208:) → Hom(
188:function
178: :
128:→
114:→
75:and not
69:category
67:(i.e. a
41:functors
35:between
29:hom-sets
1944:at the
1932:at the
1718:to the
1716:adjoint
1684:a left
1365:×
1338:to Hom(
1177:is the
1113:natural
987:is the
629:′
545:is the
530:×
380:in Hom(
254:in Hom(
186:to the
43:to the
37:objects
1911:
1878:
1853:
1690:module
1620:is an
1421:limits
1103:, the
1079:where
983:where
714:
541:where
1788:Notes
1676:be a
1602:monad
1373:is a
1307:is a
1099:is a
658:every
510:from
63:be a
1909:ISBN
1876:ISBN
1851:ISBN
1680:and
1678:ring
1672:Let
1624:and
1588:Hom(
1584:The
1119:and
598:Hom(
499:′ →
318:Hom(
192:Hom(
170:Hom(
149:maps
143:Hom(
106:Hom(
86:and
73:sets
59:Let
1949:Lab
1937:Lab
1805:Set
1756:Mod
1714:is
1704:Mod
1665:is
1654:of
1616:If
1598:Set
1594:Set
1561:in
1481:hom
1431:hom
1371:Set
1346:in
1332:Set
1302:Set
1040:Hom
1010:Hom
949:Hom
922:hom
907:Hom
826:hom
689:Set
673:Set
549:to
536:Set
520:Set
518:to
503:′.
483:to
284:in
155:in
130:Set
116:Set
90:in
79:).
19:In
1960::
1874:.
1819:).
1762:.
1754:→
1752:Ab
1750::
1712:Ab
1710:→
1669:.
1652:Ab
1608:.
1543:op
1510:op
1416:.
1386:id
1319:.
1280:.
1257:.
855:op
786:op
733:op
514:×
479:→
461::
441:→
414:.
388:).
384:,
338:,
330:,
322:,
311:→
292:,
262:).
258:,
212:,
204:,
196:,
182:→
163:,
27:,
1947:n
1935:n
1917:.
1895:.
1859:.
1817:B
1809:C
1801:C
1760:R
1758:-
1748:M
1744:R
1708:R
1706:-
1700:M
1698:(
1695:R
1688:-
1686:R
1682:M
1674:R
1663:A
1648:A
1640:A
1638:(
1635:A
1630:A
1626:A
1618:A
1590:E
1569:C
1539:C
1518:C
1506:C
1499:)
1496:X
1493:,
1487:(
1461:C
1455:C
1449:)
1443:,
1440:X
1437:(
1404:C
1398:C
1390:C
1367:C
1363:C
1356:A
1348:C
1344:A
1340:A
1328:C
1324:F
1313:A
1298:C
1294:A
1243:Y
1239:Z
1214:Z
1208:Y
1141:Z
1135:Y
1121:Z
1117:X
1064:)
1061:Z
1058:,
1055:Y
1049:X
1046:(
1034:)
1031:Z
1025:Y
1022:,
1019:X
1016:(
985:I
979:,
967:)
961:,
955:(
943:)
940:)
934:,
928:(
919:,
916:I
913:(
888:.
872:.
869:C
863:C
851:C
847::
844:)
838:,
832:(
800:C
794:C
782:C
778::
747:C
741:C
729:C
725::
721:]
707:[
685:C
666:C
647:B
643:B
639:f
627:B
623:B
619:f
608:A
604:A
600:h
586:A
582:A
578:h
556:C
551:C
543:C
532:C
528:C
516:C
512:C
501:B
497:A
493:h
489:g
485:f
481:B
477:A
473:g
455:A
451:A
447:h
443:B
439:B
435:f
427:B
423:A
412:B
402:B
386:B
382:Y
378:g
364:h
358:g
352:g
340:B
336:X
332:B
328:Y
324:B
320:h
313:Y
309:X
305:h
301:B
296:)
294:B
290:X
286:C
282:X
278:B
260:X
256:A
252:g
238:g
232:f
226:g
214:Y
210:A
206:X
202:A
198:f
194:A
184:Y
180:X
176:f
172:A
167:)
165:X
161:A
157:C
153:X
145:A
126:C
122:B
112:C
108:A
92:C
88:B
84:A
61:C
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