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Because the hom-objects need not be sets in an enriched category, one cannot speak of a particular morphism. There is no longer any notion of an identity morphism, nor of a particular composition of two morphisms. Instead, morphisms from the unit to a hom-object should be thought of as selecting an
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in some fixed monoidal category of "hom-objects". In order to emulate the (associative) composition of morphisms in an ordinary category, the hom-category must have a means of composing hom-objects in an associative manner: that is, there must be a binary operation on objects giving us at least the
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ordinary categories where the hom-set carries additional structure beyond being a set. That is, there are operations on, or properties of morphisms that need to be respected by composition (e.g., the existence of 2-cells between morphisms and horizontal composition thereof in a
1802:
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identity, and morphisms from the monoidal product should be thought of as composition. The usual functorial axioms are replaced with corresponding commutative diagrams involving these morphisms.
991:". Commutativity of the latter two diagrams is then the statement that compositions (as defined by the functions °) involving these distinguished individual "identity morphisms in
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1840:
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145:. It is motivated by the observation that, in many practical applications, the hom-set often has additional structure that should be respected, e.g., that of being a
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with the usual cartesian product, the definitions of enriched category, enriched functor, etc... reduce to the original definitions from ordinary category theory.
902:
The notion that an ordinary category must have identity morphisms is replaced by the second and third diagrams, which express identity in terms of left and right
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What is new here is that the above expresses the requirement for associativity without any explicit reference to individual morphisms in the enriched category
861:. Commutativity of the diagram is then merely the statement that both orders of composition give the same result, exactly as required for ordinary categories.
232:. Due to Mac Lane's preference for the letter V in referring to the monoidal category, enriched categories are also sometimes referred to generally as
1579:, –) to the category of sets, so any enriched category has an underlying ordinary category. In many examples (such as those above) this functor is
1739:
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category-like entities that don't themselves have any notion of individual morphism but whose hom-objects have similar compositional aspects (e.g.,
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of morphisms. In an enriched category, the set of morphisms (the hom-set) associated with every pair of objects is replaced by an
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to enriched categories. Enriched functors are then maps between enriched categories which respect the enriched structure.
68:
42:
1944:
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with
Cartesian product as the monoidal operation. (A locally small category is one whose hom-objects are small sets.)
876:— thus making the concept of associativity of composition meaningful in the general case where the hom-objects
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are enriched over themselves, where the morphisms inherit the algebraic structure "pointwise". More generally,
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in the first diagram corresponds to one of the two ways of composing three consecutive individual morphisms
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205:, where the hom-objects are numerical distances and the composition rule provides the triangle inequality).
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Enriched category theory thus encompasses within the same framework a wide variety of structures including
1716:), satisfying enriched versions of the axioms of a functor, viz preservation of identity and composition.
783:, while °, now a function, defines how consecutive morphisms compose. In this case, each path leading to
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and the vertical-composition rule that relates them correspond to the morphisms of the ordinary category
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deemed to have individual morphisms of its own, is not necessarily identifying a specific one.
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1242:, consisting of two objects and a single nonidentity arrow between them that we can write as
279:(alternatively, in situations where the choice of monoidal category needs to be explicit, a
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can be described as an ordinary category with certain additional structure or properties.
8:
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1266:) then simply deny or affirm a particular binary relation on the given pair of objects (
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which are none other than the axioms for ≤ being a preorder. And since all diagrams in
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has for each of its objects by virtue of it being (at least) an ordinary category.
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Note that there are several distinct notions of "identity" being referenced here:
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2011:
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1797:{\displaystyle T_{aa}\circ \operatorname {id} _{a}=\operatorname {id} _{T(a)},}
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1274:); for the sake of having more familiar notation we can write this relation as
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is a set whose elements may be thought of as "individual morphisms" of
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1914:
1365:, are categories enriched over the nonnegative extended real numbers
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content of the enriched category axioms for categories enriched over
868:— again, these diagrams are for morphisms in monoidal category
995:" behave exactly as per the identity rules for ordinary categories.
694:{\displaystyle g\circ _{\textbf {C}}f={^{\circ }}_{abc}(g\otimes f)}
24:
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for ordinary functors. Additionally, one demands that the diagram
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1980:
1923:. Reprints in Theory and Applications of Categories. Vol. 10.
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1963:. Reprints in Theory and Applications of Categories. Vol. 1.
1618:-categories (that is, categories enriched over monoidal category
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as the monoidal operation; the special point of a hom-object Hom(
1017:-theoretic sense, and even then only up to canonical isomorphism
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That is, the associativity requirement is now taken over by the
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1839:
1014:
983:, something we can then think of as the "identity morphism for
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934:
is the category of sets with cartesian product, the morphisms
705:
The first diagram expresses the associativity of composition:
2429:
1410:), and the existence of composition and identity translate to
216:
An enriched category with hom-objects from monoidal category
209:
In the case where the hom-object category happens to be the
1971:
1288:
immediately translate to the following axioms respectively
1238:
are categories enriched over a certain monoidal category,
1534:
as the monoidal operation (thinking of abelian groups as
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together with three commuting diagrams, discussed below.
1960:
Metric Spaces, Generalized Logic, and Closed
Categories
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is the appropriate generalization of the notion of a
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where the composition rule ensures transitivity, or
49:. Unsourced material may be challenged and removed.
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1143:Ordinary categories are categories enriched over (
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1563:can be reinterpreted as a category enriched over
1217:with Cartesian product as the monoidal operation.
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16:Category whose hom sets have algebraic structure
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1209:, by analogy, are categories enriched over (
968:, identify a particular element of each set
1913:
1250:, conjunction as the monoid operation, and
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2614:
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2003:
1989:
1920:Basic Concepts of Enriched Category Theory
1254:as its monoidal identity. The hom-objects
1155:as the monoidal operation, as noted above.
1638:is a map which assigns to each object of
109:Learn how and when to remove this message
1939:. Vol. 5 (2nd ed.). Springer.
1933:Categories for the Working Mathematician
1927:
1846:commute, which is analogous to the rule
1733:commutes, which amounts to the equation
1479:) corresponds to the zero morphism from
1467:, ∧), the category of pointed sets with
960:become functions from the one-point set
1953:
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1013:, being an identity for ⊗ only in the
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1984:
427:{\displaystyle f:I\rightarrow C(a,b)}
1723:In detail, one has that the diagram
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964:and must then, for any given object
556:, used to define the composition of
47:adding citations to reliable sources
18:
2010:
1543:Relationship with monoidal functors
899:any notion of individual morphism.
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1559:, then any category enriched over
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1815:. This is analogous to the rule
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613:{\displaystyle g:b\rightarrow c}
581:{\displaystyle f:a\rightarrow b}
376:{\displaystyle f:a\rightarrow b}
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1189:) and its own composition rule.
1138:Examples of enriched categories
1122:, which is again a morphism of
34:needs additional citations for
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1780:
1704:)) between the hom-objects of
1583:, so a category enriched over
1526:are categories enriched over (
1463:are categories enriched over (
1195:are categories enriched over (
1126:which, even in the case where
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1937:Graduate Texts in Mathematics
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1646:and for each pair of objects
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1402:are essentially distances d(
1167:category of small categories
930:Returning to the case where
141:with objects from a general
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2298:Constructions on categories
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540:for each triple of objects
170:(i.e., making the category
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2405:Higher-dimensional algebra
1567:. Every monoidal category
1087:enriched category identity
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1870:) for ordinary functors.
1551:from a monoidal category
1441:) (triangle inequality)
1359:generalized metric spaces
1207:Locally finite categories
721:of the monoidal category
176:symmetric closed monoidal
1363:pseudoquasimetric spaces
1226:closed monoidal category
1193:Locally small categories
1004:monoidal identity object
222:enriched category over M
2215:Cokernels and quotients
2138:Universal constructions
1571:has a monoidal functor
1555:to a monoidal category
203:Lawvere's metric spaces
2372:Higher category theory
2118:Natural transformation
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1811:is the unit object of
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1712:(which are objects in
1524:preadditive categories
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1213:, ×), the category of
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2655:Monoidal categories
2410:Homotopy hypothesis
2088:Commutative diagram
1510:, and the category
58:"Enriched category"
2123:Universal property
1931:(September 1998).
1929:Mac Lane, Saunders
1888:Mathematics portal
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1153:Cartesian product
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1147:, ×, {•}), the
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1956:
1955:Lawvere, F.W.
1952:
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1946:0-387-98403-8
1942:
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1657:
1653:
1649:
1645:
1642:an object of
1641:
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1598:
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1537:
1533:
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1525:
1521:
1518:over a given
1517:
1516:vector spaces
1513:
1509:
1505:
1501:
1497:
1493:
1490:The category
1489:
1486:
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1474:
1470:
1469:smash product
1466:
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1332:(reflexivity)
1331:
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1005:
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871:
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858:
854:
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814:
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806:
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795:
791:
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776:
772:
768:
763:
760:given by the
758:(×, {•}, ...)
753:
749:
745:
741:
735:
731:
726:
724:
720:
712:
708:
707:
706:
685:
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168:right adjoint
165:
160:
156:
152:
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144:
140:
137:by replacing
136:
132:
128:
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113:
110:
102:
91:
88:
84:
81:
77:
74:
70:
67:
63:
60: –
59:
55:
54:Find sources:
48:
44:
38:
37:
32:This article
30:
26:
21:
20:
2553:
2534:Categorified
2438:n-categories
2399:
2389:Key concepts
2227:Direct limit
2210:Coequalizers
2128:Yoneda lemma
2034:Key concepts
2024:Key concepts
1972:
1959:
1932:
1919:
1867:
1863:
1859:
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1170:
1162:
1159:2-Categories
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1123:
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751:
747:
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729:
727:
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704:
621:
553:
549:
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541:
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527:
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515:
511:
507:
503:
499:
495:
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472:
468:
462:
458:
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450:
445:
435:
387:as an arrow
384:
348:
344:
340:
336:
330:
326:
322:
314:
310:
306:
302:
291:
287:
283:
280:
276:
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263:
259:
255:
251:
247:
243:
234:V-categories
233:
229:
225:
221:
217:
215:
208:
180:
147:vector space
130:
120:
105:
96:
86:
79:
72:
65:
53:
41:Please help
36:verification
33:
2502:-categories
2478:Kan complex
2468:Tricategory
2450:-categories
2340:Subcategory
2098:Exponential
2066:Preadditive
2061:Pre-abelian
1915:Kelly, G.M.
1658:provides a
1215:finite sets
538:composition
127:mathematics
99:August 2019
2644:Categories
2520:3-category
2510:2-category
2483:∞-groupoid
2458:Bicategory
2205:Coproducts
2165:Equalizers
2071:Bicategory
1907:References
1538:-modules).
1530:, ⊗) with
1201:small sets
719:associator
320:an object
272:. Then an
240:Definition
230:M-category
188:2-category
69:newspapers
2569:Symmetric
2514:2-functor
2254:Relations
2177:Pullbacks
1957:(2002) .
1917:(2005) .
1757:∘
683:⊗
659:∘
636:∘
605:→
573:→
486:an arrow
441:an arrow
404:→
368:→
199:preorders
151:morphisms
2629:Glossary
2609:Category
2583:n-monoid
2536:concepts
2192:Colimits
2160:Products
2113:Morphism
2056:Concrete
2051:Additive
2041:Category
1874:See also
1672: :
1660:morphism
1581:faithful
1475:,
1448:,
1437:,
1429:,
1421:,
1406:,
1270:,
1262:,
1185:,
1096: :
1052: :
494: :
473:identity
449: :
292:category
174:or even
139:hom-sets
135:category
2619:Outline
2578:n-group
2543:2-group
2498:Strict
2488:∞-topos
2284:Modules
2222:Pushout
2170:Kernels
2103:Functor
2046:Abelian
1970:at the
1601:functor
1506:over a
1504:modules
904:unitors
732:is the
311:objects
153:, or a
83:scholar
2565:Traced
2548:2-ring
2278:Fields
2264:Groups
2259:Magmas
2147:Limits
1943:
1825:) = id
1807:where
1622:), an
1444:0 ≥ d(
1433:) ≥ d(
1425:) + d(
1211:FinSet
1165:, the
1015:monoid
224:or an
159:object
85:
78:
71:
64:
56:
2559:-ring
2446:Weak
2430:Topos
2274:Rings
1520:field
1500:R-Mod
1244:FALSE
1228:then
1224:is a
1197:SmSet
1151:with
1078:that
483:, and
309:) of
300:class
286:, or
268:be a
250:, ⊗,
129:, an
90:JSTOR
76:books
2249:Sets
1941:ISBN
1708:and
1684:) →
1650:and
1614:are
1610:and
1512:Vect
1465:Set*
1379:iff
1343:sole
1322:TRUE
1302:and
1252:TRUE
1248:TRUE
1085:the
1064:) →
1032:the
1002:the
918:and
897:have
846:and
738:(⊗,
736:and
588:and
518:) →
506:) ⊗
244:Let
62:news
2093:End
2083:CCC
1975:Lab
1819:(id
1696:),
1662:in
1654:in
1606:If
1595:An
1514:of
1502:of
1494:of
1483:to
1357:'s
1220:If
1163:Cat
1145:Set
1118:in
1009:of
987:in
624:as
620:in
552:in
532:in
491:abc
479:in
467:in
434:in
383:in
347:in
335:of
313:of
149:of
121:In
45:by
2646::
2571:)
2567:)(
1935:.
1854:)=
1852:fg
1774:id
1761:id
1680:,
1669:ab
1634:→
1630::
1528:Ab
1492:Ab
1417:d(
1396:,
1383:≥
1375:→
1328:≤
1324:⇒
1314:≤
1310:⇒
1306:≤
1298:≤
1279:≤
1246:→
1173:→
1131:is
1108:,
1100:→
1090:id
1072:,
1060:,
1046:,
1023:,
977:,
954:,
946:→
942::
936:id
906::
885:,
855:,
840:,
831:,
825:,
811:→
807:→
803:→
792:,
773:,
750:,
746:,
742:,
725:.
548:,
544:,
526:,
514:,
502:,
461:,
453:→
443:id
343:,
329:,
303:ob
298:a
262:,
258:,
254:,
236:.
2563:(
2556:n
2554:E
2516:)
2512:(
2500:n
2464:)
2460:(
2448:n
2290:)
2286:(
2280:)
2276:(
2004:e
1997:t
1990:v
1973:n
1949:.
1868:g
1866:(
1864:F
1862:)
1860:f
1858:(
1856:F
1850:(
1848:F
1834:)
1832:a
1830:(
1828:F
1822:a
1817:F
1813:M
1809:I
1792:,
1787:)
1784:a
1781:(
1778:T
1770:=
1765:a
1752:a
1749:a
1745:T
1714:M
1710:D
1706:C
1702:b
1700:(
1698:T
1694:a
1692:(
1690:T
1688:(
1686:D
1682:b
1678:a
1676:(
1674:C
1667:T
1664:M
1656:C
1652:b
1648:a
1644:D
1640:C
1636:D
1632:C
1628:T
1624:M
1620:M
1616:M
1612:D
1608:C
1585:M
1577:I
1575:(
1573:M
1569:M
1565:N
1561:M
1557:N
1553:M
1536:Z
1487:.
1485:B
1481:A
1477:B
1473:A
1452:)
1450:a
1446:a
1439:c
1435:a
1431:b
1427:a
1423:c
1419:b
1408:b
1404:a
1400:)
1398:b
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1392:(
1390:R
1385:s
1381:r
1377:s
1373:r
1368:R
1349:.
1347:2
1339:2
1330:a
1326:a
1316:c
1312:a
1308:b
1304:a
1300:c
1296:b
1286:2
1281:b
1277:a
1272:b
1268:a
1264:b
1260:a
1258:(
1256:2
1240:2
1230:C
1222:C
1187:b
1183:a
1181:(
1179:C
1175:b
1171:a
1128:C
1124:M
1120:C
1116:a
1112:)
1110:a
1106:a
1104:(
1102:C
1098:I
1093:a
1080:M
1076:)
1074:b
1070:a
1068:(
1066:C
1062:b
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1056:(
1054:C
1050:)
1048:b
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1042:(
1040:C
1037:1
1029:.
1027:)
1025:ρ
1021:λ
1019:(
1011:M
1007:I
993:C
989:C
985:a
981:)
979:a
975:a
973:(
971:C
966:a
962:I
958:)
956:a
952:a
950:(
948:C
944:I
939:a
932:M
893:C
889:)
887:b
883:a
881:(
879:C
874:C
870:M
866:C
859:)
857:d
853:c
851:(
849:C
844:)
842:c
838:b
836:(
834:C
829:)
827:b
823:a
821:(
819:C
813:d
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805:b
801:a
796:)
794:d
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786:C
781:C
777:)
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771:a
769:(
767:C
754:)
752:ρ
748:λ
744:α
740:I
730:M
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686:f
680:g
677:(
672:c
669:b
666:a
650:=
647:f
641:C
632:g
622:C
608:c
602:b
599::
596:g
576:b
570:a
567::
564:f
554:C
550:c
546:b
542:a
534:M
530:)
528:c
524:a
522:(
520:C
516:b
512:a
510:(
508:C
504:c
500:b
498:(
496:C
488:°
481:C
477:a
469:M
465:)
463:a
459:a
457:(
455:C
451:I
446:a
438:,
436:M
422:)
419:b
416:,
413:a
410:(
407:C
401:I
398::
395:f
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371:b
365:a
362::
359:f
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333:)
331:b
327:a
325:(
323:C
317:,
315:C
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305:(
290:-
288:M
284:M
277:C
266:)
264:ρ
260:λ
256:α
252:I
248:M
246:(
218:M
194:)
112:)
106:(
101:)
97:(
87:·
80:·
73:·
66:·
39:.
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