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| In [[category theory]], a branch of [[mathematics]], an '''enriched category''' generalizes the idea of a [[category (mathematics)|category]] by replacing [[hom-set]]s with objects from a general [[monoidal category]]. 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 [[vector space]] of morphisms, or a [[topological space]] of morphisms. In an enriched category, the set of morphisms (the hom-set) associated with every pair of objects is replaced by an opaque [[object (category theory)|object]] 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 structure of a [[monoidal category]], though in some contexts the operation may also need to be commutative and perhaps also to have a right adjoint (i.e., making the category [[symmetric monoidal category|symmetric monoidal]] or even [[cartesian closed]], respectively).
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| Enriched category theory thus encompasses within the same framework a wide variety of structures including
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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 [[2-category]], or the addition operation on morphisms in an [[abelian category]])
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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., [[preorder]]s where the composition rule ensures transitivity, or [[pseudoquasimetric space|Lawvere's metric spaces]], where the hom-objects are numerical distances and the composition rule provides the triangle inequality).
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| In the case where the hom-object category happens to be the [[category of sets]] with the usual cartesian product, the definitions of enriched category, enriched functor, etc... reduce to the original definitions from ordinary category theory.
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| An enriched category with hom-objects from monoidal category '''M''' is said to be an '''enriched category over M''' or an '''enriched category in M''', or simply an '''M-category'''. Due to MacLane's preference for the letter V in referring to the monoidal category, enriched categories are also sometimes referred to generally as '''V-categories'''.
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| ==Definition==
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| Let ('''M''',⊗,''I'',<math>\alpha</math>, <math>\lambda</math>, <math>\rho</math>) be a [[monoidal category]]. Then an ''enriched category'' '''C''' (alternatively, in situations where the choice of monoidal category needs to be explicit, a ''category enriched over '''M''''', or '''''M'''-category''), consists of
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| * a [[class (set theory)|class]] ''ob''('''C''') of ''objects'' of '''C''',
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| * an object '''C'''(''a'',''b'') of '''M''' for every pair of objects ''a'',''b'' in '''C''',
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| * an arrow {{math|id<sub>''a''</sub>}}:''I'' → '''C'''(''a'',''a'') in '''M''' designating an ''identity'' for every object ''a'' in '''C''', and
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| * an arrow {{math|°<sub>''abc''</sub>}}:'''C'''(''b'',''c'')⊗'''C'''(''a'',''b'') → '''C'''(''a'',''c'') in '''M''' designating a ''composition'' for each triple of objects ''a'',''b'',''c'' in '''C''',
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| together with three commuting diagrams, discussed below. The first diagram expresses the associativity of composition:
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| :[[Image:Math-enriched category associativity.svg]]
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| That is, the associativity requirement is now taken over by the [[associator]] of the hom-category.
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| For the case that '''M''' is the [[category of sets]] and {{math|(⊗,''I'',α,λ,ρ)}} is {{math|(×, {•}, …)}} is the monoidal structure given by the [[cartesian product]], the terminal single-point set, and the canonical isomorphisms they induce, then each ''C(a,b)'' is a set whose elements may be thought of as "individual morphisms" of ''C'', while °, now a function, defines how consecutive morphisms compose. In this case, each path leading to ''C(a,d)'' in the first diagram corresponds to one of the two ways of composing three consecutive individual morphisms from ''a'' → ''b'' → ''c'' → ''d'' from ''C(a,b)'',''C(b,c)'' and ''C(c,d)''. Commutativity of the diagram is then merely the statement that both orders of composition give the same result, exactly as required for ordinary categories.
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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 '''C''' — again, these diagrams are for morphisms in hom-category '''M''', and not in '''C''' — thus making the concept of associativity of composition meaningful in the general case where the hom-objects ''C(a,b)'' are abstract, and ''C'' itself need not even ''have'' any notion of individual morphism.
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| 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 [[unitor]]s:
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| :[[File:Math-enriched category identity1.svg]]
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| and
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| :[[File:Math-enriched category identity2.svg]]
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| Returning to the case where '''M''' is the category of sets with cartesian product, the morphisms {{math|id<sub>''a''</sub>: ''I'' → ''C(a,a)''}} become functions from the one-point set ''I'' and must then, for any given object ''a'', identify a particular element of each set ''C(a,a)'', something we can then think of as the "identity morphism for ''a'' in '''C'''". Commutativity of the latter two diagrams is then the statement that compositions (as defined by the functions °) involving these distinguished individual "identity morphisms in '''C'''" behave exactly as per the identity rules for ordinary categories.
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| Note that there are several distinct notions of "identity" being referenced here:
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| * the ''monoidal identity object'' {{math|I}} of '''M''', being an identity for ⊗ only in the [[monoid]]-theoretic sense, and even then only up to canonical isomorphism (λ, ρ).
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| * the ''identity morphism'' {{math|1<sub>''C(a,b)''</sub>:''C(a,b)'' → ''C(a,b)''}} that '''M''' has for each of its objects by virtue of it being (at least) an ordinary category.
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| * the enriched category ''identity'' {{math|id<sub>a</sub>:I → C(a,a)}} for each object '''a''' in '''C''', which is again a morphism of '''M''' which, even in the case where '''C''' ''is'' deemed to have individual morphisms of its own, is not necessarily identifying a specific one.
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| ==Examples of enriched categories==
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| * Ordinary categories are categories enriched over ('''Set''', ×, {•}), the [[category of sets]] with [[Cartesian product]] as the monoidal operation, as noted above.
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| * [[2-category|2-Categories]] are categories enriched over '''Cat''', the [[category of small categories]], with monoidal structure being given by cartesian product. In this case the 2-cells between morphisms ''a'' → ''b'' and the vertical-composition rule that relates them correspond to the morphisms of the ordinary category ''C(a,b)'' and its own composition rule.
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| * [[Locally small category|Locally small categories]] are categories enriched over ('''SmSet''', ×), the category of [[small set (category theory)|small sets]] with Cartesian product as the monoidal operation. (A locally small category is one whose hom-objects are small sets.)
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| * [[Locally finite category|Locally finite categories]], by analogy, are categories enriched over ('''FinSet''', ×), the category of [[finite set]]s with Cartesian product as the monoidal operation.
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| * [[Preordered set]]s are categories enriched over a certain monoidal category, '''2''', consisting of two objects and a single nonidentity arrow between them that we can write as ''FALSE'' → ''TRUE'', conjunction as the monoid operation, and ''TRUE'' as its monoidal identity. The hom-objects '''2'''(''a'',''b'') then simply deny or affirm a particular binary relation on the given pair of objects (''a'',''b''); for the sake of having more familiar notation we can write this relation as ''a''≤''b''. The existence of the compositions and identity required for a category enriched over '''2''' immediately translate to the following axioms respectively
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| ::''b'' ≤ ''c'' and ''a'' ≤ ''b'' ⇒ ''a'' ≤ ''c'' (transitivity)
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| ::''TRUE'' ⇒ ''a'' ≤ ''a'' (reflexivity)
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| :which are none other than the axioms for ≤ being a preorder. And since all diagrams in '''2''' commute, this is the ''sole'' content of the enriched category axioms for categories enriched over '''2'''.
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| * [[William Lawvere]]'s generalized metric spaces, also known as [[Metric (mathematics)#Pseudoquasimetrics|pseudoquasimetric spaces]], are categories enriched over the nonnegative extended real numbers {{math|'''R'''<sup>+∞</sup>}}, where the latter is given ordinary category structure via the inverse of its usual ordering (i.e., there exists a morphism ''r'' → ''s'' iff ''r'' ≥ ''s'') and a monoidal structure via addition (+) and zero (0). The hom-objects {{math|'''R'''<sup>+∞</sup>(''a'',''b'')}} are essentially distances d(''a'',''b''), and the existence of composition and identity translate to
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| ::d(''b'',''c'') + d(''a'',''b'') ≥ d(''a'',''c'') (triangle inequality)
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| ::0 ≥ d(''a'',''a'')
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| * Categories with [[zero morphism]]s are categories enriched over ('''Set*''', ∧), the category of pointed sets with [[smash product]] as the monoidal operation; the special point of a hom-object Hom(''A'',''B'') corresponds to the zero morphism from ''A'' to ''B''.
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| * [[preadditive category|Preadditive categories]] are categories enriched over ('''Ab''', ⊗), the [[category of abelian groups]] with tensor product as the monoidal operation.
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| ==Relationship with monoidal functors==
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| If there is a [[monoidal functor]] from a monoidal category '''M''' to a monoidal category '''N''', then any category enriched over '''M''' can be reinterpreted as a category enriched over '''N'''.
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| Every monoidal category '''M''' has a monoidal functor '''M'''(''I'', –) to the category of sets, so any enriched category has an underlying ordinary category. In many examples (such as those above) this functor is [[faithful functor|faithful]], so a category enriched over '''M''' can be described as an ordinary category with certain additional structure or properties.
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| == Enriched functors ==
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| An '''enriched functor''' is the appropriate generalization of the notion of a [[functor]] to enriched categories. Enriched functors are then maps between enriched categories which respect the enriched structure.
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| If ''C'' and ''D'' are '''M'''-categories (that is, categories enriched over monoidal category '''M'''), an '''M'''-enriched functor ''T'': ''C'' → ''D'' is a map which assigns to each object of ''C'' an object of ''D'' and for each pair of objects ''a'' and ''b'' in ''C'' provides a [[morphism]] in '''M''' ''T''<sub>''ab''</sub>: ''C''(''a'',''b'') → ''D''(''T''(''a''),''T''(''b'')) between the hom-objects of ''C'' and ''D'' (which are objects in '''M'''), satisfying enriched versions of the axioms of a functor, viz preservation of identity and composition.
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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 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.
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| In detail, one has that the diagram
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| [[Image:Enrichedidentity.png|center|300px]]
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| commutes, which amounts to the equation
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| :<math>T_{aa}\circ \operatorname{id}_a=\operatorname{id}_{T(a)},</math>
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| where ''I'' is the unit object of '''M'''. This is analogous to the rule ''F''(id<sub>''a''</sub>) = id<sub>''F''(''a'')</sub> for ordinary functors. Additionally, one demands that the diagram
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| [[Image:Enrichedmult.png|center]]
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| commute, which is analogous to the rule ''F''(''fg'')=''F''(''f'')''F''(''g'') for ordinary functors.
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| ==See also==
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| {{Portal|Category theory}}
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| * [[Internal category]]
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| ==References==
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| * [[Max Kelly|Kelly,G.M.]] [http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf "Basic Concepts of Enriched Category Theory"], London Mathematical Society Lecture Note Series No.64 (C.U.P., 1982)
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| * {{cite book |first=Saunders |last=Mac Lane |authorlink=Saunders Mac Lane|title=Categories for the Working Mathematician | edition=second |date=September 1998 |publisher=Springer |isbn=0-387-98403-8}} (Volume 5 in the series [[Graduate Texts in Mathematics]])
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| * [[F. William Lawvere|Lawvere,F.W.]] [http://tac.mta.ca/tac/reprints/articles/1/tr1.pdf "Metric Spaces, Generalized Logic, and Closed Categories"], Reprints in Theory and Applications of Categories, No. 1, 2002, pp. 1–37.
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| *{{nlab|id=enriched+category|title=Enriched category}}
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| {{DEFAULTSORT:Enriched Category}}
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| [[Category:Category theory]]
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| [[Category:Monoidal categories]]
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