Editing Enriched category (section)

==Examples of enriched categories==

* Ordinary categories are categories enriched over ('''Set''', ×, {•}), the [[category of sets]] with [[Cartesian product]] as the monoidal operation, as noted above.
* [[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'',&thinsp;''b'') and its own composition rule.
* [[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.)
* [[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.
* If ''C'' is a [[closed monoidal category]] then ''C'' is enriched in itself.
* [[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'',&thinsp;''b'') then simply deny or affirm a particular binary relation on the given pair of objects (''a'',&thinsp;''b''); for the sake of having more familiar notation we can write this relation as {{math|''a'' ≤ ''b''}}. The existence of the compositions and identity required for a category enriched over '''2''' immediately translate to the following axioms respectively
::''b'' ≤ ''c'' and ''a'' ≤ ''b'' ⇒ ''a'' ≤ ''c''    (transitivity)
::''TRUE'' ⇒ ''a'' ≤ ''a''     (reflexivity)
: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'''.
* [[William Lawvere]]'s [[generalized metric space]]s, 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'',&thinsp;''b''), and the existence of composition and identity translate to
::d(''b'',&thinsp;''c'') + d(''a'',&thinsp;''b'') ≥ d(''a'',&thinsp;''c'')   (triangle inequality)
::0 ≥ d(''a'',&thinsp;''a'') 
* 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'',&thinsp;''B'') corresponds to the zero morphism from ''A'' to ''B''.
* The category '''Ab''' of [[abelian group]]s and the category '''R-Mod''' of [[module (mathematics)|module]]s over a [[commutative ring]], and the category '''Vect''' of [[vector space]]s over a given [[field (mathematics)|field]] are enriched over themselves, where the morphisms inherit the algebraic structure "pointwise". More generally, [[Preadditive category|preadditive categories]] are categories enriched over ('''Ab''', ⊗) with [[tensor product]] as the monoidal operation (thinking of abelian groups as '''Z'''-modules).