Enriched category

In category theory, a branch of mathematics, an enriched category generalizes the idea of a category by replacing hom-sets 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 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 or even symmetric closed monoidal, respectively).
Enriched category theory thus encompasses within the same framework a wide variety of structures including
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.
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 Mac Lane's preference for the letter V in referring to the monoidal category, enriched categories are also sometimes referred to generally as V-categories.
Definition.
Let 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
The first diagram expresses the associativity of composition:
That is, the associativity requirement is now taken over by the associator of the monoidal category M.
For the case that M is the category of sets and is the monoidal structure given by the cartesian product, the terminal single-point set, and the canonical isomorphisms they induce, then each 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 in the first diagram corresponds to one of the two ways of composing three consecutive individual morphisms , i.e. elements from , and . Commutativity of the diagram is then merely the statement that both orders of composition give the same result, exactly as required for ordinary categories.
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 monoidal category M, and not in C — thus making the concept of associativity of composition meaningful in the general case where the hom-objects are abstract, and C itself need not even "have" any notion of individual morphism.
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 unitors:
and
Returning to the case where M is the category of sets with cartesian product, the morphisms become functions from the one-point set "I" and must then, for any given object "a", identify a particular element of each set , 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.
Note that there are several distinct notions of "identity" being referenced here:
Relationship with monoidal functors.
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. 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, so a category enriched over M can be described as an ordinary category with certain additional structure or properties.
Enriched functors.
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.
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 "Tab" : "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.
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.
In detail, one has that the diagram
commutes, which amounts to the equation
where "I" is the unit object of M. This is analogous to the rule "F"(id"a") = id"F"("a") for ordinary functors. Additionally, one demands that the diagram 
commute, which is analogous to the rule "F"("fg")="F"("f")"F"("g") for ordinary functors.