Enriched Category - Definition

Definition

Let (M,⊗,I,, ) 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

• a class ob(C) of objects of C,
• an object C(a,b) of M for every pair of objects a,b in C,
• an arrow id_a:I → C(a,a) in M designating an identity for every object a in C, and
• an arrow °_abc:C(b,c)⊗C(a,b) → C(a,c) in M designating a composition for each triple of objects a,b,c in C,

together with three commuting diagrams, discussed below. The first diagram expresses the associativity of composition:

That is, the associativity requirement is now taken over by the associator of the hom-category.

For the case that M is the category of sets, so that (⊗,I,α,λ,ρ) is the usual monoidal structure (cartesian product, single-point set, etc.), 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.

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.

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

For the case where M is the category of sets, the morphisms id_a: 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.

One should be careful to distinguish the various different notions of "identity" being referenced here, so:

• the monoidal identity I is an object of M, being an identity for ⊗ only in the monoid-theoretic sense, and even then only up to canonical isomorphism (λ, ρ).
• the identity morphisms 1_C(a,b):C(a,b) → C(a,b) which are actual morphisms that M has for each of its objects by virtue of it being (at least) an ordinary category.

These should be distinguished from the morphisms id_a:I → C(a,a) that define the notion of identity for objects in the enriched category C, whether or not C can be considered to have individual morphisms of its own.