Enriched category

In category theory and its applications to mathematics, an enriched category is a category whose hom-sets are replaced by objects from some other category, in a well-behaved manner.

Definition

We define here what it means for C to be an enriched category over a monoidal category $(mathbf\{M\},otimes,I)$.

We require the following structures:
*Let Ob(C) be a set (or proper class, if you prefer). An element of Ob(C) is called an "object" of C.
*For each pair ("A","B") of objects of C, let Hom("A","B") be an object of M, called the "hom-object" of "A" and "B".
*For each object "A" of C, let id_"A" be a morphism in M from $I$ to Hom("A","A"), called the "identity morphism" of "A".
*For each triple ("A","B","C") of objects of C, let:$circ:mathrm\{Hom\}(B,C)otimesmathrm\{Hom\}(A,B)\; omathrm\{Hom\}(A,C)$be a morphism in M called the "composition" morphism of "A", "B", and "C".

We require the following axioms:
*Associativity: Given objects "A", "B", "C", and "D" of C, we can go from Hom("C","D") ⊗ Hom("B","C") ⊗ Hom("A","B") to Hom("A","D") in two ways, depending on which composition we do first. These must give the same result.:
*Left identity: Given objects "A" and "B" of C, we can go from "I" ⊗ Hom("A","B") to just Hom("A","B") in two ways, either by using id_A on "I" and then using composition, or by simply using the fact that "I" is an identity for ⊗ in M. These must give the same result.:
*Right identity: Given objects "A" and "B" of C, we can go from Hom("A","B") ⊗ "I" to just Hom("A","B") in two ways, either by using id_B on "I" and then using composition, or by simply using the fact that "I" is an identity for ⊗ in M. These must give the same result.:

Then C (consisting of all the structures listed above) is a category enriched over M.

Examples

The most straightforward example is to take M to be a category of sets, with the Cartesian product for the monoidal operation.Then C is nothing but an ordinary category.If M is the category of small sets, then C is a locally small category, because the hom-sets will all be small.Similarly, if M is the category of finite sets, then C is a locally finite category.

If M is the category 2 with Ob(2) = {0,1}, a single nonidentity morphism (from 0 to 1), and ordinary multiplication of numbers as the monoidal operation, then C can be interpreted as a preordered set.Specifically, "A" ≤ "B" iff Hom("A","B") = 1.

If M is a category of pointed sets with Cartesian product for the monoidal operation, then C is a category with zero morphisms.Specifically, the zero morphism from "A" to "B" is the special point in the pointed set Hom("A","B").

If M is a category of abelian groups with tensor product as the monoidal operation, then C is a preadditive category.

A property

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 "T"_"ab": "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 diagramcommutes, which amounts to the equation:$T\_\{aa\}circ\; operatorname\{id\}\_a=operatorname\{id\}\_\{T(a)\},$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 commutes, which is analogous to the rule "F"("fg")="F"("f")"F"("g") for ordinary functors.

References

*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)

External links

* [http://www.seas.upenn.edu/~sweirich/types/archive/1988/msg00065.html An extension of Reynolds' result on the non-existence of set-models of polymorphism]
* [http://homepages.inf.ed.ac.uk/gdp/publications/sem_alg_ops.pdf Semantics for Algebraic Operations]