Enriched Category - Examples of Enriched Categories

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-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.
• Locally small categories are categories enriched over (SmSet, ×), the category of small sets with Cartesian product as the monoidal operation. (A locally small category is one whose hom-objects are small sets.)
• Locally finite categories, by analogy, are categories enriched over (FinSet, ×), the category of finite sets with Cartesian product as the monoidal operation.
• Preordered sets 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

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 spaces, also known as pseudoquasimetric spaces, are categories enriched over the nonnegative extended real numbers R+∞, 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 R+∞(a,b) are essentially distances d(a,b), and the existence of composition and identity translate to

d(b,c) + d(a,b) ≥ d(a,c) (triangle inequality)

0 ≥ d(a,a)

• Categories with zero morphisms 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.
• Preadditive categories are categories enriched over (Ab, ⊗), the category of abelian groups with tensor product as the monoidal operation.