Equivalence of Categories - Properties

Properties

As a rule of thumb, an equivalence of categories preserves all "categorical" concepts and properties. If F : C → D is an equivalence, then the following statements are all true:

• the object c of C is an initial object (or terminal object, or zero object), if and only if Fc is an initial object (or terminal object, or zero object) of D
• the morphism α in C is a monomorphism (or epimorphism, or isomorphism), if and only if Fα is a monomorphism (or epimorphism, or isomorphism) in D.
• the functor H : I → C has limit (or colimit) l if and only if the functor FH : I → D has limit (or colimit) Fl. This can be applied to equalizers, products and coproducts among others. Applying it to kernels and cokernels, we see that the equivalence F is an exact functor.
• C is a cartesian closed category (or a topos) if and only if D is cartesian closed (or a topos).

Dualities "turn all concepts around": they turn initial objects into terminal objects, monomorphisms into epimorphisms, kernels into cokernels, limits into colimits etc.

If F : C → D is an equivalence of categories, and G₁ and G₂ are two inverses of F, then G₁ and G₂ are naturally isomorphic.

If F : C → D is an equivalence of categories, and if C is a preadditive category (or additive category, or abelian category), then D may be turned into a preadditive category (or additive category, or abelian category) in such a way that F becomes an additive functor. On the other hand, any equivalence between additive categories is necessarily additive. (Note that the latter statement is not true for equivalences between preadditive categories.)

An auto-equivalence of a category C is an equivalence F : C → C. The auto-equivalences of C form a group under composition if we consider two auto-equivalences that are naturally isomorphic to be identical. This group captures the essential "symmetries" of C. (One caveat: if C is not a small category, then the auto-equivalences of C may form a proper class rather than a set.)