Equivalence of Categories - Equivalent Characterizations

Equivalent Characterizations

One can show that a functor F : C → D yields an equivalence of categories if and only if it is:

• full, i.e. for any two objects c₁ and c₂ of C, the map Hom_C(c₁,c₂) → Hom_D(Fc₁,Fc₂) induced by F is surjective;
• faithful, i.e. for any two objects c₁ and c₂ of C, the map Hom_C(c₁,c₂) → Hom_D(Fc₁,Fc₂) induced by F is injective; and
• essentially surjective (dense), i.e. each object d in D is isomorphic to an object of the form Fc, for c in C.

This is a quite useful and commonly applied criterion, because one does not have to explicitly construct the "inverse" G and the natural isomorphisms between FG, GF and the identity functors. On the other hand, though the above properties guarantee the existence of a categorical equivalence (given a sufficiently strong version of the axiom of choice in the underlying set theory), the missing data is not completely specified, and often there are many choices. It is a good idea to specify the missing constructions explicitly whenever possible. Due to this circumstance, a functor with these properties is sometimes called a weak equivalence of categories (unfortunately this conflicts with terminology from homotopy theory).

There is also a close relation to the concept of adjoint functors. The following statements are equivalent for functors F : C → D and G : D → C:

• There are natural isomorphisms from FG to I_D and I_C to GF.
• F is a left adjoint of G and both functors are full and faithful.
• F is a right adjoint of G and both functors are full and faithful.

One may therefore view an adjointness relation between two functors as a "very weak form of equivalence". Assuming that the natural transformations for the adjunctions are given, all of these formulations allow for an explicit construction of the necessary data, and no choice principles are needed. The key property that one has to prove here is that the counit of an adjunction is an isomorphism if and only if the right adjoint is a full and faithful functor.