Equivalence of categories

In category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences from many areas of mathematics. Establishing an equivalence involves demonstrating strong similarities between the mathematical structures concerned. In some cases, these structures may appear to be unrelated at a superficial or intuitive level, making the notion fairly powerful: it creates the opportunity to "translate" theorems between different kinds of mathematical structures, knowing that the essential meaning of those theorems is preserved under the translation.

If a category is equivalent to the opposite (or dual) of another category then one speaks of a duality of categories, and says that the two categories are dually equivalent.

An equivalence of categories consists of a functor between the involved categories, which is required to have an "inverse" functor. However, in contrast to the situation common for isomorphisms in an algebraic setting, the composite of the functor and its "inverse" is not necessarily the identity mapping. Instead it is sufficient that each object be naturally isomorphic to its image under this composition. Thus one may describe the functors as being "inverse up to isomorphism". There is indeed a concept of isomorphism of categories where a strict form of inverse functor is required, but this is of much less practical use than the equivalence concept.

Formally, given two categories C and D, an equivalence of categories consists of a functor F : CD, a functor G : DC, and two natural isomorphisms ε: FGID and η : ICGF. Here FG: DD and GF: CC denote the respective compositions of F and G, and IC: CC and ID: DD denote the identity functors on C and D, assigning each object and morphism to itself. If F and G are contravariant functors one speaks of a duality of categories instead.

One often does not specify all the above data. For instance, we say that the categories C and D are equivalent (respectively dually equivalent) if there exists an equivalence (respectively duality) between them. Furthermore, we say that F "is" an equivalence of categories if an inverse functor G and natural isomorphisms as above exist. Note however that knowledge of F is usually not enough to reconstruct G and the natural isomorphisms: there may be many choices (see example below).

A functor F : CD yields an equivalence of categories if and only if it is simultaneously:

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 type theory.)

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

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

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

If F : CD 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 : CC. 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.)