Main Page | See live article | Alphabetical index

Equivalence of categories

In category theory, an abstract branch of mathematics, an equivalence of categories is a relation between categories that are "essentially the same". There are numerous examples of categorical equivalences from many areas of mathematics. Establishing such an equivalence usually means to discover strong similarities between mathematical structures that formerly where considered to be unrelated or where the relation was not understood properly. The gain of this usually is a better understanding of the nature of the considered objects and the possibility to translate theorems between different kinds of mathematical structures. If a category is equivalent to the dual of another category then one speaks of a duality of categories.

An equivalence of categories consists of a functor between the equivalent categories, where this mapping is required to have an "inverse" functor. However, in contrast to the situation common for isomorphisms in an algebraic setting, the composition of the functor and its "inverse" is not necessarily the identity mapping. Instead it is sufficient if each object is naturally isomorphic to its image under this composition. Thus one may describe the functors as being "inverse up to isomorphism".

Table of contents
1 Definition
2 Equivalent Characterizations
3 Examples


Formally, given two categories C and D, a functor F : C -> D, an equivalence of categories is a functor F such that there is a functor G : D -> C with the composition FG naturally isomorphic to ID, and GF naturally isomorphic to IC. Here ID denotes the identity functor D -> D that assigns every object and every morphism to itself.

In this situation, we say that the categories C and D are equivalent. If F and G are contravariant functors, then one speaks instead of a duality of categories.

Equivalent Characterizations

The above defition is probably the easiest one of many equivalent statements, some of which are listed below. Most importantly, there is a close relation to the concept of adjoint functors.

The following are equivalent:


A categorical equivalence of the above form, connecting classes of ordered sets to classes of topological spaces, is sometimes called Stone's duality.