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 |
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.
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:
Definition
Equivalent Characterizations
Examples
A categorical equivalence of the above form, connecting classes of
ordered sets to classes of topological spaces, is
sometimes called Stone's duality.