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

Definition

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 I_D, and GF naturally isomorphic to I_C. Here I_D 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:

• The functors F : C -> D and G : D -> C form an equivalence of categories.
• F is a left adjoint of G and both functors are full and faithful.
• F is full and faithful and each object d in D is isomorphic to an object of the form Fc, for c in C.
• The conditions obtained by exchanging F and G in the above statements.

Examples

• In pointless topology the category of spatial locales is known to be equivalent to the dual of the category of sober spaces.

• In lattice theory, there are a number of famous dualities, based on representation theorems that connect certain classes of lattices to classes of topological spaces. Probably the most well-known theorem of this kind is Stone's representation theorem for Boolean algebras. Each Boolean algebra B is mapped to a specific topology on the set of ultrafilters of B. Conversely, for any topology the clopen (i.e. closed and open) subsets yield a Boolean algebra. One obtains a duality between the category of Boolean algebras (with their homomorphisms) and Stone spaces (with continuous mappings).

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

• One of the central themes of algebraic geometry is the duality of the category of affine schemes and the category of commutative rings. The functor G associates to every commutative ring its spectrum, the scheme defined by the prime ideals of the ring. Its adjoint F associates to every affine scheme its ring of global sections.

• In functional analysis the category of commutative C*-algebras with identity is contravariantly equivalent to the category of compact Hausdorff spaces. Under this duality, every compact Hausdorff space X is associated with the algebra of continuous complex-valued functions on X, and every commutative C*-algebra is associated with the space of its maximal ideals. This is the Gelfand representation.