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 |

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:

- 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.

- 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).

- 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.