# Equivalence of categories

An extension of the concept of an isomorphism of categories brought about, first of all, by the presence of classes of isomorphic objects.

Two categories and are called equivalent if there are one-place covariant functors and such that the product is naturally equivalent to the identity functor and the product to the functor ; in other words, the categories and are equivalent if there are functors and "almost" inverse to one another. Two categories are equivalent if and only if their skeletons are isomorphic (see Skeleton of a category).

Pontryagin's duality theorem establishes the equivalence of the category of Abelian groups and the category that is dual to that of topological Abelian groups; the category of Boolean algebras is equivalent to the category that is dual to that of Boolean spaces; the category of binary relations over the category of sets is equivalent to the Kleisli category for the triple defined by the functor of taking the set of subsets.

#### Comments

Cf. Triple for that notion, and the editorial comments to Category for the notion of a Kleisli category of a triple.

#### References

[a1] | B. Mitchell, "Theory of categories" , Acad. Press (1965) |

**How to Cite This Entry:**

Equivalence of categories. M.Sh. Tsalenko (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Equivalence_of_categories&oldid=12063