View source for Equivalence of categories

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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 $\mathfrak{K}$ and $\mathfrak{L}$ are called equivalent if there are one-place covariant functors $F : \mathfrak{K} \rightarrow \mathfrak{L}$ and $G : \mathfrak{L} \rightarrow \mathfrak{K}$ such that the product $FG$ is naturally equivalent to the identity functor $\mathrm{Id}_{\mathfrak{L}}$ and the product $GF$ to the functor $\mathrm{Id}_{\mathfrak{K}}$; in other words, the categories $\mathfrak{K}$ and $\mathfrak{L}$ are equivalent if there are functors $F$ and $G$  "almost"  inverse to one another. Two categories are equivalent if and only if their [[Skeleton of a category|skeletons]] are isomorphic.

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 (cf. the editorial comments to [[Category]] for the notion of a Kleisli category of a triple).

====References====
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Mitchell,   "Theory of categories" , Acad. Press  (1965)</TD></TR>
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