View source for 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036050/e0360501.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036050/e0360502.png" /> are called equivalent if there are one-place covariant functors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036050/e0360503.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036050/e0360504.png" /> such that the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036050/e0360505.png" /> is naturally equivalent to the identity functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036050/e0360506.png" /> and the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036050/e0360507.png" /> to the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036050/e0360508.png" />; in other words, the categories <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036050/e0360509.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036050/e03605010.png" /> are equivalent if there are functors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036050/e03605011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036050/e03605012.png" />  "almost"  inverse to one another. Two categories are equivalent if and only if their skeletons are isomorphic (see [[Skeleton of a category|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|Triple]] for that notion, and the editorial comments to [[Category|Category]] for the notion of a Kleisli category of a triple.

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