Equivalence of Categories

In Category Theory, two categories are said to be equivalent if there are [[Adjoint Functor]]s between them whose compositions are naturally isomorphic to the identity endofunctors.

More explicitly, let \(C\) and \(D\) be categories, and \(F\) and \(G\) be functors such that \(F: C \to D\) and \(G: D \to C\). If there exist natural isomorphisms \(1_C \cong GF\) and \(FG \cong 1_D\), this information is said to comprise an equivalence of categories, and \(C\) and \(D\) are said to be equivalent.

Contractible Groupoids

A category which is equivalent to the terminal category \(1\) is called a contractible groupoid.