In category theory, two categories C and D are isomorphic if there exist functors F : C → D and G : D → C that are mutually inverse to each other, i.e. FG = 1<sub>D</sub> (the identity functor on D) and GF = 1<sub>C</sub>. This means that both the objects and the morphisms of C and D stand in a one-to-one correspondence with each other. Two isomorphic categories share all properties defined solely in category theory; for all practical purposes, they are identical and differ only in the notation of their objects and morphisms.

Isomorphism of categories is a strong condition and is rarely satisfied in practice. Much more important is the notion of equivalence of categories; roughly speaking, for an equivalence of categories, we don't require that <math>FG</math> be equal to <math>1_D</math>, but only naturally isomorphic to <math>1_D</math>, and likewise that <math>GF</math> be naturally isomorphic to <math>1_C</math>.

Properties

As is true for any notion of isomorphism, we have the following general properties formally similar to an equivalence relation:

  • any category C is isomorphic to itself
  • if C is isomorphic to D, then D is isomorphic to C
  • if C is isomorphic to D and D is isomorphic to E, then C is isomorphic to E.

A functor F : C → D yields an isomorphism of categories if and only if it is bijective on objects and morphism sets.