An inverse of a morphism f:XYf : X \to Y in a category (or an element of a monoid) is another morphism f 1:YXf^{-1} : Y \to X which is both a left-inverse (a retraction) as well as a right-inverse (a section) of ff, in that

ff 1:YY f \circ f^{-1} : Y \to Y

equals the identity morphism on YY and

f 1f:XX f^{-1} \circ f : X \to X

equals the identity morphism on XX.


  • A morphism which has an inverse is called an isomorphism.

  • The inverse f 1f^{-1} is unique if it exists.

  • The inverse of an inverse morphism is the original morphism, (f 1) 1=f(f^{-1})^{-1} = f.

  • An identity morphism, ii, is a morphism which is its own inverse: i 1=ii^{-1} = i.

  • A category in which all morphisms have inverses is called a groupoid.

  • An amusing exercise is to show that if f,g,hf,g,h are morphisms such that fg,ghf\circ g,\; g\circ h are defined and are isomorphisms, then f,g,hf,g,h are all isomorphisms.

  • In a balanced category, such as a topos or more particularly Set, every morphism that is both monic and epic is an isomorphism and thus has an inverse. A partial order is an unbalanced category where every morphism is both monic and epic. Only its identity morphisms have inverses.

In non-associative contexts

These can be a little more complicated; see quasigroup for some discussion of the one-object version.

Last revised on September 17, 2017 at 20:15:48. See the history of this page for a list of all contributions to it.