nLab
inverse

Definition

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

$$f \circ f^{-1} : Y \to Y$$

equals the identity morphism on $Y$ and

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

equals the identity morphism on $X$.

Remarks

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

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

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

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

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

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

  • This is a special case of the two-out-of-six property which is satisfied by the weak equivalences in any homotopical category.

  • When this is applied to a homotopy category such as that of Top for the standard model structure on topological spaces it implies the construction of and formulae for certain homotopies.

• 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.

Related concepts

• inverse morphism, inverse function

• weak inverse

• inversion involution