nLab
inverse

Contents

Contents

Definition

An inverse of a morphism f:XYf : X \to Y in a category or unital magmoid (or an element of a monoid or unital magma) 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.

Remarks

  • 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-unital contexts

In a magmoid or semicategory (or an element of a semigroup or magma), a morphism f:abf:a \to b has a unique retraction f 1:baf^{-1}:b \to a if

  • for every morphism g:bcg:b \to c, g(f 1f)=gg \circ (f^{-1} \circ f) = g,

  • for every morphism g:cag:c \to a, (f 1f)g=g(f^{-1} \circ f) \circ g = g,

and a morphism f:abf:a \to b has a unique section f 1:baf^{-1}:b \to a if

  • for every morphism g:acg:a \to c, g(ff 1)=gg \circ (f \circ f^{-1}) = g,

  • for every morphism g:acg:a \to c, (ff 1)g=g(f \circ f^{-1})\circ g = g,

A morphism f:abf:a \to b has a unique inverse if it has a retraction that is also a section.

Last revised on May 25, 2021 at 09:36:53. See the history of this page for a list of all contributions to it.