left inverse

left inverse (plural left inverses)

1. (mathematics) A related function that, given the output of the original function, returns the input that produced that output.

   In order for a function to have a left inverse it must be injective.

2. (category theory) For a given morphism f : X → Y, its right inverse (if it has one) is a morphism s : Y → X such that ${\displaystyle s\circ f={\mbox{id}}_{X}}$.

Given two functions ${\displaystyle f:X\to Y}$ and ${\displaystyle g:Y\to X}$, ${\displaystyle g}$ is the left inverse of ${\displaystyle f}$ iff for all ${\displaystyle x}$ in ${\displaystyle X}$, ${\displaystyle g(f(x))=x}$. The term is only used if there exists a ${\displaystyle y}$ in ${\displaystyle Y}$ such that ${\displaystyle f(g(y))!=y}$, and an unqualified inverse is preferred if no such ${\displaystyle y}$ exists.

• right inverse

• inverse