# Contents

## Definition

Let $\sim$ be the relation of being homotopic (for example between morphisms in the category Top). Let $f:X\to Y$ and $g:Y\to X$ be two morphisms. We say that $g$ is a left homotopy inverse to $f$ or that $f$ is a right homotopy inverse to $g$ if $g\circ f\sim {\mathrm{id}}_{X}$. A homotopy inverse of $f$ is a map which is simultaneously a left and a right homotopy inverse to $f$.

$f$ is said to be a homotopy equivalence if it has a left and a right homotopy inverse. In that case we can choose the left and right homotopy inverses of $f$ to be equal. To show this denote by ${g}_{L}$ the left and by ${g}_{R}$ the right homotopy inverse of $f$. Then

${g}_{L}\sim {g}_{L}\circ \left(f\circ {g}_{R}\right)=\left({g}_{L}\circ f\right)\circ {g}_{R}\sim {g}_{R}.$g_L \sim g_L\circ (f\circ g_R) = (g_L\circ f)\circ g_R \sim g_R.

Hence

$f\circ {g}_{L}\sim f\circ {g}_{R}\sim {\mathrm{id}}_{X},$f\circ g_L\sim f\circ g_R\sim id_X,

therefore ${g}_{L}$ is not only a left, but also a right, homotopy inverse to $f$.

This makes sense in any category equipped with an equivalence relation $\sim$, which is compatible with the composition (and with the equality of morphisms).

## Examples

Revised on May 29, 2012 06:28:05 by Urs Schreiber (131.130.239.199)