nLab
homotopy inverse

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Definition

Let \sim be the relation of being homotopic (for example between morphisms in the category Top). Let f:XYf:X\to Y and g:YXg:Y\to X be two morphisms. We say that gg is a left homotopy inverse to ff or that ff is a right homotopy inverse to gg if gfid Xg\circ f\sim id_X. A homotopy inverse of ff is a map which is simultaneously a left and a right homotopy inverse to ff.

ff 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 ff to be equal. To show this denote by g Lg_L the left and by g Rg_R the right homotopy inverse of ff. Then

g Lg L(fg R)=(g Lf)g Rg R. g_L \sim g_L\circ (f\circ g_R) = (g_L\circ f)\circ g_R \sim g_R.

Hence

fg Lfg Rid X, f\circ g_L\sim f\circ g_R\sim id_X,

therefore g Lg_L is not only a left, but also a right, homotopy inverse to ff.

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)