Theorems

Contents

Definition

Let ∼\sim be the relation of being homotopic (for example between morphisms in the categoryTop). Let f:X→Yf:X\to Y and g:Y→Xg: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 g∘f∼idXg\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 gLg_L the left and by gRg_R the right homotopy inverse of ff. Then