How to show that : $$ \mathrm{Hot} \simeq W_{ \infty }^{-1} \mathcal{T} \mathrm{op} $$
For information :
- $ \mathrm{Hot} $ is the homotopy category defined as being the category where the objects are $ \mathrm{CW} $ - complexes, and the morphismes are homotopy classes of continuous maps between $ \mathrm{CW} $ - complexes.
- $ \mathcal{T} \mathrm{op} $ is the category of topological spaces, and continuous maps between them.
- $ W^{-1} \mathcal{T} \mathrm{op} $ is the category built by inversing formally the morphisms belonging to $ W $.
- $ W_{ \infty } $ is the subset of $ \mathrm{Fl} ( \mathcal{T} \mathrm{op} ) $ formed of weak topological equivalences.
- A week topological equivalence is a continuous map $ f : X \to Y $ where :
--- $ \pi_0 (f) \ : \ \pi_0 ( X ) \to \pi_0 ( Y ) $ is a bijective map.
--- $ \forall n \geq 0 \ \forall x \in X $ : $ \pi_n ( f , x ) \ : \ \pi_n (X,x) \to \pi_n ( Y , f(x)) $ is an isomorphism of groups.
Thanks in advance for your help.