Quillen equivalence

Remarks

Equivalently, the derived adjunction $\mathbf{L}F \colon \operatorname{Ho}(\mathcal{C}) \rightleftarrows \operatorname{Ho}(\mathcal{D}) \colon \mathbf{R}G$ is an equivalence of categories.

Example: geometric realization $\abs{\cdot}$ from simplicial sets to topological spaces is left adjoint to the singular complex functor $\operatorname{Sing}$. It is a Quillen equivalence between the Kan–Quillen model structure and the Quillen model structure. Every simplicial set is cofibrant and every space is fibrant. The check reduces to two facts: the unit $K \to \operatorname{Sing}\abs{K}$ is a weak equivalence, and the counit $\abs{\operatorname{Sing} Y} \to Y$ is a weak homotopy equivalence.

Non-example: give the category of sets the model structure whose weak equivalences are the bijections and in which every map is a cofibration and a fibration. The functor $\pi_0$ from simplicial sets to sets is left adjoint to the functor that sends a set to the discrete simplicial set on it. This is a Quillen adjunction. It is not a Quillen equivalence. Take the simplicial circle $K = \Delta^1 / \partial \Delta^1$; the identity map of the one-point set $\pi_0 K$ is a bijection. Its adjoint $K \to \Delta^0$ is not a weak equivalence, since $\abs{K}$ is a circle.