nLab derived adjunction

Stetment

Proposition

(derived adjunction)

For $\mathcal{C} \underoverset{\underset{R}{\longrightarrow}}{\overset{L}{\longleftarrow}}{{}_{\phantom{Qu}}\bot_{Qu}}\mathcal{D}$ a Quillen adjunction between model categories, also the corresponding left and right derived functors form a pair of adjoint functors

$$Ho(\mathcal{C}) \underoverset {\underset{\mathbb{R}R}{\longrightarrow}} {\overset{\mathbb{L}L}{\longleftarrow}} {\bot} Ho(\mathcal{D})$$

between the corresponding homotopy categories.

Moreover, the adjunction unit and adjunction counit of this derived adjunction are the images of the derived adjunction unit and derived adjunction counit of the original Quillen adjunction.

(Quillen 67, I.4 theorem 3)

For full proof see also at Introduction to Homotopy Theory this Prop., or at geometry of physics – categories and toposes this Prop..

References

• Daniel Quillen, chapter I Axiomatic homotopy theory in: Homotopical Algebra, Lecture Notes in Mathematics 43, Springer 1967(doi:10.1007/BFb0097438)