nLab
derived adjunction

Contents

Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for rational \infty-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\infty,1)-sheaves / \infty-stacks

Contents

Stetment

Proposition

(derived adjunction)

For 𝒞 Qu QuRL𝒟\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(𝒞)R𝕃LHo(𝒟) 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

Last revised on July 12, 2021 at 08:58:30. See the history of this page for a list of all contributions to it.