Contents

model category

for ∞-groupoids

# Contents

## Stetment

###### Proposition

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.

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

## References

Created on August 23, 2020 at 07:44:04. See the history of this page for a list of all contributions to it.