nLab opposite model structure

Definition

Definition

(opposite model categories)

If a category $C$ carries a model category structure, then the opposite category $C^{op}$ carries the opposite model structure:

• its weak equivalences are those morphisms whose dual was a weak equivalence in $C$,

• its fibrations are those morphisms that were cofibrations in $C$

• its cofibrations are those that were fibrations in $C$.

Properties

Proposition

(opposite Quillen adjunction)
Given a Quillen adjunction

$$\mathcal{D} \underoverset {\underset{R}{\longrightarrow}} {\overset{L}{\longleftarrow}} {\;\;\;\;\; \bot_{\mathrlap{{}_{Qu}}} \;\;\;\;\;} \mathcal{C} \,,$$

its opposite adjunction is a Quillen adjunction

$$\mathcal{D}^{op} \underoverset {\underset{L^{op}}{\longrightarrow}} {\overset{R^{op}}{\longleftarrow}} {\;\;\;\;\; \bot_{\mathrlap{{}_{Qu}}} \;\;\;\;\;} \mathcal{C}^{op}$$

between the opposite model categories (Def. ).

Related concepts

• slice model structure

• opposite category, opposite 2-category

• opposite (∞,1)-category

References

Textbook accounts:

• Philip Hirschhorn, Proposition 7.1.10 of: Model Categories and Their Localizations, AMS Math. Survey and Monographs Vol 99 (2002) (ISBN:978-0-8218-4917-0, pdf toc, pdf)