Contents

model category

for ∞-groupoids

# Contents

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

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