# nLab opposite model structure

Contents

### Context

#### Model category theory

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

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

Model structures

for $\infty$-groupoids

for ∞-groupoids

for equivariant $\infty$-groupoids

for rational $\infty$-groupoids

for rational equivariant $\infty$-groupoids

for $n$-groupoids

for $\infty$-groups

for $\infty$-algebras

general $\infty$-algebras

specific $\infty$-algebras

for stable/spectrum objects

for $(\infty,1)$-categories

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

for $(\infty,1)$-operads

for $(n,r)$-categories

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

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

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