nLab
opposite model structure

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 equivariant \infty-groupoids

for rational \infty-groupoids

for rational equivariant \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

Definition

Definition

(opposite model categories)

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

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

  • its fibrations are those morphisms that were cofibrations in CC

  • its cofibrations are those that were fibrations in CC.

Properties

Proposition

(opposite Quillen adjunction)
Given a Quillen adjunction

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

its opposite adjunction is a Quillen adjunction

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

between the opposite model categories (Def. ).

References

Textbook accounts:

Last revised on July 20, 2021 at 07:44:17. See the history of this page for a list of all contributions to it.