nLab
model structure on modules over an algebra over an operad

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 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

Higher algebra

Contents

Definition

Theorem

Let

Then then category Mod P(A)Mod_P(A) of modules over an algebra over an operad carries the transferred model structure along the forgetful functor U:Mod P(A)U : Mod_P(A) \to \mathcal{E}.

Every morphism of cofibrant PP-algebras f:ABf : A \to B induced a Quillen adjunction

(f !f *):Mod P(B)f *f !Mod P(A) (f_! \dashv f^*) : Mod_P(B) \stackrel{\overset{f_!}{\leftarrow}}{\underset{f^*}{\to}} Mod_P(A)

which is a Quillen equivalence if ff is a weak equivalence.

This is (BergerMoerdijk, theorem 2.6).

Examples

References

  • Benoit Fresse, Modules over operads and functors Springer Lecture Notes in Mathematics, (2009) (pdf)
Revised on February 11, 2013 01:36:37 by Urs Schreiber (89.204.137.65)