# nLab model structure on modules over an algebra over an operad

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

#### Higher algebra

higher algebra

universal algebra

# Contents

## Definition

###### Theorem

Let

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

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

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

which is a Quillen equivalence if $f$ is a weak equivalence.

This is (BergerMoerdijk, theorem 2.6).

## References

• Benoit Fresse, Modules over operads and functors Springer Lecture Notes in Mathematics, (2009) (pdf)

Last revised on February 11, 2013 at 01:36:37. See the history of this page for a list of all contributions to it.