nLab model structure on modules over an algebra over an operad

Definition

Theorem

Let

• $\mathcal{E}$ be a cofibrantly generated symmetric monoidal model category;

• $P$ an admissible $\Sigma$-cofibrant operad in $\mathcal{E}$ (see model structure on operads);

• $A$ a cofibrant $P$-algebra (see model structure on algebras over an operad).

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).

Examples

• model structure on dg-modules

Related concepts

• (∞,1)-operad, model structure on operads

  • algebra over an (∞,1)-operad, model structure on algebras over an operad

    • module over an algebra over an (∞,1)-operad, model structure on modules over an algebra over an operad

• operad for modules over an algebra

References

• Clemens Berger, Ieke Moerdijk, On the derived category of an algebra over an operad (arXiv)

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