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

model category

## Model structures

for ∞-groupoids

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

#### Higher algebra

higher algebra

universal algebra

# Contents

## Definition

###### Theorem

Let

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

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

$\left({f}_{!}⊣{f}^{*}\right):{\mathrm{Mod}}_{P}\left(B\right)\stackrel{\stackrel{{f}_{!}}{←}}{\underset{{f}^{*}}{\to }}{\mathrm{Mod}}_{P}\left(A\right)$(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)
Revised on February 11, 2013 01:36:37 by Urs Schreiber (89.204.137.65)