nLab model structure on modules in a monoidal model category

Idea

For $\mathcal{C}$ a monoidal model category and $A \in Mon(\mathcal{C})$ a monoid in $\mathcal{C}$, there is under mild conditions a natural model category structure on its category of modules over $A$. (Schwede & Shipley 2000, Thm. 4.1 (3.1 in the preprint)).

Statement

Proposition

Let $(\mathcal{C}, \otimes, \mathbb{1})$ be a cofibrantly generated monoidal model category and let $R \,\in\, Mon(\mathcal{C}, \otimes, \mathbb{1})$ be a monoid object whose underlying object is cofibrant in $\mathcal{C}$. Then:

1. The category $R Mod(\mathcal{C}, \otimes, \mathbb{1})$ of internal $R$-module objects carries a cofibrantly generated model structure whose weak equivalences and fibrations are those whose underlying maps are so in $\mathcal{C}$, hence which is right transferred along the forgetful functor $U$:

   $$R Mod(\mathcal{C}) \underoverset {\underset{ U }{\longrightarrow}} {\overset{F}{\longleftarrow}} {\;\;\;\;\;\; \bot_{\mathrlap{Qu}} \;\;\;\;\;\;} \mathcal{C} \,.$$

2. If $R$ is in addition a commutative monoid object then the tensor product of modules makes $R Mod(\mathcal{C})$ itself into a monoidal model category.

(Schwede & Shipley 2000, Thm. 3.1 with Rem. 3.2)

Related concepts

• model structure on monoids in a monoidal model category

  • model structure for ring spectra

• model structure on commutative monoids in a symmetric monoidal model category

• model structure on algebras over an operad

• model structure on algebras over a monad

• model structure on simplicial T-algebras

References

General discussion:

• Stefan Schwede, Brooke Shipley, Algebras and modules in monoidal model categories, Proc. London Math. Soc. 80 2 (2000) 491-511 [arXiv:math/9801082, doi:10.1112/S002461150001220X]

Examples:

• Jordan Williamson, §2.1 in: Algebraic models of change of groups functors in (co)free rational equivariant spectra, J. Pure Appl. Algebra 226 11 (2022) 107108 [doi:10.1016/j.jpaa.2022.107108, arXiv:2003.12412]

The special case of the model structure on modules in a functor category with values in a closed symmetric monoidal model category is (re-)derived (see the discussion here) in:

• Angelos Anastopoulos, Marco Benini, Sec. 2.4 of Homotopy theory of net representations, Rev. Math. Phys. (2023) [arXiv:2201.06464, doi:10.1142/S0129055X23500083]

(there for the purpose of desribing representations of nets of observables in homotopical algebraic quantum field theory).