# nLab model structure on modules in a monoidal model category

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

#### Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

#### Higher algebra

higher algebra

universal algebra

# Contents

## 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.

## References

General discussion:

Examples:

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:

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

Last revised on November 8, 2023 at 08:52:46. See the history of this page for a list of all contributions to it.