# 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

## 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)).

## References

General discussion:

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 $[$arXiv:2201.06464$]$

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

Last revised on May 31, 2022 at 08:55:46. See the history of this page for a list of all contributions to it.