nLab
monoidal model category

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Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (∞,1)-categories

Model structures

for ∞-groupoids

for ∞-groupoids

for n-groupoids

for ∞-groups

for ∞-algebras

general

specific

for stable/spectrum objects

for (∞,1)-categories

for stable (∞,1)-categories

for (∞,1)-operads

for (n,r)-categories

for (∞,1)-sheaves / ∞-stacks

Monoidal categories

monoidal categories

With symmetry

With duals for objects

With duals for morphisms

With traces

  • trace

  • traced monoidal category?

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Examples

Theorems

In higher category theory

Contents

Idea

A monoidal model category is a model category which is also a closed monoidal category in a compatible way. In particular, its homotopy category inherits a closed monoidal structure.

Definition

A (symmetric) monoidal model category is a category equipped with

such that

  • the pushout-product axiom is satisfied, and

  • For any cofibrant object X, the map Qe⊗X→e⊗X≅X is a weak equivalence, where e is the unit object of the monoidal structure and Qe→e is a cofibrant resolution for it. This is automatically satisfied if e is cofibrant, as it is in most (but not all) cases.

Properties

The central fact about a monoidal model category is that its homotopy category inherits a closed monoidal structure.

Examples

Model structure on G-objects

Assumption

Let ℰ be a category equipped with the structure of

such that

Proposition

Under these conditions there is for each finite group G the structure of a monoidal model category on the category ℰ BG of objects in ℰ equipped with a G-action, for which the forgetful functor

ℰ BG→ℰ\mathcal{E}^{\mathbf{B}G} \to \mathcal{E}

preserves and reflects fibrations and weak equivalences.

See for instance (BergerMoerdijk 2.5).

Model structure on monoids

See model structure on monoids in a monoidal model category.

References

A general standard reference is

The monoidal model structure on ℰ BG is discussed for insztance in

Revised on June 8, 2013 22:50:45 by Adeel Khan (129.173.251.24)
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