nLab simplicial monoidal model category

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Contents

Context

Model category theory

model category, model \infty -category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for equivariant \infty-groupoids

for rational \infty-groupoids

for rational equivariant \infty-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general \infty-algebras

specific \infty-algebras

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\infty,1)-sheaves / \infty-stacks

Enriched category theory

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

Contents

Idea

A simplicial monoidal model category is a sSet Qu sSet_{Qu} -enriched monoidal model category, namely a model category equipped with

  1. the structure of a monoidal model category,

  2. the structure of a simplicial model category,

  3. the compatibility structure that makes the underlying monoidal and sSet-enriched structure an

    sSet-enriched monoidal category.

Examples

Example

It ought to be true that the Bousfield localization at the “realization equivalences” (see this Prop.) of the Reedy model structure on simplicial objects in chain complexes is simplicial monoidal under the objectwise tensor product of chain complexes.

References

The notion may be implicit in many discussions of monoidal model category such as used, notably, to build symmetric monoidal smash products of spectra. One reference making more explicit the need to specify compatibility structure is:

with a precursor in

but the actual compatibility is maybe not made very explicit there either, for this see the references at enriched monoidal category.

Last revised on May 13, 2023 at 13:51:13. See the history of this page for a list of all contributions to it.