nLab
pushout-product axiom

Contents

Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

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

Model structures

for \infty-groupoids

for ∞-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

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

Monoidal categories

monoidal categories

With symmetry

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

The pushout-product axiom is a compatibility condition between

  1. a closed symmetric monoidal structure

  2. a model category structure

on a category.

Closedsymmetric monoidal categories satisfying the pushout-product axiom, together with a unit condition, are called monoidal model categories and hence are in particular closed monoidal homotopical categories.

This is relevant in enriched homotopy theory, which pairs enriched category theory with homotopy theory.

Definition

Let CC be a closed symmetric monoidal category equipped with a model category structure.

Then CC satisfies the pushout-product axiom if for any pair of cofibrations f:XYf : X \to Y and f:XYf' : X' \to Y' their pushout-product, hence the induced morphism out of the coproduct

(XY) XX(YX)YY, (X \otimes Y') \coprod_{X \otimes X'} (Y \otimes X') \to Y \otimes Y' \,,

is itself a cofibration, which, furthermore, is acyclic if ff or ff' is.

This means that the tensor product

:C×CC \otimes : C \times C \to C

is a left Quillen bifunctor.

Remarks

  • This implies in particular that tensoring with cofibrant objects preserves cofibrations and acyclic cofibrations.

  • However the tensor product of two (acyclic) cofibrations is in general not an (acyclic) cofibration.

  • The pushout-product axiom makes sense more generally in the context of a two-variable adjunction between model categories. This is important in enriched homotopy theory.

Last revised on March 30, 2016 at 11:18:17. See the history of this page for a list of all contributions to it.