pushout-product axiom


Model category theory

model category



Universal constructions


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



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



The pushout-product axiom is a compatibility condition between

  1. a closed symmetric monoidal structure

  2. a model category structure

on a category.

Closed symmetric 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.


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.


  • 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.

Revised on March 30, 2016 11:18:17 by Urs Schreiber (