nLab pushout-product axiom



Model category theory

model category, model \infty -category



Universal constructions


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

Monoidal categories

monoidal categories

With symmetry

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products



Internal monoids



In higher category theory



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 and having a model category structure, that satisfies 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' \colon 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 \colon C \times C \to C

is a left Quillen bifunctor.

If the monoidal category underlying CC is a closed monoidal category, one can dually define the pullback-power axiom to mean that if f:ABf \colon A \to B is a cofibration and g:XYg \colon X \to Y is a fibration, their pullback power

[B,X][A,X]× [A,Y][B,Y], [B, X] \to [A, X] \times_{[A,Y]} [B, Y] \,,

is a fibration, which, furthermore, is acyclic if ff or gg is.

However, by Joyal-Tierney calculus, the pullback-power axiom holds if and only if the pushout-power axiom holds.


  • 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.


Discussion for model \infty -categories (such as with homotopy Kan fibrations):

Last revised on July 7, 2022 at 12:43:11. See the history of this page for a list of all contributions to it.