nLab pushout-product axiom

Idea

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.

Definition

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

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

$$(X \otimes Y') \coprod_{X \otimes X'} (Y \otimes X') \to Y \otimes Y' \,,$$

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

This means that the tensor product

$$\otimes \colon C \times C \to C$$

is a left Quillen bifunctor.

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

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

is a fibration, which, furthermore, is acyclic if $f$ or $g$ is.

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

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.

Related concepts

• monoidal model category

• dually: pullback-power axiom in enriched model categories

• Joyal-Tierney calculus

References

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

• Aaron Mazel-Gee, around Def. 4.3 in: Model ∞-categories II: Quillen adjunctions, New York Journal of Mathematics 27 (2021) 508-550. [arXiv:1510.04392, nyjm:27-21]