nLab pushout-product axiom

Idea

The pushout-product axiom for a two-variable adjunction between model categories is a condition on pushout-products of certain cofibrations which ensures (together with its equivalent dual pullback-power axiom) that the two-variable adjunction is homotopically meaningful in a way (see the remark here) analogous to how the axioms of a Quillen adjunction ensure that an ordinary adjunction between model categories is. Therefore one refers to such two-variable adjunctions also as Quillen bifunctors.

In particular, in the definition of enriched model categories the pushout-product axiom ensures that the enriched tensoring/cotensoring is homotopically meaningful.

Specialized to the case of simplicial model categories this is the origin of the notion of the pushout-product axiom, in its dual guise as the pullback-power axiom as “axiom SM7” in Quillen (1967).

Specialized, alternatively, to the case of self-enrichment the pushout-product axiom for monoidal model categories ensures that the tensor product in two-variable adjunction with its internal hom-functor is homotopically well-behaved.

This situation of monoidal model categories has come to be the case where the pushout-product axiom is most prominently discussed in the literature, and where it has received its name, see the references there.

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

$$(X \otimes Y') \overset {X \otimes X'} {\amalg} (Y \otimes X') \longrightarrow Y \otimes Y' \,,$$

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.

By Joyal-Tierney calculus, the pullback-power axiom holds if and only if the pushout-product axiom holds.

Remark

The pushout-product axiom implies in particular that tensoring with cofibrant objects preserves cofibrations and acyclic cofibrations.

However the plain tensor product of a pair of (acyclic) cofibrations is in general not an (acyclic) cofibration.

Remark

In a cofibrantly generated model category the pushout product axiom holds as soon as it holds for (acyclic) generating cofibration (see here).

Remark

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

The pullback-power axiom in its role in enriched model categories, specifically in simplicial model categories originates in:

• Daniel Quillen, axiom “SM7” in §2.2 of: Homotopical Algebra, Lecture Notes in Mathematics 43, Springer (1967) [doi:10.1007/BFb0097438]

For more see the references at:

• monoidal model category

• enriched model category

• pushout-product.

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]