nLab pushout-product axiom



Model category theory

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In higher category theory



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.


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

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


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.


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


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.


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

For more see the references at:

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

Last revised on May 21, 2023 at 07:17:38. See the history of this page for a list of all contributions to it.