nLab
pushout-product axiom

Idea

The pushout-product axiom is a compatibility condition between

1. a closed symmetric monoidal structure

2. and 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.

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 $f\prime :X\prime \to Y\prime$ the induced map

$$(X\otimes Y\prime ){\cup }_{X\otimes X\prime }(Y\otimes X\prime )\to Y\otimes Y\prime$$(X \otimes Y') \cup_{X \otimes X'} (Y \otimes X') \to Y \otimes Y'

is a cofibration which is acyclic if $f$ or $f\prime$ is.

This means that the tensor product

$$\otimes :C\times C\to C$$\otimes : C \times C \to C

is a left Quillen bifunctor.

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.