Model category theory
Producing new model structures
Presentation of -categories
for stable/spectrum objects
for stable -categories
for -sheaves / -stacks
With duals for objects
With duals for morphisms
Special sorts of products
In higher category theory
The pushout-product axiom is a compatibility condition between
a closed symmetric monoidal structure
a 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.
Let be a closed symmetric monoidal category equipped with a model category structure.
Then satisfies the pushout-product axiom if for any pair of cofibrations and their pushout-product, hence the induced morphism out of the coproduct
is itself a cofibration, which, furthermore, is acyclic if or is.
This means that the tensor product
is a left Quillen bifunctor.
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.
Revised on March 30, 2016 11:18:17
by Urs Schreiber