model category, model $\infty$-category
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Morphisms
Universal constructions
Refinements
Producing new model structures
Presentation of $(\infty,1)$-categories
Model structures
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on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
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category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
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 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.
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
is itself a cofibration, which, furthermore, is acyclic if $f$ or $f'$ is.
This means that the tensor product
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
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.
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.
dually: pullback-power axiom in enriched model categories
Discussion for model $\infty$-categories (such as with homotopy Kan fibrations):
Last revised on July 7, 2022 at 12:43:11. See the history of this page for a list of all contributions to it.