nLab
pushout-product axiom
Context
Model category theory
model category

Definitions
Morphisms
Universal constructions
Refinements
Producing new model structures
Presentation of $(\infty,1)$ -categories
Model structures
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for ∞-groupoids

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Monoidal categories
monoidal categories

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Closed structure
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Semisimplicity
Morphisms
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Examples
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In higher category theory
Contents
Idea
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 .

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 $f' : X' \to Y'$ their pushout-product , hence the induced morphism out of the coproduct

$(X \otimes Y') \coprod_{X \otimes X'} (Y \otimes X')
\to
Y \otimes Y'
\,,$

is itself a cofibration, which, furthermore, is acyclic if $f$ or $f'$ is.

This means that the tensor product

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

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
(195.37.209.180)