nLab
pushout-product axiom

Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\infty,1)-sheaves / \infty-stacks

Monoidal categories

Contents

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 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' : X' \to Y' the pushout-product morphis, the induced morphism out of the coproduct

(XY) XX(YX)YY (X \otimes Y') \coprod_{X \otimes X'} (Y \otimes X') \to 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 : 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.

Revised on March 23, 2012 08:16:38 by Urs Schreiber (82.172.178.200)