Splitting Monoidal Stable Model Categories

TL;DR

This paper demonstrates that in a monoidal stable model category, an idempotent element induces a splitting of the category into a product of localized categories, with a strong monoidal Quillen equivalence.

Contribution

It proves that the splitting of the homotopy category via an idempotent extends to a splitting of the model category itself under certain conditions.

Findings

The homotopy category splits according to idempotents in the endomorphism ring of the unit.

The splitting of the model category is a strong monoidal Quillen equivalence.

The result applies when the localized model structures exist.

Abstract

If C is a stable model category with a monoidal product then the set of homotopy classes of self-maps of the unit S forms a commutative ring. An idempotent e of this ring will split the homotopy category. We prove that provided the localised model structures exist, this splitting of the homotopy category comes from a splitting of the model category, that is, C is Quillen equivalent to the product of C localised at the object eS and C localised at the object (1-e)S. This Quillen equivalence is strong monoidal and is symmetric when the monoidal product of C is.