Homotopy theory of Well-generated algebraic triangulated categories

TL;DR

This paper constructs a model structure on DG categories stable under key operations and demonstrates how it naturally enhances the category of well-generated algebraic triangulated categories, advancing the understanding of their homotopy theory.

Contribution

It introduces a cofibrantly generated Quillen model structure on DG categories with specific stability properties and connects it to the framework of well-generated algebraic triangulated categories.

Findings

Established a new model structure on DG categories

Connected the model to well-generated algebraic triangulated categories

Provided tools for homotopical analysis of triangulated categories

Abstract

For every regular cardinal $\alpha$, we construct a cofibrantly generated Quillen model structure on a category whose objects are essentially DG categories which are stable under suspensions, cosuspensions, cones and $\alpha$-small sums. Using results of Porta, we show that the category of well-generated (algebraic) triangulated categories in the sense of Neeman is naturally enhanced by our Quillen model category.