How to construct a Hovey triple from two cotorsion pairs

James Gillespie¹

¹ Ramapo College of New Jersey School of Theoretical and Applied Science 505 Ramapo Valley Road Mahwah, NJ 07430, U.S.A.

Fundamenta Mathematicae, Tome 230 (2015) no. 3, pp. 281-289.

Résumé

Let $\mathcal{A}$ be an abelian category, or more generally a weakly idempotent complete exact category, and suppose we have two complete hereditary cotorsion pairs $(\mathcal{Q}, \widetilde{\mathcal{R}})$ and $(\widetilde{\mathcal{Q}}, \mathcal{R})$ in $\mathcal{A}$ satisfying $\widetilde{\mathcal{R}} \subseteq \mathcal{R}$ and $\mathcal{Q} \cap \widetilde{\mathcal{R}} = \widetilde{\mathcal{Q}} \cap \mathcal{R}$. We show how to construct a (necessarily unique) abelian model structure on $\mathcal{A}$ with $\mathcal{Q}$ (resp. $\widetilde{\mathcal{Q}}$) as the class of cofibrant (resp. trivially cofibrant) objects, and $\mathcal{R}$ (resp. $\widetilde{\mathcal{R}}$) as the class of fibrant (resp. trivially fibrant) objects.

Détail

DOI : 10.4064/fm230-3-4

Keywords: mathcal abelian category generally weakly idempotent complete exact category suppose have complete hereditary cotorsion pairs mathcal widetilde mathcal widetilde mathcal mathcal mathcal satisfying widetilde mathcal subseteq mathcal mathcal cap widetilde mathcal widetilde mathcal cap mathcal construct necessarily unique abelian model structure mathcal mathcal resp widetilde mathcal class cofibrant resp trivially cofibrant objects mathcal resp widetilde mathcal class fibrant resp trivially fibrant objects