Derived, coderived, and contraderived categories of locally presentable abelian categories

TL;DR

This paper develops new model structures for derived and coderived categories of locally presentable abelian categories, establishing their generation by key objects and ensuring well-behaved Hom sets under certain conditions.

Contribution

It constructs projective and injective derived model structures for locally presentable and Grothendieck abelian categories, and introduces a framework for derived categories of exact categories with object size functions.

Findings

Existence of enough homotopy projective complexes of projective objects.

Derived categories are generated by projective or injective generators.

Derived categories of locally presentable abelian categories have Hom sets.

Abstract

For a locally presentable abelian category $\mathsf B$ with a projective generator, we construct the projective derived and contraderived model structures on the category of complexes, proving in particular the existence of enough homotopy projective complexes of projective objects. We also show that the derived category $\mathsf D(\mathsf B)$ is generated, as a triangulated category with coproducts, by the projective generator of $\mathsf B$. For a Grothendieck abelian category $\mathsf A$, we construct the injective derived and coderived model structures on complexes. Assuming Vopenka's principle, we prove that the derived category $\mathsf D(\mathsf A)$ is generated, as a triangulated category with products, by the injective cogenerator of $\mathsf A$. More generally, we define the notion of an exact category with an object size function and prove that the derived category of any such exact category with exact $\kappa$-directed colimits of chains of admissible monomorphisms has Hom sets. In particular, the derived category of any locally presentable abelian category has Hom sets.