Approximations and adjoints in homotopy categories

TL;DR

This paper establishes a criterion for right approximations in cocomplete additive categories and uses it to construct adjoint functors in homotopy categories, with applications to pure derived categories and their properties.

Contribution

It generalizes a known result to provide a new criterion for approximations and constructs adjoints in homotopy categories, advancing the understanding of pure derived categories.

Findings

Pure derived category of any module category is compactly generated.

Provided a criterion for the existence of right approximations in cocomplete additive categories.

Constructed adjoint functors in homotopy categories.

Abstract

We provide a criterion for the existence of right approximations in cocomplete additive categories; it is a straightforward generalisation of a result due to El Bashir. This criterion is used to construct adjoint functors in homotopy categories. Applications include the study of (pure) derived categories. For instance, it is shown that the pure derived category of any module category is compactly generated.