Purity and almost split morphisms in abstract homotopy categories: A unified approach via Brown representability
The main purpose of this paper is to develop a theory of purity and a theory of existence of almost split morphisms in an abstract homotopy category (an additive category which has coproducts, weak cokernels and a minimal Whitehead subcategory). The main tool for doing this is the classical Brown representability theorem. The theory is applied in the last section to important examples of abstract homotopy categories, namely those of module categories, locally finitely presented categories with products and compactly generated triangulated categories. In all these examples, the author unifies the main results on purity and the existence of almost split morphisms, sequences and triangles.
scientific article; zbMATH DE number 1879216