The orthogonal subcategory problem and the small object argument

Abstract

A classical result of P. Freyd and M. Kelly states that in “good” categories, the Orthogonal Subcategory Problem has a positive solution for all classes H of morphisms whose members are, except possibly for a subset, epimorphisms. We prove that under the same assumptions on the base category and on H, the generalization of the Small Object Argument of D. Quillen holds - that is, every object of the category has a cellular H-injective weak reflection. In locally presentable categories, we prove a sharper result: a class of morphisms is called quasi-presentable if for some cardinal ë every member of the class is either ë-presentable or an epimorphism. Both the Orthogonal Subcategory Problem and the Small Object Argument are valid for quasi-presentable classes. Surprisingly, in locally ranked categories (used previously to generalize Quillen’s result), this is no longer true: we present a class H of morphisms, all but one being epimorphisms, such that the orthogonality subcategory H? is not reflective and the injectivity subcategory InjH is not weakly reflective. We also prove that in locally presentable categories, the Injectivity Logic and the Orthogonality Logic are complete for all quasi-presentable classes.