The Orthogonal Subcategory Problem and the Small Object Argument

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 lambda every member of the class is either lambda-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(perpendicular to) is not reflective and the injectivity subcategory Inj H 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.