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 ${\mathcal {H}}$ of morphisms whose members are, except possibly for a subset, epimorphisms. We prove that under the same assumptions on the base category and on ${\mathcal {H}}$ , the generalization of the Small Object Argument of D. Quillen holds—that is, every object of the category has a cellular ${\mathcal {H}}$ -injective weak reflection. In locally presentable categories, we prove a sharper result: a class of morphisms is called quasi-presentable if for some cardinal λ every member of the class is either λ-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 ${\mathcal {H}}$ of morphisms, all but one being epimorphisms, such that the orthogonality subcategory ${\mathcal {H}}^\perp$ is not reflective and the injectivity subcategory Inj $\,{\mathcal {H}}$ 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.