, Volume 17, Issue 3, pp 211-246
Date: 25 Jul 2008

The Orthogonal Subcategory Problem and the Small Object Argument

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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.

Financial support by Centre for Mathematics of University of Coimbra and by School of Technology of Viseu is acknowledged by the third author.