Applied Categorical Structures

, Volume 17, Issue 3, pp 211–246

The Orthogonal Subcategory Problem and the Small Object Argument


    • Department of Theoretical Computer ScienceTechnical University of Braunschweig
  • Michel Hébert
    • Mathematics DepartmentThe American University in Cairo
  • Lurdes Sousa
    • Departamento de Matemática da Escola Superior de Tecnologia de ViseuCampus Politécnico
    • CMUCUniversity of Coimbra

DOI: 10.1007/s10485-008-9153-4

Cite this article as:
Adámek, J., Hébert, M. & Sousa, L. Appl Categor Struct (2009) 17: 211. doi:10.1007/s10485-008-9153-4


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.


Orthogonal subcategory problemSmall object argumentInjectivity logicPresentable morphismOrthogonality logic

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© Springer Science+Business Media B.V. 2007