Article

Applied Categorical Structures

, Volume 17, Issue 3, pp 211-246

First online:

The Orthogonal Subcategory Problem and the Small Object Argument

  • Jiří AdámekAffiliated withDepartment of Theoretical Computer Science, Technical University of Braunschweig Email author 
  • , Michel HébertAffiliated withMathematics Department, The American University in Cairo
  • , Lurdes SousaAffiliated withDepartamento de Matemática da Escola Superior de Tecnologia de Viseu, Campus PolitécnicoCMUC, University of Coimbra

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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.

Keywords

Orthogonal subcategory problem Small object argument Injectivity logic Presentable morphism Orthogonality logic