The Orthogonal Subcategory Problem and the Small Object Argument |
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Authors: | Jiří Adámek Michel Hébert Lurdes Sousa |
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Institution: | (1) Department of Theoretical Computer Science, Technical University of Braunschweig, Postfach 3329, 38023 Braunschweig, Germany;(2) Mathematics Department, The American University in Cairo, P.O. Box 2511, Cairo, 11511, Egypt;(3) Departamento de Matemática da Escola Superior de Tecnologia de Viseu, Campus Politécnico, 3504-510 Viseu, Portugal;(4) CMUC, University of Coimbra, 3001-454 Coimbra, Portugal |
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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 of morphisms whose members are, except possibly for a subset, epimorphisms. We prove that under the same assumptions on the
base category and on , the generalization of the Small Object Argument of D. Quillen holds—that is, every object of the category has a cellular
-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 of morphisms, all but one being epimorphisms, such that the orthogonality subcategory is not reflective and the injectivity subcategory Inj 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. |
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Keywords: | Orthogonal subcategory problem Small object argument Injectivity logic Presentable morphism Orthogonality logic |
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