The Orthogonal Subcategory Problem and the Small Object Argument

Our paper is devoted to two classical problems of category theory: the Orthogonal Subcategory Problem which asks, given a class H of morphisms, whether the subcategory H⊥ of all objects orthogonal to every member of H is reflective. And the Small Object Argument which asks whether the subcategory InjH of all objects injective to every member of H is weakly reflective and, moreover, the weak reflection maps can be chosen to be cellular (that is: they lie in the closure of H under transfinite composition and pushout). The orthogonal subcategory problem was solved for cocomplete categories A with a factorization system (E ,M) satisfying some mild side conditions by Peter Freyd and Max Kelly [7]: the answer is affirmative whenever the class H is ”almost small” by which we mean that H is a union of a set and a subclass of E . We call categories satisfying the conditions formulated by Freyd and Kelly locally bounded. Example: the category Top of topological spaces is locally bounded for (Epi, StrongMono). This is, of course, the best factorization system for the above result. Also the category of topological spaces with an additional binary relation is locally bounded; here we present a class for which the orthogonal subcategory is not reflective. Thus, the orthogonal subcategory problem does not work for arbitrary classes. This contrasts to the result of Jǐŕı Rosický and the first author that in locally presentable categories the orthogonal subcategory problem has, assuming the large cardinal Vopěnka’s Principle, always an affirmative answer, see [5]. Now locally presentable categories are locally bounded for (StrongEpi,Mono), which is a weaker factorization system for our purposes; it does not seem to be known whether they are bounded for (Epi, StrongMono) in general. We prove that, nevertheless, independent of set theory, in locally presentable categories