Definable orthogonality classes in accessible categories are small

Joan Bagaria, Carles Casacuberta, Adrian R. D. Mathias & Jiří Rosický
We lower substantially the strength of the assumptions needed for the validity of certain results in category theory and homotopy theory which were known to follow from Vopenka's principle. We prove that the necessary large-cardinal hypotheses depend on the complexity of the formulas defining the given classes, in the sense of the Lévy hierarchy. For example, the statement that, for a class $\mathcal{S}$ of morphisms in an accessible category $\mathcal{C}$, the orthogonal class of objects...
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