Definable orthogonality classes in accessible categories are small

Abstract

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 Vopěnka'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 a locally presentable category $\mathcal{C}$ of structures, the orthogonal class of objects is a small-orthogonality class (hence reflective) can be proved in ZFC if $\mathcal{S}$ is ${\mathbf{\Sigma }}_1$, while it follows from the existence of a proper class of supercompact cardinals if $\mathcal{S}$ is ${\mathbf{\Sigma }}_2$, and from the existence of a proper class of what we call $C(n)$-extendible cardinals if $\mathcal{S}$ is ${\mathbf{\Sigma }}_{n+2}$ for $n\ge 1$. These cardinals form a new hierarchy, and we show that Vopěnka's principle is equivalent to the existence of $C(n)$-extendible cardinals for all $n$. As a consequence of our approach, we prove that the existence of cohomological localizations of simplicial sets, a long-standing open problem in algebraic topology, is implied by the existence of arbitrarily large supercompact cardinals. This follows from the fact that $E^{\ast }$-equivalence classes are ${\mathbf{\Sigma }}_2$, where $E$ denotes a spectrum treated as a parameter. In contrast with this fact, $E_{\ast }$-equivalence classes are ${\mathbf{\Sigma }}_1$, from which it follows (as is well known) that the existence of homological localizations is provable in ZFC.