dc.contributor.author Bagaria, Joan dc.contributor.author Casacuberta, Carles dc.contributor.author Mathias, Adrian R. D. dc.contributor.author Rosický, Jiří dc.date.accessioned 2011-03-20T12:01:10Z dc.date.accessioned 2016-10-05T14:14:19Z dc.date.available 2011-03-20T12:01:10Z dc.date.available 2016-10-05T14:14:19Z dc.date.issued 2011-05-15 dc.identifier.uri http://publications.mfo.de/handle/mfo/1187 dc.description Research in Pairs 2008 en_US dc.description.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 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 $\mathcal{S}^\bot$ is a small-orthogonality class (hence reflective, if $\mathcal{C}$ is cocomplete) is provable in ZFC if $\mathcal{S}$ is $\Sigma_1$, while it follows from the existence of a proper class of supercompact cardinals if $\mathcal{S}$ is $\Sigma_2$, and from the existence of a proper class of what we call $C(n)$-extendible cardinals if $\mathcal{S}$ is $\Sigma_{n+2}$ for $n \geq 1$. These cardinals form a new hierarchy, and we show that Vopenka's principle is equivalent to the existence of $C(n)$-extendible cardinals for all $n$. As a consequence, we prove that the existence of cohomological localizations of simplicial sets, a long-standing open problem in algebraic topology, follows from the existence of sufficiently large supercompact cadianls, since $E*$-equivalences are $\Sigma_2$-definable for every cohomology theory $E*$. On the other hand, $E*$-equivalences are $\Sigma_1$-definable, from which it follows (as is well known) that the existence of homological localizations is provable in ZFC. en_US dc.language.iso en en_US dc.publisher Mathematisches Forschungsinstitut Oberwolfach en_US dc.relation.ispartofseries Oberwolfach Preprints;2011,14 dc.title Definable orthogonality classes in accessible categories are small en_US dc.type Preprint en_US dc.identifier.doi 10.14760/OWP-2011-14 local.scientificprogram Research in Pairs 2008 local.series.id OWP-2011-14 local.subject.msc 03 local.subject.msc 18 local.subject.msc 55 dc.identifier.urn urn:nbn:de:101:1-201105249914 dc.identifier.ppn 1650929501
﻿