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dc.contributor.authorBagaria, Joan
dc.contributor.authorCasacuberta, Carles
dc.contributor.authorMathias, Adrian R. D.
dc.contributor.authorRosický, Jiří
dc.date.accessioned2011-03-20T12:01:10Z
dc.date.accessioned2016-10-05T14:14:19Z
dc.date.available2011-03-20T12:01:10Z
dc.date.available2016-10-05T14:14:19Z
dc.date.issued2011-05-15
dc.identifier.urihttp://publications.mfo.de/handle/mfo/1187
dc.descriptionResearch in Pairs 2008en_US
dc.description.abstractWe 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.isoenen_US
dc.publisherMathematisches Forschungsinstitut Oberwolfachen_US
dc.relation.ispartofseriesOberwolfach Preprints;2011,14
dc.titleDefinable orthogonality classes in accessible categories are smallen_US
dc.typePreprinten_US
dc.identifier.doi10.14760/OWP-2011-14
local.scientificprogramResearch in Pairs 2008
local.series.idOWP-2011-14
local.subject.msc03
local.subject.msc18
local.subject.msc55
dc.identifier.urnurn:nbn:de:101:1-201105249914
dc.identifier.ppn1650929501


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