Definable orthogonality classes in accessible categories are small

Details Definable orthogonality classes in accessible categories are small Definable orthogonality classes in accessible categories are small

2011-01-14

arxiv.org/abs/1101.2792v2

Mathematics

Category Theory

32 pages

Scientific paper

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 Levy hierarchy. For example, the statement that, for a class S of morphisms in an accessible category C, the orthogonal class of objects is a small-orthogonality class (hence reflective, if C is cocomplete) is provable in ZFC if S is \Sigma_1, while it follows from the existence of a proper class of supercompact cardinals if S is \Sigma_2, and from the existence of a proper class of what we call C(n)-extendible cardinals if S is \Sigma_{n+2} for n bigger than or equal to 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 cardinals, since cohomology equivalences are \Sigma_2 definable for every cohomology theory. On the other hand, homology equivalences are \Sigma_1 definable, from which it follows (as is well known) that the existence of homological localizations is provable in ZFC.

Affiliated with

Also associated with

No associations

Rating

Definable orthogonality classes in accessible categories are small does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Definable orthogonality classes in accessible categories are small, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Definable orthogonality classes in accessible categories are small will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-79266