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 a locally presentable category C of
structures, the orthogonal class of objects is a small-orthogonality class
(hence reflective) 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, is implied by the existence
of arbitrarily large supercompact cardinals. This result follows from the fact
that cohomology equivalences are \Sigma_2. In contrast with this fact, homology
equivalences are \Sigma_1, from which it follows (as is well known) that the
existence of homological localizations is provable in ZFC.Comment: 38 pages; some results have been improved and former inaccuracies
have been correcte