Cohomological localizations and set-theoretical reflection

Abstract

Homological localizations of spaces and spectra have been a fundamental tool in algebraic topology since the decade of 1970, especially in the setting of chromatic homotopy. However, it is unknown whether the existence of cohomological localizations can be proved in ZFC or not. Although this is apparently a homotopy-theoretical problem, it turned out to be closely related with set-theoretical reflection principles and therefore with the existence of large cardinals. In this note we present the state of the art with enough background so that proofs of results are readable by both topologists and set theorists.