Fibrations and nullifications

Abstract

The paper considers some aspects of the preservation of a fibration F → E → B by a localization functor L, meaning that LF → LE → LB is also a fibration. We obtain necessary and sufficient conditions for this, in case L is equivalent to a nullification functor Pw. Applications include the fact that if such a fibration is preserved by a nullification, then so too are all of its pullbacks. Another application is a homological criterion for a fibration to be preserved. No similar results are known for general localization functors.