Conditionally flat functors on spaces and groups

TL;DR

This paper investigates the behavior of certain flat functors on spaces and groups, showing that only nullification functors preserve flatness under pullbacks, with implications for nilpotent group extensions.

Contribution

It characterizes which localization functors preserve Gamma_{c+1}-flatness in group extensions and spaces, identifying nullifications as the only such functors.

Findings

Nullification functors are the only homotopical localizations that preserve pullbacks.

Nullifications and certain epireflections preserve flatness in group extensions.

The positive answer applies specifically to nilpotent group quotients.

Abstract

Consider an extension of groups 1 -> K -> G -> Q -> 1 which enjoys the property that the quotient by the lower central series Gamma_{c+1} produces another extension 1 -> K/ Gamma_{c+1} K -> G /Gamma_{c+1} G -> Q / Gamma_{c+1} Q -> 1, of nilpotent groups of class c. We say that the extension is Gamma_{c+1}-flat. Let us pull back the original extension along any homomorphism X -> Q. Does the pullback extension enjoy the same Gamma_{c+1}-flatness property? To answer this question we consider not only quotients by the lower central series, but any localization functor in the category of groups. In fact we start by studying the analogous question for spaces, where we replace extensions by fibration sequences. We prove that the only homotopical localization functors which behave well under pull-backs are nullifications. In the category of groups, nullifications also enjoy this property, and so do all epireflections arising from a variety of groups. In particular the answer to the question about the nilpotent quotients is positive.