Non-existence of fiberwise localization for crossed modules

TL;DR

This paper demonstrates that fiberwise localization functors do not always exist for crossed modules, providing explicit examples and showing limitations of nullification functors in this context.

Contribution

It characterizes the conditions for the existence of fiberwise localization in crossed modules and shows that certain nullification functors lack fiberwise versions.

Findings

Fiberwise localization does not always exist for crossed modules.

Some nullification functors have non-acyclic fibers.

Explicit examples and counter-examples are provided.

Abstract

We prove that localization functors of crossed modules of groups do not always admit fiberwise (or relative) versions. To do so we characterize the existence of a fiberwise localization by a certain normality condition and compute explicit examples and counter-examples. In fact, some nullification functors do not behave well and we also prove that the fiber of certain nullification functors, known as acyclization functors in other settings such as groups or spaces, is not acyclic.