Cellular covers of local groups

TL;DR

This paper demonstrates that in the category of groups, the composition of cellularization and localization functors can fail to be idempotent, answering a question posed by Emmanuel Dror Farjoun negatively.

Contribution

It shows that the composition of cellularization and localization functors is not necessarily idempotent in the category of groups, providing a counterexample to a longstanding question.

Findings

Composition of cellularization and localization is not always idempotent.

Provides a negative answer to Farjoun's question.

Highlights limitations of functor compositions in group theory.

Abstract

We prove that, in the category of groups, the composition of a cellularization and a localization functor need not be idempotent. This provides a negative answer to a question of Emmanuel Dror Farjoun.