An extension theory for partial groups and localities

Abstract

A partial group is a generalization of the concept of group recently introduced by A. Chermak. By considering partial groups as simplicial sets, we propose an extension theory for partial groups using the concept of (simplicial) fibre bundle. This way, the classical extension theory for groups naturally extends to an extension theory of partial groups. In particular, we show that the category of partial groups is closed by extensions. We also describe the cohomological obstructions for existence and uniqueness of extensions, generalizing the usual obstructions for group extensions. The second part of the paper considers extensions of (finite) localities, which are a particular type of partial group, mimicking the p-local structure of finite groups. The goal here is to give sufficient conditions for an extension of localities to produce a new locality.