Limits and colimits, generators and relations of partial groups

Abstract

The purpose of this short paper is to analyse limits and colimits in the category P art of partial groups, algebraic structures introduced by A. Chermak. The class of partial groups contains a subclass of objects corresponding to the class of transporter systems as defined by Oliver and Ventura. Special cases of such transporter systems are the centric linking systems associated to saturated fusion systems that were defined by Broto, Levi and Oliver. We will prove that P art is both complete and cocomplete and, in addition, that the full subcategory of finite partial groups is both finitely complete and finitely cocomplete. A. González has proven that P art is equivalent to a full subcategory of the category sSet of simplicial sets. We will show that such subcategory is not closed under formation of colimits in sSet, so that a different construction must be taken into consideration.