Garside Categories, Periodic Loops and Cyclic Sets

Garside groupoids, as recently introduced by Krammer, generalise Garside groups. A weak Garside group is a group that is equivalent as a category to a Garside groupoid. We show that any periodic loop in a Garside groupoid G may be viewed as a Garside element for a certain Garside structure on another Garside groupoid Gm, which is equivalent as a category to G. As a consequence, the centraliser of a periodic element in a weak Garside group is a weak Garside group. Our main tool is the notion of divided Garside categories, an analog for Garside categories of Bökstedt-Hsiang-Madsen’s subdivisions of Connes’ cyclic category. This tool is used in our separate proof of the K(π, 1) property for complex reflection arrangements.