Generalization of the Dehornoy-Lafont order complex to categories. Application to exceptional braid groups

The homology of a Garside monoid, thus of a Garside group, can be computed efficiently through the use of the order complex defined by Dehornoy and Lafont. We construct a categorical generalization of this complex and we give some computational techniques which are useful for reducing computing time. We then use this construction to complete results of Salvetti, Callegaro and Marin regarding the homology of exceptional complex braid groups. We most notably study the case of the Borchardt braid group $B(G_{31})$ through its associated Garside category.