Partial categories and directed path spaces -- a proposed construction of orbit categories for p-local finite groups

Given a finite group G the orbit category of G consists of all transitive G-sets and equivariant maps between them. Aside from algebra the orbit category has also proved very useful in topology for describing G-equivariant homotopy theory.
For a saturated fusion system/p-local finite group F with Sylow subgroup S the existing constructions of an orbit category for F only makes sense for subgroups of S that are "sufficiently large". In this talk I will propose a construction of an orbit category for F that works for all subgroups of S, but the result will be a "partial" category (to be defined during the talk) where composition of morphisms is only partially defined. The construction builds upon the theorem of Andy Chermak that a saturated fusion system is always realized by a (suitably unique) partial group.
Every partial category gives rise to a actual category enriched in simplicial sets via a very explicit procedure using a sort of directed path spaces. We shall see that the proposed orbit category restricts to a classical category when considering "sufficiently large" subgroups of S and that we recover the old definition of an orbit category for F.
This joint project with Rémi Molinier is very much a work in progress, and as such the talk will contain many more conjectures than theorems.