Title:  Groupoids with weak orders

Abstract:  We discuss certain groupoids equipped with partial orders satisfying properties which abstract those of weak orders of Coxeter groups (the resulting structures are not "partially ordered groupoids" in the usual sense). In particular, we describe braid presentations of the underlying groupoids and some of the very strong closure properties of these structures under natural categorical constructions. Applied even to familiar examples such as the symmetric groups, these constructions produce interesting new structures.