  We extend properties of the weak order on finite Coxeter groups to Weyl
groupoids admitting a finite root system. In particular, we determine the
topological structure of intervals with respect to weak order, and show that
the set of morphisms with fixed target object forms an ortho-complemented meet
semilattice. We define the Coxeter complex of a Weyl groupoid with finite root
system and show that it coincides with the triangulation of a sphere cut out by
a simplicial hyperplane arrangement. As a consequence, one obtains an algebraic
interpretation of many hyperplane arrangements that are not reflection
arrangements.
