Product systems over right-angled Artin semigroups

We build upon Mac Lane's definition of a tensor category to introduce the concept of a product system that takes values in a tensor groupoid . We show that the existing notions of product systems fit into our categorical framework, as do the -graphs of Kumjian and Pask. We then specialize to product systems over right-angled Artin semigroups; these are semigroups that interpolate between free semigroups and free abelian semigroups. For such a semigroup we characterize all product systems which take values in a given tensor groupoid . In particular, we obtain necessary and sufficient conditions under which a collection of -graphs form the coordinate graphs of a -graph.