Product systems over right-angled Artin semigroups

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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 G \mathcal G . We show that the existing notions of product systems fit into our categorical framework, as do the k k -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 G \mathcal G . In particular, we obtain necessary and sufficient conditions under which a collection of k k 1 1 -graphs form the coordinate graphs of a k k -graph.