Product systems over right-angled Artin semigroups

Fowler, Neal; Sims, Aidan

doi:10.1090/S0002-9947-01-02911-7

Product systems over right-angled Artin semigroups
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by Neal J. Fowler and Aidan Sims PDF

Trans. Amer. Math. Soc. 354 (2002), 1487-1509 Request permission

Abstract:

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

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