Quivers and path semigroups characterized by locality conditions

Path algebras from quivers are a fundamental class of algebras with wide applications. Yet it is challenging to describe their universal properties since their underlying path semigroups are only partially defined. A new notion, called locality structures, was recently introduced to deal with partially defined operation, with motivation from locality in convex geometry and quantum field theory. We show that there is a natural correspondence between locality sets and quivers which leads to a concrete class of locality semigroups, called Brandt locality semigroups, which can be obtained by the paths of quivers. Further these path Brandt locality semigroups are precisely the free objects in the category of Brandt locality semigroups with a rigidity condition. This characterization gives a universal property of path algebras and at the same time a combinatorial realization of free rigid Brandt locality semigroups.