Groupoids and C*-Algebras for Categories of Paths

In this paper we describe a new method of defining C^*-algebras from oriented combinatorial data, thereby generalizing the construction of algebras from directed graphs, higher-rank graphs, and ordered groups. We show that only the most elementary notions of concatenation and cancellation of paths are required to define versions of Cuntz-Krieger and Toeplitz-Cuntz-Krieger algebras, and the presentation by generators and relations follows naturally. We give sufficient conditions for the existence of an AF core, hence of the nuclearity of the C^*-algebras, and for aperiodicity, which is used to prove the standard uniqueness theorems.