Discrete Conduché fibrations and $C^*$-algebras

Abstract

The $k$-graphs in the sense of Kumjian and Pask~\cite {KP} are discrete Conduch\'{e} fibrations over the monoid~$\mathbb {N}^k$, satisfying a finiteness condition. We examine the generalization of this construction to discrete Conduch\'{e} fibrations with the same finiteness condition and a lifting property for completions of cospans to commutative squares, over any category satisfying a strong version of the right Ore condition, including all categories with pullbacks and right Ore categories in which all morphisms are monic.