  Categories of paths are a generalization of several kinds of oriented
discrete data that have been used to construct $C^*$-algebras. The techniques
introduced to study these constructions apply almost verbatim to the more
general situation of left cancellative small categories. We develop this theory
and derive the structure of the $C^*$-algebras in the most general situation.
We analyze the regular representation, and the Wiener-Hopf algebra in the case
of a subcategory of a groupoid.
