Path categories and resolutions

摘要

Path categories are defined, and their basic properties are described, for simplicial and cubical sets. A calculational method for describing the path category $P(K)$ of a finite oriented simplicial complex $K$ is introduced, which involves a finite $2$-category which can be specified by generators and relations. This method specializes to higher dimensional automata via the triangulation functor from cubical to simplicial sets, and leads to calculations of their associated execution paths.