Path categories and resolutions (Q607285)

The path category functor is left adjoint to the nerve functor from small categories to simplicial sets. The path category of a simplicial set is equal to the free category generated by the \(1\)-simplices modulo the relations generated by the \(2\)-simplices. This method specializes to cubical sets via the triangulation functor from cubical to simplicial sets. The path category of a cubical set is then isomorphic to the free category generated by the \(1\)-skeleton modulo the relations arising from all \(2\)-squares. So it leads to calculations of the fundamental category. Note that the notion of simplicial set is directed in the sense that there is a natural orientation of the simplices and that the triangulation functor from cubical sets to simplicial sets preserves the direction of time contained in a cubical set (an \(n\)-cube modeling the concurrent execution of \(n\) actions).