Spaces of directed paths on pre-cubical sets II
Krzysztof Ziemiański · Journal of Applied and Computational Topology · 2019
For a given pre-cubical set ( \(\square \) -set) K with two distinguished vertices \(\mathbf {0}\) , \(\mathbf {1}\) , we prove that the space \(\vec {P}(K)_\mathbf {0}^\mathbf {1}\) of d-paths on the geometric realization of K with source \(\mathbf {0}\) and target \(\mathbf {1}\) is homotopy equivalent to its subspace \(\vec {P}^t(K)_\mathbf {0}^\mathbf {1}\) of tame d-paths. When K is the underlying \(\square \) -set of a Higher Dimensional Automaton A , tame d-paths on K represent step executions of A . Then, we define the cube chain category of K and prove that its nerve is weakly homotopy equivalent to \(\vec {P}(K)_\mathbf {0}^\mathbf {1}\) .