2019/01/16 by Ziemiański, Krzysztof
#55P15 #55U05 #68Q99 #Algebraic Topology (math.AT) #FOS: Mathematics
paper · doi:10.48550/arxiv.1901.05206
For a given pre-cubical set (\square--set) K with two distinguished vertices \bO, \bI, we prove that the space \vP(K)_\bO^\bI of d-paths on the geometric realization of K with source \bO and target \bI is homotopy equivalent to its subspace \vPt(K)_\bO^\bI 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 \vP(K)_\bO^\bI.