Spaces of directed paths on pre-cubical sets II

16 Jan 2019  ·  Ziemiański Krzysztof ·

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 $\vP^t(K)_\bO^\bI$ 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 $\vP(K)_\bO^\bI$. read more

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Algebraic Topology