Strictifying and taming directed paths in Higher Dimensional Automata

Directed paths have been used by several authors to describe concurrent executions of a program. Spaces directed in an appropriate state space contain with all possible legal schedulings. It is interesting investigate whether one obtains different topological properties such if restricts attention schedulings "nice" properties, eg involving synchronizations. This note shows that this not the case, ie may operate nice without inflicting any harm. Several results had previously obtained Ziemianski. We attempt make them accessible for wider audience giving easier proof these findings application quite elementary from algebraic topology; notably nerve lemma.