T-Homotopy and Refinement of Observation—Part II: Adding NewT-Homotopy Equivalences

This paper is the second part of a series of papers about a newnotion of T-homotopy of flows. It is proved that the old definitionof T-homotopy equivalence does not allow the identification of thedirected segment with the 3-dimensional cube. This contradicts aparadigm of dihomotopy theory. A new definition of T-homotopyequivalence is proposed, following the intuition of refinement ofobservation. And it is proved that up to weak S-homotopy, an oldT-homotopy equivalence is a new T-homotopy equivalence. Theleft properness of the weak S-homotopy model category of flows isalso established in this part. The latter fact is usedseveral times in the next papers of this series