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

This paper is the second part of a series of papers about a new notion of T-homotopy of flows. It is proved that the old definition of T-homotopy equivalence does not allow the identification of the directed segment with the 3-dimensional cube. This contradicts a paradigm of dihomotopy theory. A new definition of T-homotopy equivalence is proposed, following the intuition of refinement of observation. And it is proved that up to weak S-homotopy, an old T-homotopy equivalence is a new T-homotopy equivalence. The left properness of the weak S-homotopy model category of flows is also established in this part. The latter fact is used several times in the next papers of this series.