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.