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International Journal of Mathematics and Mathematical Sciences
Volume 2007, Article ID 87404, 20 pages
Research Article

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

Preuves Programmes et Systèmes, Université Paris 7–Denis Diderot, Case 7014, 2 Place Jussieu, Paris Cedex 05 75251, France

Received 11 October 2006; Revised 24 January 2007; Accepted 31 March 2007

Academic Editor: Monica Clapp

Copyright © 2007 Philippe Gaucher. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.