<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>T</mml:mi></mml:math>-Homotopy and Refinement of Observation—Part II: Adding New <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>T</mml:mi></mml:math>-Homotopy Equivalences

Gaucher, Philippe

doi:https://doi.org/10.1155/2007/87404

International Journal of Mathematics and Mathematical Sciences

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Research Article | Open Access

Volume 2007 | Article ID 087404 | https://doi.org/10.1155/2007/87404

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

Philippe Gaucher¹

Academic Editor: Monica Clapp

Received11 Oct 2006

Revised24 Jan 2007

Accepted31 Mar 2007

Published24 May 2007

Abstract

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.

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Copyright

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.

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