1 9 Ja n 20 06 T - HOMOTOPY AND REFINEMENT OF OBSERVATION ( I ) : INTRODUCTION

This paper is the extended introduction of a series of papers about modelling T-homotopy by refinement of observation. The notion of T-homotopy equivalence is discussed. A new one is proposed and its behaviour with respect to other construction in dihomotopy theory is explained. The main feature of the two algebraic topological models of higher dimensional automata (or HDA) introduced in [GG03] and in [Gau03] is to provide a framework for modelling continuous deformations of HDA corresponding to subdivision or refinement of observation. Globular complexes and flows are specially designed to model the weak S-homotopy equivalences (the spatial deformations) and the T-homotopy equivalences (the temporal deformations). The first descriptions of spatial deformation and of temporal deformation dates back from the informal and conjectural paper [Gau00]. Let us now explain a little bit what the spatial and temporal deformations consist of before presenting the results. The computer-scientific and geometric explanations of [GG03] must of course be preferred for a deeper understanding. In dihomotopy theory, processes running concurrently cannot be distinguished by any observation. For instance in Figure 1, each axis of coordinates represents one process and the two processes are running concurrently. The corresponding geometric shape is a full 2-cube.