\(T\)-homotopy and refinement of observation. II: Adding new \(T\)-homotopy equivalences (Q925393)

In an earlier series of papers [Homology Homotopy Appl. 5, No. 1, 549--599, electronic only (2003; Zbl 1069.55008)], the author introduced the notion of \(T\)-homotopy as a tool for studying higher dimensional automata. This original definition fails to allow the identification of a directed segment with a three-dimensional cube. The author introduces a new definition of \(T\)-homotopy based on the previously introduced idea of refinement of observation. It is shown that up to weak \(S\)-homotopy a \(T\)-homotopy equivalence in the old sense is a \(T\)-homotopy equivalence in this new sense. The new definition overcomes the previously mentioned problem of failing to identify the directed segment with a three-dimensional cube. The paper is self contained in that it can be read without reference to the earlier one. Proofs rely on a category theoretic approach allied with a notion of globular complex -- a complex whose cells carry flows on them.