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Comparing globular complex and flow. (English) Zbl 1090.55002

The category {glCW} of globular CW-complexes [P. Gaucher and E. Goubault, Homology Homotopy Appl. 5, No. 2, 39–82, electronic only (2003; Zbl 1030.68058)] and the category {Flow} of flows [P. Gaucher, Homology Homotopy Appl. 5, No. 1, 549–599, electronic only (2003; Zbl 1069.55008)] were both introduced to model concurrent computations. Where the model {glCW} may be seen as a space in which some of the paths are executions, the category {Flow} models the space of allowed paths {P}\(X\) without an underlying space, except that there is a discrete space \(X_0\) which gives the initial and final point of each path. In {glCW}, there is a notion of \(T\)-homotopy equivalence, which models the allowed deformations along the time direction, and \(S\)-homotopy equivalence, which contains deformations “transversely” to the time direction. The category {Flow} has \(S\)-homotopy and in [P. Gaucher, loc. cit.] it was given a model structure.
In the present paper, the author constructs a functor \(cat\): {glCW} \(\to\) {Flow}, which induces an equivalence of categories from {glCW}\([\mathcal{SH}^{-1}]\), the localization with respect to \(\mathcal{SH}\), the class of \(S\)-homotopy equivalences, and {Flow}[\(\mathcal{S}^{-1}]\), the localization at weak \(S\)-homotopy equivalences. A notion of \(T\)-homotopy in {Flow} is then introduced, and it is proven, that this corresponds to the \(T\)-homotopies in {glCW} via the functor \(cat\), i.e., that there exists a \(T\)-homotopy equivalence, up to weak \(S\)-homotopy, of globular CW-complexes \(f:X\to Y\), if and only if there exists a \(T\)-homotopy equivalence \(g:cat(X)\to cat(Y)\) of flows. In the last part of the paper, the underlying homotopy type of a flow is studied. The result is, that the functor which takes the underlying topological space of a globular CW complex induces a functor from {glCW}\([\mathcal{SH}^{-1}]\) to {HoTop}. The composite functor {Flow} \(\to\) {Flow}\([\mathcal{S}^{-1}]\simeq\) {glCW}\([\mathcal{SH}^{-1}] \to \){HoTop} is the underlying homotopy type of {Flow}. This is then a dihomotopy invariant.

MSC:

55P99 Homotopy theory
68Q85 Models and methods for concurrent and distributed computing (process algebras, bisimulation, transition nets, etc.)
55P15 Classification of homotopy type
55U05 Abstract complexes in algebraic topology
55U35 Abstract and axiomatic homotopy theory in algebraic topology
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