Comparing globular complex and flow (Q2581070)

The category {\textbf{glCW}} of globular CW-complexes [\textit{P. Gaucher} and \textit{E. Goubault}, Homology Homotopy Appl. 5, No. 2, 39--82, electronic only (2003; Zbl 1030.68058)] and the category {\textbf{Flow}} of flows [\textit{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 {\textbf{glCW}} may be seen as a space in which some of the paths are executions, the category {\textbf{Flow}} models the space of allowed paths {\textbf{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 {\textbf{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 {\textbf{Flow}} has \(S\)-homotopy and in [\textit{P. Gaucher}, loc. cit.] it was given a model structure. In the present paper, the author constructs a functor \(cat\): {\textbf{glCW}} \(\to\) {\textbf{Flow}}, which induces an equivalence of categories from {\textbf{glCW}}\([\mathcal{SH}^{-1}]\), the localization with respect to \(\mathcal{SH}\), the class of \(S\)-homotopy equivalences, and {\textbf{Flow}}[\(\mathcal{S}^{-1}]\), the localization at weak \(S\)-homotopy equivalences. A notion of \(T\)-homotopy in {\textbf{Flow}} is then introduced, and it is proven, that this corresponds to the \(T\)-homotopies in {\textbf{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 {\textbf{glCW}}\([\mathcal{SH}^{-1}]\) to {\textbf{HoTop}}. The composite functor {\textbf{Flow}} \(\to\) {\textbf{Flow}}\([\mathcal{S}^{-1}]\simeq\) {\textbf{glCW}}\([\mathcal{SH}^{-1}] \to \){\textbf{HoTop}} is the underlying homotopy type of {\textbf{Flow}}. This is then a dihomotopy invariant.