Components of the Fundamental Category II

In this article we carry on the study of the fundamental category (Goubault and Raussen, 2002; Goubault, 2003) of a partially ordered topological space (Nachbin, 1965; Johnstone, 1982), as arising in e.g. concurrency theory (Fajstrup et al., 2006), initiated in (Fajstrup et al., 2004). The “algebra” of dipaths modulo dihomotopy (the fundamental category) of such a po-space is essentially finite in a number of situations. We give new definitions of the component category that are more tractable than the one of (Fajstrup et al., 2004), as well as give definitions of future and past component categories, related to the past and future models of (Grandis, 2005). The component category is defined as a category of fractions, but it can be shown to be equivalent to a quotient category, much easier to portray. A van Kampen theorem is known to be available on fundamental categories (Grandis, 2003; Goubault, 2003), we show in this paper a similar theorem for component categories (conjectured in (Fajstrup et al., 2004)). This proves useful for inductively computing the component category in some circumstances, for instance, in the case of simple PV mutual exclusion models (Goubault and Haucourt, 2005), corresponding to partially ordered subspaces of IR minus isothetic hyperrectangles. In this last case again, we conjecture (and give some hints) that component categories enjoy some nice adjunction relations directly with the fundamental category.