Components of the Fundamental Category II
Résumé
In this article we carry on the study of the fundamental category of a partially ordered topological space, as arising in e.g. concurrency theory, initiated in [APCS 12/1, 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 [APCS 12/1, 2004], as well as give definitions of future and past component categories, related to the past and future models of [Grandis 2003]. 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 2001, Goubault 2001], we show in this paper a similar theorem for component categories (conjectured in [APCS 12/1, 2004]). This proves useful for inductively computing the component category in some circumstances, for instance, in the case of simple PV mutual exclusion models [CONCUR 2005], corresponding to partially ordered subspaces of R^n 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.