Components of the Fundamental Category II

Abstract

In this article we carry on the study of the fundamental category (Goubault and Raussen, Dihomotopy as a tool in state space analysis. ). The "algebra" of dipaths modulo dihomotopy (the fundamental category) of such a pospace 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. (APCS, 12(1): 81-108, 2004), as well as give definitions of future and past component categories, related to the past and future models of Grandis (Theory Appl. Categ., 15(4): 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, Cahiers Topologie Géom. Différentielle Catég., 44: 281-316, 2003; Goubault, Homology, Homotopy Appl., 5(2): 95-136, 2003), we show in this paper a similar theorem for component categories (conjectured in Fajstrup et al. (APCS, 12(1): 81-108, 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, A practical application of geometric semantics to static analysis of concurrent programs. In: Abadi, M., de Alfaro, L. (eds.) CONCUR 2005 -Concurrency Theory: 16th International to partially ordered subspaces of IR 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.

Invariants of Directed Spaces

“…The article [7] describes a method of "compressing" information in a small category by quotienting out a subcategory of so-called weakly invertible morphisms satisfying certain properties. This method has since been refined in [11] and related to a quotient approach based on general congruences on small categories that was described earlier in [2].…”

“…Yoneda invertible morphisms are at the base of the quotienting process described and applied to the fundamental category of a d-space in [7,11]. While these seem to be very adequate for categories where all isomorphisms are identities (as in d-spaces arising from a partial order), the presence of loops causes serious problems.…”

“…Having defined the wide category C F of C, we can proceed along the lines of [7] or of [11] to arrive at a quotient category identifying objects and morphisms that are linked to each other by C F -morphisms. In order to get a consistent construction, it is usually necessary to restrict the morphisms furthermore to a wide subcategory Σ ⊆ C F ⊆ C: …”

“…In both the preorder and the factorization category, only the identities are isomorphisms. Moreover, the factorization category is loopfree (This property is essential for the later parts of [11].) In these cases, C F will be the subcategory consisting of concatenation morphisms that induce homotopy equivalences, resp.…”

“…Morphisms in the indexing category are then regarded as (weakly) invertible if they are mapped into isomorphisms by the functor. As a result, one can apply the localization method leading to components suggested in [7] and modified in [11] to the indexing category.…”