A Framework for Component Categories

This paper provides further developments in the study of the component categories which have been introduced in [Fajstrup, L., E. Goubault, E. Haucourt and M. Raußen, Component categories and the fundamental category, APCS 12 (2004), pp. 81-108]. In particular, the component category functor is seen as a left adjoint hence preserves the pushouts. This property is applied to prove a Van Kampen like theorem for component categories. This last point is very important to make effective calculations. The original purpose of component categories is to suitably reduce the size of the fundamental categories which are the directed counterpart of classical fundamental groupoids (see [Higgins, P.J., "Categories and Groupoids," Mathematical Studies 32, Van Nostrand Reinhold, 1971, libre d'accs sur internet l'adresse http://www.tac.mta.ca/tac/reprints/]). In concrete examples, the fundamental category is as "big" as R while the component category is "finitely generated". We take advantage of this fact to define the cohomology of a directed geometrical shape as the cohomology of its component category. The cohomology of small categories is defined in [Baues, H.J. and G. Wirsching, Cohomology of small categories, Journal of Pure and Applied Algebra 38 (1985)] and [Baues, H.J., "Combinatorial Homotopy and 4-Dimensional Complexes," De Gruyter expositions in Mathematics 2, Walter de Gruyter, 1991]. Still, in the recent paper [Husainov, A.A., On the homolgy of small categories and asynchronous transition systems, Homology, Homotopy and Applications 6 (2004), pp. 439-471], the homology of small categories is defined in a very similar way and applied to the study of asynchronous transition systems.