A Framework for Component Categories

This paper provides further developments in the study of the component categories which have been introduced in [7]. 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 [15]). 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 [2] and [1]. Still, in the recent paper [18], the homology of small categories is defined in a very similar way and applied to the study of asynchronous transition systems.