concurrent program P, the topology of X can be seen as the 'local commutations' of actions in the program P. In classical models of concurrency like 'asynchronous graphs' of 'Mazurkiewicz traces', the local commutations are provided by a relation over the arrows or the actions. The fundamental category The fundamental category of a directed space is defined by mimicking the construction of the fundamental groupoid of a topological space. More precisely, given a directed space we consider the (small) category of directed paths over up to monotonic reparametrisation and define the fundamental category of as the quotient . This construction gives rise to a functor from the category of directed spaces to the category of small categories. Some properties The fundamental category functor satisfies some kind of Seifert–van Kampen theorem. The fundamental category functor preserves binary products. As a consequence of the antisymmetry, the fundamental category C of a pospace is loop-free i.e. for all objects x and y, if both homsets C(x,y) and C(y,x) are nonempty, then x = y and C(x,x) is a singleton. Two directed paths γ and δ sharing the same image i.e. {γ(t) | t∊dom(γ)} = {δ(t) | t∊dom(δ)} are dihomotopic i.e. γ ~ δ. This property obviously fails in algebraic topology e.g. consider paths winding around the circle. Given X the model of some concurrent program P, the homsets of the fundamental category of X are countable. In addition, if no looping instruction occurs in P, then the homsets of X are finite. This is the case when P is a PV program in the sense originally given by Dijkstra. In comparison, all the nontrivial homsets of the category of directed paths DX are uncountable. The category of components While the fundamental category construction drastically reduces the size of the homsets of DX, it leaves its collection of objects unchanged. And yet, if X is the geometric model of some concurrent program P, this collection is uncountable. The category of components was introduced to find a full subcategory of the fundamental category with as few objects as possible though it contains all the relevant information from the original. If is a loop-free category, then its category of components can be described in the language of category theory without assuming is the fundamental category of some directed space. In this case the intuitive notion of insignificant morphisms is formalised as a collection of morphisms of