The idea of the following text is to begin with a category of objects presumed undirected and construct from that a supercategory of directed objects, analogous to how Marco Grandis developed directed topological spaces out of the usual undirected ones.
Let be a subset of the set of co-span-endomorphisms of . Let be a subset of the hom-set .
Then we call the pair an object with directed path space (or directed object) if the following conditions (attributed to Marco Grandis) are satisfied:
(Constant paths) Every map is directed;
(Reparametrisation) If , , then . If e.g , then this condition means that is a sieve in .
(Concatenation) Let be consecutive wrt. (i.e. equals ), let denote the pushout of and , then by the universal property of the pushout there is a map . By definition of the interval object (described there in the section “Intervals for Trimble -categories”) there is a unique morphism . Then the composition of and is defined by . Then shall be closed under composition of consecutive paths.
We define a morphism of objects with directed path space to be a morphism of their underlying objects that preserves directed paths (i.e. if , , then ). Objects with directed path space and morphisms thereof define a category denoted by .