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.
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 .
is a subcategory of .
The definition and study of directed topological spaces was undertaken in
Applications of categories regarded as models for directed spaces are discussed in