Had a thought that maybe the right way to think about directed topologies is to simply slap multiple topological structures on the same set and require morphisms to be continuous along all of them.
Exercise: Show that the following two conditions on a function f : R → R are equivalent:
- f is continuous when the the lower limit topology is used for the domain and codomain, and also when the upper limit topology is used for both
- f is continuous for the usual topology on R, and monotone nondecreasing.
The general idea when constructing a "directed topological space" from a category in this way (not that this coincides obviously with the other notions of directed topology in the literature that I know of) is that the closure operation of one of your topologies works like "include the final endpoint of all paths" and the other looks like "include the initial endpoint". For higher-dimensional categories, you include more closure operations for the higher-dimensional notions of boundary.