This is essentially a reference request, or a request for an explanation of why this cannot be done in a useful or interesting way (i.e. an explanation of why no such reference exists!).
If I have a directed space, perhaps modeled by something like a globular complex or complicial set, is there any way to compute "homology" or "cohomology" of it with coefficients in a (commutative) monoid?
A particularly simple example might just be to think of $S^1$ as a directed 1-type, and ask whether or not there's some homology theory such that $H_1(S^1,\mathbb{N})\cong\mathbb{N}$, or something along those lines.
This seems like too simple of an idea to not have been attempted. Does anyone have a reference? If not, does anyone know what goes wrong?
One possibly silly attempt at answering this might be: write down the directed space $X$ as some kind of $(\infty,\infty)$-category, then write down the sequence of deloopings $(M,BM,B^2M,...)$ where we have to take the $n$th "delooping" here to be something like the $(\infty,n)$-category with one object and an $M$'s worth of $n$-morphisms. Then define $H^n(X,M)$ to be something like functors of $(\infty,\infty)$-categories $X\to B^nM$ modulo some notion of equivalence of $(\infty,\infty)$-functors.
