I was wondering if there was some work in directed algebraic topology to define a *classifying directed space* of a category? By that I mean the following.

In (undirected) algebraic topology, we define the classifying space of a category as the geometric realization of its nerve. This space has the particularity property that its fundamental groupoid is equivalent to the free groupoid generated by this category. I guess (I actually do not know, I have never seen a complete proof of this statement, and I would be interested in a reference for that), this can be proved in two steps:

- Proving that the fundamental category of the nerve is isomorphic to the initial category. I think this is proved in Joyal and Tierney's notes.
- Proving that the fundamental groupoid of a simplicial set is equivalent to the fundamental groupoid of its geometric realization. I think I see how to construct the equivalence, but proving it actually works needs to prove something like ``a homotopy between paths in the geometric realization can be approximate by a simplicial homotopy''. I would be grateful if someone knows an elementary reference for that.

Anyway, my real question comes now. I think we can use these steps to prove a similar result in directed algebraic topology.

- This step is still relevant.
- Here we need a
*directed*geometric realization of a simplicial set such that its fundamental category is equivalent to the fundamental category of the simplicial set. I think I can provide such a realization by equipping standard geometric simplexes with a structure of $d$-space. For example, I can prove that the fundamental category of the standard geometric $n$-simplex with this d-space structure is equivalent to the poset $\{0,1,...,n\}$ with the natural order, as expected. But to wrap up, I get the same issues as in the undirected case.

This question seems very basic, so I guess someone should have already addressed this problem. Do you know of a work in that direction ?

Edit: I am adding the d-space structure on the standard geometric simplexes that I am considering.

If $\Delta_n = \{(t_0, t_1, \ldots, t_n) \in [0, 1]^{n+1} \mid \sum_i t_i = 1\}$, for $i \in \{0, \ldots, n\}$, define $$D_i = \{(t_0, \ldots, t_n) \in \Delta_n \mid \forall j < i, t_j < t_i \wedge \forall j > i, t_j \leq t_i\}.$$

We will say that a continuous map $\gamma: [0,1] \longrightarrow \Delta_n$ is a dipath of $\Delta_n$ if there exist $k \geq 1$, $0 \leq i_1 < \ldots < i_k \leq n$ integers and $0 < t_1 < \ldots < t_{k-1} < t_k = 1$ real numbers with:

- $\forall t \in [0,t_1]$, $\gamma(t) \in D_{i_1}$,
- $\forall j \in \{2, \ldots, k\}$, $\forall t \in ]t_{j-1},t_j]$, $\gamma(t) \in D_{i_j}$