TITLE: Is every category equivalent to the fundamental category of a directed space?
QUESTION [8 upvotes]: 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:

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

REPLY [2 votes]: The obvious subdivision to try is the ordinal one (see paper by Phil Ehlers and myself). I don't guarantee that it will do the job but ....  
You might look at things on quasi-categories as well.  
There is also the idea of exit paths and Lurie has something on stratified spaces and this has been explored further by 
D. Ayala, J. Francis and N. Rozenblyum, 2015, A stratified homotopy hypothesis, arXiv preprint arXiv:1502.01713.  I hope this helps. (Do contact me by e-mail if you want to discuss those sources in more depth.)