# Is every category equivalent to the fundamental category of a directed space?

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:

1. 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.
2. 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.

1. This step is still relevant.
2. 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:

1. $\forall t \in [0,t_1]$, $\gamma(t) \in D_{i_1}$,
2. $\forall j \in \{2, \ldots, k\}$, $\forall t \in ]t_{j-1},t_j]$, $\gamma(t) \in D_{i_j}$
• What is the $n$-directed simplex ? Simplicial sets have something directed in their definition (unlike symmetric simplicial sets). So is it just the $n$-simplex ? Jan 11, 2018 at 9:05
• Sorry. I meant the standard geometric n-simplex with my structure of d-space. I am editing my question. Jan 11, 2018 at 9:11
• There is a sort of obvious notion if a directed two complex and it is clearly equivalent to a presentation of a category by generators and relations. Jan 11, 2018 at 13:09
• Gabriel and Zisman give in their book the relation between the fundamental groupoid of the nerve of a category and universal groupoid of the category. Jan 11, 2018 at 13:10
• There is some discussion of directed geometric realization in the papers and books of Marco Grandis, see dima.unige.it/~grandis/Dht1.pdf and the book dima.unige.it/~grandis/Bk.XXDATXX.pdf (in particular Chapter 3). The picture seems to be that it works better for cubical sets than simplicial sets (because the obvious d-structure on simplicial sets does not respect barycentric subdivision), but then we run into trouble because there is no obvious way to construct a cubical set from a category, analogous to the simplicial nerve construction. Jan 11, 2018 at 15:38

• I played with poset-stratified spaces. One annoying point about them is colimits. I tried to define a geometric realization in those stratified spaces, using the standard stratification on simplexes, but that would not work. A simple test is the directed loop, defined by identifying the end points of a segment. If the stratification on the segment splits $[0,\frac{1}{2}]$ and $]\frac{1}{2},1]$ (that is the standard stratification), then the stratification on the circle obtained by colimit is the trivial stratification. That is, you obtain the undirected circle from the directed segment. Jan 24, 2018 at 9:15