# 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 ? – Philippe Gaucher Jan 11 '18 at 9:05
• Sorry. I meant the standard geometric n-simplex with my structure of d-space. I am editing my question. – Jeremy Jan 11 '18 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. – Benjamin Steinberg Jan 11 '18 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. – Benjamin Steinberg Jan 11 '18 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. – Mark Grant Jan 11 '18 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. – Jeremy Jan 24 '18 at 9:15