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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 $$D_i = \{(t_0, \ldots, t_n) \in \Delta_n \mid \forall j < i, t_j < t_i \wedge \forall j > i, t_j \leq t_i\}.$$

enter image description here

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. $\forall j \in \{2, \ldots, k\}$, $\forall t \in ]t_{j-1},t_j]$, $\gamma(t) \in D_{i_j}$
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  • $\begingroup$ 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 ? $\endgroup$ – Philippe Gaucher Jan 11 '18 at 9:05
  • $\begingroup$ Sorry. I meant the standard geometric n-simplex with my structure of d-space. I am editing my question. $\endgroup$ – Jeremy Jan 11 '18 at 9:11
  • $\begingroup$ 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. $\endgroup$ – Benjamin Steinberg Jan 11 '18 at 13:09
  • $\begingroup$ 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. $\endgroup$ – Benjamin Steinberg Jan 11 '18 at 13:10
  • $\begingroup$ 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. $\endgroup$ – Mark Grant Jan 11 '18 at 15:38
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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.)

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  • $\begingroup$ Thank you for the references. Ideas related to exit paths are on my to-do list for quite some time, I guess it is time to consider them seriously. Thank you for the proposal. I will try to digest them by myself, and I am coming back to you then. $\endgroup$ – Jeremy Jan 12 '18 at 7:58
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    $\begingroup$ I think it would be worth reformulating the notion of directed n-simplex in terms of a stratified space. $\endgroup$ – Tim Porter Jan 12 '18 at 11:49
  • $\begingroup$ You are right. It seems that my directed structure is precisely the one you get from the standard stratification. I felt like it was ad hoc, but it seems not. That's a good news. $\endgroup$ – Jeremy Jan 12 '18 at 18:00
  • $\begingroup$ 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. $\endgroup$ – Jeremy Jan 24 '18 at 9:15
  • $\begingroup$ You need to do things relative to the base poset. $\endgroup$ – Tim Porter Jan 24 '18 at 12:38

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