Let $X$ be topological realization of a (finite) 
[Delta set][1], $G$ a finite group and $p: P \to X$ a 
principal $G$-bundle.


Let's recall the standard fact that more generally any 
numerable principal G-bundle $p: P \to X$ over paracompact 
base space $X$ after having fixed a trivializing cover
$(U_i)_{i \in I}$ of $p$ the bundle can be encoded (up to 
isomorphism) by cocycle datum 
$(g_{ij}: U_i \cap U_j \to G)_{j,i \in I}$ satisfying 
cocycle condition. Now from practical reason it might be in 
general hard to find a trivializing cover of $p$.


My question is if for a "simple enough" base space - a finite 
Delta complex $X$, that's will be the "price" we are ready pay
to obtain a simpler description - the cocycle 
data can be constructed more or less from the combinatorical 
data of the given Delta set $X$? Or even better if it depends only 
on the $1$-skeleton of $X$?

More concretely I would like to know if following construction
might work:

Let $X_0$ the set von vertices and $X_1$ the set of edges or
$1$-simplices of $X$. Fix for any vertex $v \in X_0$ any 
open subset $U_v \subset X$ over which $p:P \to X$ trivialize,
$P_{\vert{U_v}} \cong U_v \times G$ and such that $v$ is the 
only vertex which $U_v$ contains.
Note that we not require
that the $(U_v)_{v \in X_0}$ cover $X$, just the the set of vertices.

Next let $e_{vw} \in X_1$ be an edge or $1$-simplex with vertices
$v,w \in X_0$. Note that in contrast to a simplicial complex
the two vertices of a $1$-simplex of a Delta complex
 can be identified, that's becomes important. 
We distinguish two cases: $v \neq w$ and $v=w$, ie $e_{vw}$ 
is a circle in $X$.

Case $v \neq w$: Then the $1$-simplex $e_{vw}$ is a line 
isomorphic to interval $[0,1]$ and therefore contractible.
Then we can extend the trivializing neighborhood $U_v$ of $v$
to contain $[0,3/4)$ and $U_w$ of $w$
to contain $(1/4,1]$ such that the intersetion $U_v \cap U_w$ 
contain the $(1/4,3/4)$ middle piece of $e_{vw}$. 

Let $U_{e_{vw}} $ a connected component containing this
$(1/4,3/4)$ piece of $e_{vw}$ but not the middle pieces of over
$1$-simplices $e'_{vw}$ with possibly same vertices. 

We consider the restricted map $  U_{e_{vw}} \to G  $ induced 
usual cocycle associated to trivializations 
$P_{\vert{U_v}} \cong U_v \times G$. This map is constant and 
therefore we obtain a $g_{e_{vw}} \in G$ is it's image.


Case $v=w$: Then $e_{vw}$ is a cirle. We next pullback 
$p: P \to X$ along inclusion $e_{vw} \subset X$ and 
obtain obtain induced $G$-bundle over $e_{vw} \cong S^1$.
This can be encoded by a single $g_{e_{vw}} \in G$. Let's take it and
we obtain a datum

$$ ( g_{e_{vw}})_{e_{vw} \in X_1}   $$

I would like to call it "$1$-skelton cocycle" and this raises the 
question if this datum suffice to reconstruct up to isom our
principal $G$-bundle $p: P \to X$ as long as our space $X$ is
"simple enough" to be a Delta set.

Next step is to use the Delta set description of a model of 
classifing space $BG$ -the "bar construction" -which can be found e.g. in A. Hatcher's 
Algebraic Topology on p 89.
We observe that it's $1$-simplices can be identified with $G$.

Let's do the most naive thing we can do now: we restrict to
$1$-skeleta, map every vertex $v$ of $X$ to unique vertex 
of $BG$ and every $1$-simplex $e_{vw}$ to $g_{e_{vw}}$.

This gives a map $X_1 \to BG_1$ on level of $1$-skeleta.
This raises the natural question can this map be extended to
$f_p: X \to BG$ canonically. If yes then in the next step we temper
to compare original $p:P \to X$ with pullback of universal
bundle $U:EG \to BG$ along $f_p$.


First of all, I would be happy about any feedback about
this construction. Does it work at all, are there unavoidable
gaps, eg the construction step with handling circles, the 
extension from $1$-skeleton to the whole Delta complex, and even 
if that's possible, do I retain up to isom my $G$-bundle back?

Of course it's not always possible to extend any map of 
Delta sets from $1$-skeleton, but my hope that it might be work 
here comes from that the target space
$BG$ for finite $G$ (or more general "simple" class of groups) 
is a model for 1-homotopy type, 
so it reasonable to expect that the neccessary data for construction 
of classifiyng map associated to principal G-bundle $p:P \to X$ 
over $X$ 
"simple enough" - here a finite Delta set - lying in 1-skeleton 
might be enough to fully build up $f_p:X \to BG$
or is my intuition  leading me astray. The question is closely connected to [/463641][2]


  [1]: https://en.wikipedia.org/wiki/Delta_set
  [2]: https://mathoverflow.net/questions/463627/combinatorical-description-of-classifying-map-for-principal-g-bundle-over-delt/463641#463641