Let $X$ be topological realization of a (finite) Delta set, $G$ 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
