combinatorical description of classifying map for principal $G$-bundle over Delta set

Question

Let $X$ be topological realization of a (finite) Delta set, $G$ a finite group and $p: P \to X$ a principal $G$-bundle. It's standard that the isom' classes of such principal $G$-bundles are classified by an (up to homotopy unique) classifying space $BG$, which posses also a model as a Delta set: see eg A. Hatcher's "Algebraic Topology" page 89 for concrete construction.

That means that there exist a functorially compatible bijection

$$C:[X, BG]_{\text{Htpy}} \to \text{PrinBun}_G(X)/ \sim, \ \ [f] \mapsto f^{*}U $$

given explicitly by pulling back the universal bundle $U: EG \to BG$ along a homotopy a map $f: X \to BG$ representing it's homotopy class $[f]$.
This raises the natural question if that's possible to write in explicit terms the map in the other direction, the "classifying map", namely for any $p: P \to X$ a principal $G$-bundle, can we construct a map $f_p:X \to BG$ whose pullback $f_p^*U$ of universal bundle is isomorphic to $p$ explicitly?

At first glace one obstruction for finding such map is that $BG$ is only unique up to a homotopy class therefore there exist several models of $BG$ giving homotopically the same object.
Nevertheless there exist two "distinguished" ways to construct a model of $BG$, one directly as topological space, the "Milnor Construction" (can be found eg Husemoeller's "Fiber Bundles", p 54), the other firstly on simplicial level, the bar construction (eg in Hatcher's AT. p 89), then passing to realization.

So one might pose the question about explicit for explicit construction after having fixed a concrete model for $BG$.

Husemoeller's book (Thm. 12.2, p 57) - is working with Milnor's model of $EG$ and $BG$ -the explicit construction of the classfying map is based on countable partition of unity and works for numerable principal $G$-bundles over paracompact base spaces $X$.

So using analytic methods it's possible to write down this map explicitly up to finding the partition of unity.

My question is if it's also possible to write down the classifying map purely combinatorically or simplicially: Assume that our base space $X$ is
a Delta set, and in following we work with Delta set model of $EG$ and $BG$ from Hatcher book. Hatcher's bar model of $BG$ is a Delta set with $n$-simplices $[g_1 | g_2 |... |g_n]$ with boundaries $[g_1 | ... | g_i g_{i+1} | ... |g_n]$.

Is it possible to describe explicitly the classifying map associated to a principal $G$-bundle $p:P \to X$ map $f_p:X \to BG$ combinatorically?

Here an approach looking promissing but contains hurdles I not know how to overcome:

Let $p: P \to X$ be a principal $G$-bundle with base Delta set $X$. Let $X_0$ be the set of vertices, and $X_1$ the set of $1$-simplices - the "edges" - of Delta set $X$. If we could find a covering family $(U_v)_{v \in X_0}$ of open connected $U_v \subset X$ over which $p$ trivialize $P_{\vert{U_v}} \cong U_v \times G$ parametrized by vertices $X_0$ satisfying:

-vertex $v \in X_0$ lies in exactly one open set $U_v$
-every edge $e_{vw}$ with vertices $v$ and $w$ (note: there might be several edges with identical vertices) lies in $U_v \cup U_w$ for unique pair of vertices $v,w \in X_0$ and the intersection $e_{vw} \cap U_v \cap U_w$ is isomorphic to an open interval

If such covering family $(U_v)_{v \in X_0}$ exists, then we can try to construct the classifying map as follows:

This covering family $(U_v)_{v \in X_0}$ would give us a cocycle $(g_{vw}: U_v \cap U_w \to G)_{v,w \in X_0}$ and we could try to construct from this cocycle combinatorically the classifying map $f_p: X \to BG$ which maps each vertex to the unique vertex of $BG$, and every edge/ $1$-simplex of $e_{vw} $ to the $1$-simplex of $BG$ determined by $g_{vw} \in G$ from the cocycle. Note that the $1$-simplices of $BG$ it Hatcher's bar construction correspond to elements of $G$.

Problems: Does such covering $(U_v)_{v \in X_0}$ with these properties always exist for a Delta set $X$?
If yes, can this map I described on level of vertices and $1$-simplices always be extended to the total Delta complex?

If $X$ would be a simplicial complex then the answer should be yes because every $n$-simplex of a simplicial complex is determinded by it's vertices, but for Delta sets the glueing of boundaries could become more complicated. For example several vertices of same $n$-simplex could be identified, for example a $1$-simplex could become a circle when it's two vertices are identified. Note that such glueings is allowed for Delta sets, but not for for simplicial complexes.

On the other hand we can applying barycentric divisions to make a Delta set $X$ to a simplicial complex. But this raises another problem: Even if $X$ would be a simplicial complex, the $BG$ is a Delta set (all $1$-simplices are circles) and if we would resolve it with barycentric subdivisions to a simplicial complex, it would not have this structure from Hatcher's bar construction anymore, so it's $1$-simplices wouldn't be identified with elements of $G$ anymore and we couldn't associate $1$-simplices of $X$ to those of $BG$ using cocycle representation of $p: P \to X$.

Could these issues be remedied and the classifying map $f_p: X \to BG$ associated to $G$-bundle $p: P \to X$ over base $X$ Delta set be described combinatorically in that or a similar way?

Answer 1

Since the question uses semisimplicial sets, it makes sense to point out the following rather elegant model for the classifying space $\def\B{{\sf B}}\B G$ as a semisimplicial set: declare the set of $n$-simplices of $\B G$ to be the set of principal $G$-bundles over the $n$-simplex $Δ^n$ (the same $n$-simplex that is used in the construction of geometric realizations of semisimplicial sets). The face maps restrict a principal $G$-bundle over $Δ^n$ to its faces.

Now given a semisimplicial set $K$ together with a principal $G$-bundle $P→|K|$, the classifying map $|K|→|\B G|$ is the geometric realization of the simplicial classifying map $K→\B G$ that sends an $n$-simplex $σ$ in $K$ to the restriction of the principal $G$-bundle $P→|K|$ to $|σ|⊂|K|$.

As usual, to ensure that $\B G$ is a simplicial set and not merely a simplicial class, any of the standard tricks can be deployed. If the cardinality of $K$ is limited (the question talks specifically about finite $K$), then we can simply require the underlying sets of total spaces of principal $G$-bundles to be subsets of a fixed set, e.g., for a finite (or countable) $K$ we can take a set of cardinality continuum.

Concerning the question about using an open cover and its pairwise intersections to present a principal $G$-bundle, it is true at least in the contexts where good open covers are available, e.g., smooth manifolds. This follows from Proposition 4.13 in Numerable open covers and representability of topological stacks and Lemma 3.3.28 in Equivariant principal infinity-bundles, which show that given an ∞-sheaf of groups on the site of cartesian spaces, its delooping as an ∞-presheaf is in fact an ∞-sheaf, so the descent datum for a principal $G$-bundle can be expressed as a morphism from the Čech nerve of a good open cover to the delooping $\B G$ as an ∞-presheaf (i.e., objectwise).