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Consider a sufficiently nice topological space $X$ as well as topological groups $G$ and $H$. Consider the functor $F$ that associates to $X$ the set of all isomorphism classes of all principal $H$-bundles $\pi_H\colon P_H\twoheadrightarrow P_G$ on all principal $G$-bundles $\pi_G\colon P_G\twoheadrightarrow X$ on $X$. I think that this would be representable by the Brown representability theorem, i.e. there is a topological space $C_{G,H}$ such that $[X,C_{G,H}]$ is in bijection with the isomorphism classes of such bundles-over-bundles (modulo pointed-vs-unpointed spaces). But is there an explicit description/characterisation of it?

I am specifically interested in the case where $G=\operatorname U(1)^n$ is a torus (and $H$ is compact Lie); the motivation is that this would describe the so-called Kaluza–Klein reductions of gauge fields on nontrivially fibred torus compactifications.

As a special case, the subset of $F(X)$ that corresponds to $H$-bundles on the trivial $n$-torus bundle $\operatorname U(1)^n\times X$ is classified by the $n$-fold iterated free loop space $\mathcal L^n\mathrm BG$; and in the case $n=1$ this object is well studied. But I don't know how to generalise to arbitary principal torus bundles.

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  • $\begingroup$ Your notion of "isomorphism classes" is what one would guess? $\endgroup$
    – Konrad Waldorf
    Commented Oct 13, 2021 at 19:01
  • $\begingroup$ @KonradWaldorf: yes, the obvious notion (in the unpointed case). $\endgroup$
    – Zislu R.
    Commented Oct 13, 2021 at 19:26

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If I understand the question right, I think the classifying space can be described like this. Let $Map(G,BH)$ be the space of all continuous maps from $G$ to $BH$. Make the group $G$ act continuously on that function space (using the usual free transitive action of $G$ on the space $G$), and form the associated bundle over $BG$: $$ EG\times_G Map(G,BH). $$

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    $\begingroup$ This looks good, I was just about to write this. $\endgroup$
    – Konrad Waldorf
    Commented Oct 13, 2021 at 19:06
  • $\begingroup$ I might be worthwhile to point out that when G is finite, then this is a wreath-product construction: the above is the classifying space a wreath product with normal subgroup $H^{|G|}$, where $G$ permutes the factors according to left multiplication on itself. $\endgroup$
    – Charles Rezk
    Commented Oct 21, 2021 at 15:54
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    $\begingroup$ There's a variant $B(H\wr \Sigma_n)=E\Sigma_n \times_{\Sigma_n} Map(n, BH)$, where n is the finite set, which classifies $P\to Y\to X$ where $Y\to X$ is an n-fold cover and $P\to Y$ a principal H-bundle. $\endgroup$
    – Charles Rezk
    Commented Oct 21, 2021 at 15:56

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