# Classifying space of bundles over bundles

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.

New contributor
Zislu R. is a new contributor to this site. Take care in asking for clarification, commenting, and answering. Check out our Code of Conduct.
• Your notion of "isomorphism classes" is what one would guess? Oct 13 at 19:01
• @KonradWaldorf: yes, the obvious notion (in the unpointed case). Oct 13 at 19:26

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 $$BG$$: $$EG\times_G Map(G,BH).$$