classifying $\infty$-toposes for topological/localic groups?

Question

Let $G$ be a locally compact topological group (or more generally a localic group). Is there an infinity topos which classify principal $G$ bundles ?

More precisely, is there an $\infty$-topos $BG$ such that for every localic topos $\mathcal{L}$ the category of geometric morphism from $\mathcal{L}$ to $BG$ is equivalent to the category of $G$ principal bundle over $\mathcal{L}$, where a $G$-principal bundle over $\mathcal{L}$ is a locale $\mathcal{X}$ endowed with a $G$ action and an invariant map $p: \mathcal{X} \rightarrow \mathcal{L}$

such that:

1)$p$ is an open surjection.

2) The canonical map $\mathcal{X} \times G \rightarrow \mathcal{X} \times_{\mathcal{L}} \mathcal{X}$ is an isomorphism.

I am especially interested in the cases where $G= \mathbb{U}$ (the group of complex number of module $1$) and $G=\mathbb{R}$.

Of course, if $G$ is pro-discrete, then the answer is yes: it suffice to consider the infinity topos associated to the $1$-topos of continuous $G$ set. In the general case, one should look for an infinity topos of spaces endowed with a $G$ action (up to homotopy), but my knowledge of homotopy theory is not enough to see if this trivially work/does not work or if it is a difficult question...

Answer 1

The projection map $p: \mathbf{R} \rightarrow \ast$ induces a fully faithful embedding of sheaf categories $p^{\ast}: Shv(\ast) \rightarrow Shv( \mathbf{R} )$. This is equally true for sheaves of sets, sheaves of spaces, or other variants. It follows that for any topos (or $\infty$-topos) $\mathcal{X}$, the ($\infty$-)category of geometric morphisms from $\ast$ to $\mathcal{X}$ embeds fully faithfully into the ($\infty$-)category of geometric morphisms from $\mathbf{R}$ into $\mathcal{X}$. As a consequence, there can't be a classifying topos (or $\infty$-topos) for $U(1)$-bundles in the sense you describe (the category of $U(1)$-bundles on a point does not embed fully faithfully in the category of $U(1)$-bundles on $\mathbf{R}$). The same counterexample works if you replace $U(1)$ by any topological group $G$ for which there exists a nonconstant continuous map $\mathbf{R} \rightarrow G$.

Answer 2

[I would comment, but I don't have enough points yet!]

Marta Bunge (*) shows that for any open localic group $G$, $BG$ classifies the principal bundles of the etale completion of $G$. I think you are asking about the case where $G$ is not necessarily etale complete. Here the only result I am aware of is that the category of stably Frobenius adjunctions from $\bf{Loc}$ to $[G , \bf{Loc}]$ (i.e. to the presheaf category, seeing $G$ as an internal category of $\bf{Loc}$) and over $\bf{Loc}$, classify principal bundles (by a general argument about cartesian categories). Since, in the case when $G$ is etale complete, these adjunctions correspond exactly to geometric morphisms, I think that's as good as you are going to get.

(*) An application of descent to a classification theorem for toposes. Math Proc Camb Phil Soc 1990

Answer 3

I think I had confounded the category of CGWH spaces with something I had read by Flach. In Section 2.2 of "The Weil-Etale Cohomology of Arithmetic Schemes," Flach works in a topos $T$ of sheaves on the site $(Top^{lc}, J_{op})$ of locally compact topological spaces with the usual open-cover topology. Flach notes that the category of compact spaces is a generating full subcategory of $T$. Flach goes on (in Section 2.3 of loc. cit.) to work with classifying spaces, etc., in the topos $T$. This might be more helpful to the poster of the question.

I think I had confused the topos $T$ (generated by compact spaces) with the CGWH spaces I read about in Strickland's notes. In any case, I hope the work of Flach is more helpful.

------------------Original answer below-------------

I'll give this answer, but I'm unsure of the details since I'm no expert in topos theory. Let's stick with $G$ being a locally compact (= Hausdorff too) topological group.

Let $T$ be the topos of compactly-generated weakly Hausdorff (CGWH) spaces. Strickland has some nice notes on CGWH spaces at http://math.mit.edu/~mbehrens/18.906/cgwh.pdf

Then $G$ is a group in this topos, and one can look at the 1-topos $BG$ of objects in $T$ endowed with $G$-actions in $T$.

I believe this should work, but I am a bit too time-constrained to work out the details/references. Also, I have no experience with infinity-categories, so I'll leave it to others to "take the associated infinity-category" if needed.