Hello,

i'd like to ask a very hard question on a very general level. I have understood that the statement of the so-called Baum-Connes conjecture involved the classifying space of proper actions of a group. But the classifying space is a particular case of a classifying topos. So, is there a statement of the Baum-Connes conjecture in the setting of toposes ?

I do not think that a topos-theoretic interpretation of this conjecture has ever been formulated, but a good reference for the relationship between classifying toposes and classifying spaces is this monograph by I. Moerdijk; see also this paper.

You could start by characterizing the property of an action of a group on a topological space being proper in terms of suitable topos-theoretic invariants. For instance, as stated in Exercise 10 of Chapter 3 of Sheaves in Geometry and Logic, if the action of a discrete group $G$ on a topological space $X$ is proper then the canonical morphism from the topos of étale $G$-spaces to the topos of sheaves on the orbit space is an equivalence. It would be interesting to investigate to which extent this implication can be reversed.

About the reverse implication (coincidentally, I looked at this last week):

Suppose that $G$ is a discrete group acting on a topological space $X$, and let $\mathbf{Sh}_G(X)$ be the topos of $G$-equivariant sheaves on $X$. The localic reflection of $\mathbf{Sh}_G(X)$ is given by $\mathbf{Sh}(X/G)$. This means that, if $\mathbf{Sh}_G(X)$ is equivalent to a topos of sheaves on some topological space $Y$, then necessarily $Y = X/G$, with the quotient topology.

The claim is that if $\mathbf{Sh}_G(X) = \mathbf{Sh}(X/G)$, then the $G$-action is free and the quotient map is étale. Let's assume that $X$ is sober (e.g. $X$ is Hausdorff).

First, consider the category of points of $\mathbf{Sh}_G(X)$. The objects are the points of $X$, and the morphisms $x \to y$ correspond to the elements $g \in G$ such that $x$ lies in the closure of $g \cdot y$. If $\mathbf{Sh}_G(X) = \mathbf{Sh}(X/G)$, then the category of points is necessarily a preorder, so this shows that the $G$-action is free.

Further, there is an étale geometric morphism $\mathbf{Sh}(X) \to \mathbf{Sh}_G(X)$. So if $\mathbf{Sh}_G(X) = \mathbf{Sh}(X/G)$, then we that the geometric morphism $\mathbf{Sh}(X) \to \mathbf{Sh}(X/G)$ is étale, which in turn implies that the continuous map $X \to X/G$ is étale.

If the $G$-action on $X$ is free and the quotient map $X/G$ is étale, then it is not too difficult to prove that the action is properly discontinuous. Conversely, if the action is properly discontinuous, then it follows that the action is free, and that the quotient map $X \to X/G$ is a covering projection (see here), in particular the quotient map is étale.

Hi Jens, coincidentally I have been working on the reverse implication too.

The result I have is the same: that if and only if (i.e. the action is free). Note that the condition that is sober isn't necessary.

The method relies on saying that some certain commutative squares are pullbacks (which unfortunately can't be typeset on here) if and only if . I've been writing the proof up in for my own research diary, but since other people may be interested once I've finished I'll be sure to upload it here too.

Interestingly, since the proof is in mostly categorical terms, it may generalise readily to actions by a topological group, or indeed to a point-free setting (whatever a free group action on a locale may look like). Please don't hesitate to contact me if you think any of this sounds interesting.

@ocaramello Many thanks for your comments Professor. I think the references you gave will be very helpful. I hope to ask more precise questions after reading them.

Hi again, I've attached the notes promised in the post above. Let me know if there's any mistakes. Otherwise, this gives necessary and sufficient conditions for the topological and discrete group action cases: namely the group action must be free.

Hi Jens and Joshua,

it's very kind from you to have tried to help. Many thanks for Joshua to have typed such notes. Actually, i'm really a beginner with all this stuff. But your contributions will help me to progress !!

Hi all, attached are some updated notes representing some work that myself and Jens have been conducting behind the scenes. It includes a corrected version of the notes posted above.

What would be an interesting question is, given a topological group *G* acting continuously on spaces *X* and *Y*, satisfying my condition 2.1 in the notes (name suggestions are also welcome), with an étale map, whether or not is also étale. I have a feeling it shouldn't be, but haven't been able to come up with a counter-example yet either.

All comments and corrections are very welcome.