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Baum-Connes conjecture and topos


Jean-Jacques Brahim
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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 ?


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Olivia Caramello
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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. 


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Jens Hemelaer
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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.


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

The result I have is the same: that {\bf Sh}_G(X) \simeq {\bf Sh}(X/G) if and only if \forall x \in X, \, \forall g \in G \setminus \{\,e\,\} \ \, g \cdot x \neq x (i.e. the action is free).  Note that the condition that X 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 {\bf Sh}_G(X) \simeq {\bf Sh}(X/G).  I've been writing the proof up in \LaTeX 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.

 


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Jean-Jacques Brahim
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@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.

 

 


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Joshua Wrigley
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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.

 


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Jean-Jacques Brahim
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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 !!


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Joshua Wrigley
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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  p \colon Y \to X an étale map, whether or not  p_{/G} \colon Y/G \to X/G 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.

 


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Jean-Jacques Brahim
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Hello,

i must apologize because i did not improve my understanding about the question i've asked. Nethertheless, i just have seen that moduli spaces are also a great generalization on classifying spaces. So do someone know what relations one can make between moduli problem and toposes ?


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Jens Hemelaer
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Hi!

Topological spaces can often be interpreted as solutions to classification problems. For example, the complex projective line "classifies" lines through the origin in the complex plane, in the sense that for each such line there is a point on the complex projective line.

However, I wouldn't call a topological space like this a moduli space, because the name "moduli space" is used almost only in Algebraic Geometry (there the moduli space is typically a scheme or a stack).

There are analogies though: if you think of a scheme as the analogue of a topological space, then a stack is the analogue of a topos (for me, this is just an analogy, I don't know how to make this precise). If you have a topological space with a group action, then the formal quotient by this group action will be a topos. Similarly, if you have a scheme with an algebraic group action, then the formal quotient will be a stack.

It's true that the Baum–Connes conjecture features the classifying space of proper group actions, but there are also other ingredients to it that might be less related to topos theory. For example, it also needs equivariant K-homology, and it is not clear whether or not this has a topos-theoretic interpretation. Translating the setting of the Baum–Connes conjecture to topos theory sounds exciting, but I think it will be very difficult as well, and it doesn't sound like the easiest path to learning about this conjecture. The main prerequisite seems to be algebraic topology (CW-complexes, spectra, generalized cohomology theories). But I'm not the best person to ask, since I'm not very familiar with the Baum–Connes conjecture.


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