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For the experts on profinite groups among you, here is a question which I am trying to figure out. Quite likely it doesn’t actually take much expertise at all but just very elementary profinite yoga.
Over at modular equivariant elliptic cohomology is recalled something that almost looks like the orbit category of $GL_2$ with coefficients in the profinite integers $\widehat{\mathbb{Z}}$ and I am wondering if with some massaging we might not get something that is the actual orbit category of some profinite group, or something.
So the almost-orbit category there is that whose objects are given by pairs $(n,\Gamma)$ with $n \in \mathbb{N}$ and $\Gamma$ a subgroup of $GL_2(\mathbb{Z}/n\mathbb{Z})$.
To define the morphisms, first with $p_n \colon GL_2(\widehat{\mathbb{Z}})\to GL_2(\mathbb{Z}/n\mathbb{Z})$ denoting the canonical projection, let then $p_n^{-1}(\Gamma) \subset GL_2(\widehat{\mathbb{Z}})$ be the pre-image of $\Gamma$. The idea is then to consider only orbits which are cosets by such $p_n^{-1}(\Gamma)$ and to allow maps only between orbits where the $n$s relate suitably:
$Hom((n_1,\Gamma_1), (n_2,\Gamma_2)) \coloneqq \left\{ \array{ Hom_{GL_2(\widehat{\mathbb{Z}})}( GL_2(\widehat{\mathbb{Z}})/p_{n_1}^{-1}(\Gamma_1), \; GL_2(\widehat{\mathbb{Z}})/p_{n_2}^{-1}(\Gamma_2) ) & if\; n_2|n_1 \\ \emptyset & otherwise } \right.$That’s the definition which I would like to see realized as a genuine orbit category. I might be very much not seeing the obvious here, please bear with me. But my question would be what is – or what is the “closest” – way to make this a genuine orbit category.
(I gather that $GL_2(\widehat{\mathbb{Z}})\simeq \underset{\leftarrow}{\lim}_n GL_2(\mathbb{Z}/n\mathbb{Z})$?)
Ah I see that there is something at least closely related to this question in
This characterizes the maximal closed subgroups of $GL_2(\widehat{\mathbb{Z}})$ and finds (prop. 2.5, cor. 2.7) that they are given by maximal closed subgroups $H \subset GL_2(\mathbb{Z}_p)$ (adic $p$), being of the form $H\times \prod_{p'\neq p} GL_2(\mathbb{Z}_{p'}) \hookrightarrow GL_2(\widehat{\mathbb{Z}})$. Then furthermore corollary 2.14 there says how the maximal closed subgroups of $GL_2(\mathbb{Z}_p)$ come from maximal closed subgroups of a $GL_2(\mathbb{Z}/p^k \mathbb{Z})$-stage.
I suppose that goes at least some way to answering my question: I guess I want to be looking at the orbits which are cosets by maximal closed subgroups.
What are the properties that you want in an orbit category? Do you need an Elmendorf theorem or what?
There is paper that might give some help: G. Quick, Continuous group actions on profinite spaces, J. Pure Appl. Alg., 215, (2011), 1024–1039. It does not get near to your question but looks at some related ideas.
One minor point of wording on that page to which you link. How does one take a general linear group with coefficients in a group? Was a ring intended?
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