Re: Profinite groupoids

[Hans Halvorson <hans.halvorson@gmail.com>]

This question is answered in the negative in Moerdijk and Vermeulen,
"Proper maps of toposes" (Remark 4.2, page 81).

Hans Halvorson

On Mon, Mar 7, 2011 at 5:13 AM, Prof. Peter Johnstone
<P.T.Johnstone@dpmms.cam.ac.uk> wrote:
> I've recently received an enquiry from a graduate student (below)
> about whether the question "is every Stone topological groupoid
> a profinite groupoid?" is still open. I've never considered the
> question myself, and I don't recall seeing anything published on
> the subject. Does anyone on this list know of a reference?
>
> Peter Johnstone
>
> ---------- Forwarded message ----------
> Date: Sat, 5 Mar 2011 08:46:27 +0200
> From: Alexandru Chirvasitu <chirvasitua@gmail.com>
> To: P.T.Johnstone@dpmms.cam.ac.uk
> Subject: a technical question
>
> Dear Professor Johnstone,
> My name is Alexandru Chirvasitu, and I am a second year graduate student at
> UC Berkeley. I apologize for the imposition, but there aren't many people  I
> know who could assist with the problem I had questions about. Being familiar
> with some of your work, and since the question is one on Stone spaces, I
> thought I'd ask.
>
> It's well known that a topological group whose underlying space is profinite
> is automatically the limit of an inverse system of finite groups. The
> question, briefly, is whether or not the same is true of profinite
> groupoids. By this I mean that the spaces of arrows and objects are
> profinite, and the source and target maps are continuous. Is it then the
> case that the groupoid is an inverse limit (in the category of groupoids)  of
> finite groupoids? Is this problem open to your knowledge?
> The only direct reference I could find in the literature is in the Magid's
> paper
>
> Magid, Andy R.
> The separable closure of some commutative rings.
> Trans. Amer. Math. Soc. 170 (1972), 109?124
>
> where he develops the Galois theory of commutative rings. He says that it
> seems to be open, but that was a long time ago.
>
> In the course of a joint project with a fellow graduate student here at
> Berkeley (having to do with Galois theory), we stumbled upon this problem.
> We think we can construct, rather naturally, plenty of counterexamples; what
> we do not know is whether or not the problem was previously open.
>
> This was pretty much it. My apologies again for taking up your time. I'm
> sure any input you might have will be very valuable.
>
>
> Thank you,
>
> Alexandru


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