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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