RE: Re: fundamental localic groupoid?

RE: Re: fundamental localic groupoid?

[Marta Bunge]

Dear Peter, Dear David, Dear All,

I was going to reply, but was too busy with non-mathematical things at the moment (a big move).  

The following are the papers relevant to David's question, beginning with the one that Peter mentions:

M. C. Bunge, Classifying toposes and fundamental localic groupoids,in: Category Theory 1991, CMS Conference Procedings 13, American Mathematical Society (1992), 75-96. 
M. C. Bunge, Universal Covering Localic Toposes,Comptes Rendues Societe Royale du Canada 24 (1992), 245-250.
M. C. Bunge and I.Moerdijk,On the construction of the Grothendieck fundamental group of a topos by paths,J. Pure and Applied Algebra 116 (1997), 99-113. 

Comments will have to wait, I'm afraid. 

Cordial regards,

Marta
************************************************
Marta Bunge
Professor Emerita
Dept of Mathematics and Statistics 
McGill UniversityBurnside Hall, Office 1005
805 Sherbrooke St. West
Montreal, QC, Canada H3A 2K6
Office: (514) 398-3810/3800  
Home: (514) 935-3618
marta.bunge@mcgill.ca 
http://www.math.mcgill.ca/~bunge/
************************************************



> From: P.T.Johnstone@dpmms.cam.ac.uk
> To: droberts@maths.adelaide.edu.au; categories@mta.ca
> Subject: categories: Re: fundamental localic groupoid?
> Date: Sun, 2 May 2010 10:49:59 +0100
> 
> I'm surprised that no-one has yet replied to David's original question
> by citing the work of Marta Bunge; she has a paper called (I think)
> "Classifying toposes and fundamental localic groupoids" dating from the
> early 1990s. (Being away from home at present, I don't have the exact
> reference to hand; I was hoping Marta would provide it.)
> 
> Peter Johnstone
> 
> On Apr 28 2010, David Roberts wrote:
> 
>>Hi all,
>>
>> it is (or should be) well known that the fundamental groupoid of a
>> locally connected, semilocally 1-connected space X can be given a
>> topology such that it is a topological groupoid (composition continuous
>> etc). Without these assumptions on X there are counterexamples to the
>> continuity of composition (c.f. misguided attempts to build a topological
>> fundamental group). I was wondering, though, if there is a localic
>> fundamental groupoid of an arbitrary space. One could presumably pass the
>> the topos Sh(X) and then consider the fundamental groupoid of that, but I
>> was wondering if there was a way to pass directly from the description of
>> the arrows of Pi_1(X) as a set of classes of paths to a locale of classes
>> of paths, and thence to a localic groupoid. My only 'evidence' that this
>> might be the case is that the product in Loc is different to the product
>> in Top, and so this may provide a work around the non-continuity of
>> composition.
>>
>>Thanks,
>>
>>David Roberts
>>
 		 	   		  

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Re fundamental localic groupoid?

[Joyal, André]

Dear Marta,

I thank you for the information.
It seems that you have constructed the factorisation system:

(connected morphisms of toposes, totally disconnected localic morphisms)

It depends on identifying correctly the internal notion
of totally disconnected locale. I will read your papers:

> Marta Bunge, On two non-discrete localic generalizatins of \Pi_0, Cahiers Issue on the celebration of the 100th anniversary of the birth of Charles Ehresmann.  

>Marta Bunge, Fundamental Pushout Toposes,Theory and Applications of Categories 20 (2008) 186-214.

Best,
André


-------- Message d'origine--------
De: martabunge@hotmail.com de la part de Marta Bunge
Date: dim. 02/05/2010 12:17
À: Joyal, André; David Roberts
Cc: Robert Rosebrugh; Eduardo Dubuc; Peter Johnstone
Objet : RE: RE : categories: Re: fundamental localic groupoid?
 

Dear Andre, Dear All,
I just sent a message to categories in response to the specific question of Robert Davis, prompted by remarks made by Peter Johnstone. 
It concerned of a locally connected Grothendieck topos, in particular in the localic case. The three papers mentioned in my previous message were from the 1990's. 

More relevant to the message by Andre is the following reference on Galois toposes in the locally connected case.
Marta Bunge, Galois groupoids and covering morphisms in topos theory,Proceedings of the Fields Institute: Workshop on Descent, Galois Theory and Hopf Algebras,Fields Institute Communications, American Mathematical Society, 2004, 131-162.

In the general case (not necessarily locally connected) the the totally disconnected and zero-dimensional reflections are constructed in:

Marta Bunge, On two non-discrete localic generalizatins of \Pi_0, Cahiers Issue on the celebration of the 100th anniversary of the birth of Charles Ehresmann.  

Marta Bunge, Fundamental Pushout Toposes,Theory and Applications of Categories 20 (2008) 186-214.

As I explained in my previous message, I have no time for commnents right now. 
Cordial regards,
Marta





************************************************
Marta Bunge
Professor Emerita
Dept of Mathematics and Statistics 
McGill UniversityBurnside Hall, Office 1005
805 Sherbrooke St. West
Montreal, QC, Canada H3A 2K6
Office: (514) 398-3810/3800  
Home: (514) 935-3618
marta.bunge@mcgill.ca 
http://www.math.mcgill.ca/~bunge/
************************************************


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]