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/ ]