you suggest to neglect the higher order aspect of toposes, i.e. when one has a got a site in a logos then one may form sheaves over this internal site. But the result will be neither cc nor have a soc if the
Dear Andre, I agree with you in that in that one must distinguish the notions of an elementary topos and that of a Grothendieck topos. However, I see no need for a change in terminology, as the latter (Grothendieck topos) is recovered from the former by means of the notion of a bounded geometric morphism. Topos theory is a well-established subject. A change as the one you suggest ("logos") - or any other change for that matter, is not only not necessary but in my opinion it will bring useless discussions as the one we are having now. After all, the original meaning of the name of a subject is not always adequate, but this matters little if everybody knows what is meant by it. Think that "geometry" means "measurement of the Earth", or that "Analysis" is a what was left from "Infinitesimal Analysis" after the then undesirable infinitesimals were banned and replaced by epsilons and deltas. Suppose you adopt "logos" instead of "elementary topos" and use "topos" to mean "sheaves on a site in a logos". Okay, then what? Those of us who work in Topos Theory would have the choice of accepting or rejecting your suggestion. Suppose we accept it. Then I can well imagine any paper in the area of "Logos Theory" starting with the proviso that we use "logos" in the sense of Joyal, but that it is the same concept as an elementary topos in the sense of Lawvere and Tierney. What about morphisms of logoi (logoses, logi) - geometric, logical as before, or you will also need new names for those? This is getting nowhere and I have nothing else to say on the subject. With very best regards, Marta ________________________________ From: Joyal, André <joyal.andre@uqam.ca> Sent: November 5, 2016 12:35:22 PM To: Thomas Streicher Cc: Marta Bunge; Martin Escardo; categories@mta.ca; Steve Vickers Subject: RE: categories: Re: Grothendieck toposes Dear Thomas, You wrote: base logos was or has not< Sorry if I was unclear. A logos is just a different name for "elementary topos". I wish to distinguish clearly the notion of elementary "topos" from that of (Grothendieck) topos by calling the first a logos and the second a topos. Best, André [For admin and other information see: http://www.mta.ca/~cat-dist/ ]