Dear All, Just a few remarks: Grothendieck was always very careful with terminology. The name "topos" is an explicit reference to the idea of space. The notion of geometric morphism between toposes is taken from the notion of continuous maps between topological spaces. The idea of an "elementary topos" is a child of categorical logic, especially of the axiomatisation of the category of sets by Lawvere. The natural notion of morphism between elementary toposes is that of logical functor. It is marvelous that the two notions should be so related. But it is be better to keep them appart before uniting them. Otherwise the miracle disappear in confusion. Best, André ________________________________________ From: Marta Bunge [martabunge@hotmail.com] Sent: Sunday, October 30, 2016 4:17 PM To: categories@mta.ca Cc: Steve Vickers Subject: categories: Re: Grothendieck toposes Dear Steve, When an elementary (base) topos S is specified, I use "S-bounded topos" to mean the pair (E, e), with E an elementary topos and e: E--> S a (bounded) geometric morphism. When S = Set (a model of ZFC) and E an arbitrary elementary topos, then there is a most one geometric morphism e:E---> Set, so in that case the latter need not be specified. I therefore use "E is a Grothendieck topos" to mean "E is an elementary topos bounded over Sets". The latter has been shown to be equivalent to what Grothendieck meant by it. Best regards, Marta ************************************************ Marta Bunge Professor Emerita Dept of Mathematics and Statistics McGill University Montreal, QC, Canada H3A 2K6 Home: (514) 935-3618 marta.bunge@mcgill.ca http://www.math.mcgill.ca/people/bunge ************************************************ [For admin and other information see: http://www.mta.ca/~cat-dist/ ]