Dear Olivia, Could you perhaps explain why you say that the notion of S-bounded elementary topos is "certainly less controversial" than the Grothendieckian notion of category of sheaves on a site? As you know, by a theorem of Diaconescu, the two points of view are equivalent: an elementary topos is S-bounded if and only if it is equivalent to the category of S-valued sheaves on an internal site in S. In light of this result, I find it very natural to refer to bounded S-toposes as "Grothendieck toposes over S", and I have noticed that this use is quite widespread in the literature. Of course agree with you that S-bounded topos and S-valued sheaves on a site in S, for S an elementary topos, are equivalent notions by a theorem of Diaconescu. By "certainly less controversial" I was referring to a previous posting of mine in response to Steve Vickers, in which by a Grothendieck topos I had meant therein a category of Set-valued sheaves on a site in Set rather than on an arbitrary (elementary) topos S. I agree with André that it is important to clearly distinguish the concept of Grothendieck topos from that of elementary topos also terminologically, since the presence of sites of definition is a distinctive feature which was central in Grothendieck's view and usage of toposes. Sites (or other kinds of presentations for bounded toposes) are essential for studying 'concrete' mathematical problems (not just in algebraic geometry or topology but in virtually any branch of mathematics) from a topos-theoretic point of view. Whilst general results about bounded toposes should be preferably proved without referring to their presentations and even at the elementary topos level whenever possible, the essential ambiguity given by the fact that a Grothendieck topos admits in general an infinite number of different sites of definition can be exploited to generate a great number of interesting notions and results arising from the 'calculation' of topos-theoretic invariants in terms of these different presentations. Once again, there seems to be a misunderstanding, as I too have pointed out the distinction between the notion of a Grothendieck topos (as, say, S-valued sheaves on a site in S, for S an elementary topos) and that of an elementary topos, such as S. What I was arguing against was the need to change the terminology in such a way that by "topos" one meant "sheaves on a site" and that "elementary topos" ought to be relabelled "logical topos" considering, according to Joyal, that "the natural notion of morphism between elementary toposes is that of a logical morphism", suggesting by it that the notion of an elementary topos came from (or is suitable to) logic and not from (or suitable to) geometry. Now, this is simply not the case. The very fact that such categories were called "(elementary) toposes" already suggests otherwise. Moreover, the discovery (by Lawvere) that all of higher-order logic could be interpreted in an elementary topos came afterwards, and so it turned out that both geometry and logic were present in it. The only way to distinguish them is therefore by means of the morphisms adopted in each case - that is, either geometric or logical morphisms. Best wishes, Marta ________________________________ From: Olivia Caramello <olivia@oliviacaramello.com> Sent: November 2, 2016 11:49:44 AM To: 'Marta Bunge'; categories@mta.ca Subject: R: categories: Re: Grothendieck toposes Dear Marta, Could you perhaps explain why you say that the notion of S-bounded elementary topos is "certainly less controversial" than the Grothendieckian notion of category of sheaves on a site? As you know, by a theorem of Diaconescu, the two points of view are equivalent: an elementary topos is S-bounded if and only if it is equivalent to the category of S-valued sheaves on an internal site in S. In light of this result, I find it very natural to refer to bounded S-toposes as "Grothendieck toposes over S", and I have noticed that this use is quite widespread in the literature. I agree with André that it is important to clearly distinguish the concept of Grothendieck topos from that of elementary topos also terminologically, since the presence of sites of definition is a distinctive feature which was central in Grothendieck's view and usage of toposes. Sites (or other kinds of presentations for bounded toposes) are essential for studying 'concrete' mathematical problems (not just in algebraic geometry or topology but in virtually any branch of mathematics) from a topos-theoretic point of view. Whilst general results about bounded toposes should be preferably proved without referring to their presentations and even at the elementary topos level whenever possible, the essential ambiguity given by the fact that a Grothendieck topos admits in general an infinite number of different sites of definition can be exploited to generate a great number of interesting notions and results arising from the 'calculation' of topos-theoretic invariants in terms of these different presentations. Best wishes, Olivia [For admin and other information see: http://www.mta.ca/~cat-dist/ ]