Dear Marta and Steve, I'm not at all a specialist in topos theory, and learned about them quite late when I was already engaged in algebraic topology ... but let me give an "external" point of view on the subject. I believe that Grothendieck did never have the "modern point of view" of elementary toposes, but he always thought of a topos as being a category of sheaves on a site. For instance, in "Pursuing Stacks" he calls the subobject classifier the "Lawvere object" and he uses the latter with homotopical purposes (as a good interval object) but he never uses it as a foudational structure for defining a topos. Therefore, I find it a little artificial to employ the term "Grothendieck topos" in contexts where the topos cannot be represented as a category of sheaves on a site. It might of course be possible to give a sense ``internal to a given elementary topos S'' of what it means to be a category of sheaves on a site, but this is beyond my competence. Working over S=Sets I always found it quite nice that Grothendieck toposes can be characterized among elementary toposes as those which are accessible, because this gives in a very precise sense what is needed in order to represent the elementary topos as a category of sheaves on a site. Moreover, there is an analogous characterization of accessible quasi-toposes due to Borceux et al. Of course, when working over an arbitrary elementary topos S, one should define what it means to be S-accessible, but I guess that this has already been done. All the best, Clemens. Le 2016-10-31 12:27, Steve Vickers a ??crit??:
Dear Marta,
Thanks for your reply. My question was a survey of usage, so what follows is not meant disputatiously, but I thought your reasoning raised some interesting issues.
As I see it you give two reasons here for taking Grothendieck toposes as being over Set = ZFC, both pertinent: (1) what Grothendieck meant, and (2) the (essential) uniqueness of geometric morphisms to Set.
(1) is delicate, given Grothendieck's underlying implicit definition of topos as "that of which topology is the study".
My understanding of this (you may know more - my knowledge of Grothendieck's work is almost all second-hand) is that there are two ways of viewing it, somewhat akin (respectively) to algebraic and general topology.
The first is that he meant a topos to be a category with which to do sheaf cohomology, by forming an (exact) injective resolution of Abelian groups, taking global sections (becoming non-exact), and then extracting the ker/im cohomology groups. This is perhaps an algebraic topologist's idea of what topology means. Classicality of Set seems then to be needed in order to get injective hulls in the category of sheaves over a site.
On the other hand, there is also the idea that global points can be recovered as sections of the geometric morphism to Set, and the topos simultaneously embodies both the points and the topology on them. Is this also part of what Grothendieck meant? It is closer to general topology, points and their continuous transformations.
This idea then generalizes well to elementary toposes, replacing Set by S. Elementary toposes are not generalized spaces in themselves, but bounded geometric morphisms between them are, and many topological properties are redefined for geometric morphisms.
(2) raises the question of why Set should have its privileged property. Technically, it is that every object of Set is a colimit of copies of 1, and that is preserved up to unique isomorphism by the inverse image part of any geometric morphism. But is that not because ZFC provides a 2-level structure of sets and classes, and we are implicitly using ZFC classes for our toposes? As we explore the foundational options then we should expect this uniqueness property for Set = ZFC to evaporate.
All the best,
Steve.
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