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Madeleine Birchfield (Jan 31 2024 at 23:45):What does one call an -category with
The term "elementary -topos" usually refers to something much stronger.
Kevin Arlin (Feb 01 2024 at 01:49):I’m not sure it has a name because it’s not obviously a very natural notion. Do you have examples in mind other than -toposes for some ?
Mike Shulman (Feb 01 2024 at 05:20):One could coin a phrase like "1-elementary -topos" perhaps.
David Kern (Feb 02 2024 at 01:00):This (well, turning the question around) is actually something I've never understood in the definition of elementary -topoi: why do we have to specifically require the existence of a subobject classifier in addition to that of sufficiently many univalent fibrations?
Mike Shulman (Feb 02 2024 at 01:29):If you mean why doesn't it follow from the existence of univalent fibrations, then the answer is that from a univalent fibration you can construct a classifier of some subobjects, but since no single univalent fibration can classify all objects, you can't get a classifier of all subobjects this way.
Mike Shulman (Feb 02 2024 at 01:31):If you mean why do we choose to add this additional assumption to the definition of elementary -topos rather than being content with sufficiently many univalent fibrations, for me it's in order to get as close as possible to the terminology in 1-category theory, where the existence of a subobject classifier (for all subobjects) is an essential distinction between a "topos" and various kinds of "pretopos".
David Kern (Feb 02 2024 at 01:58):Right, thank you for the explanations, that's what I kinda suspected it had to be but I was hoping that since monomorphisms are special in that their fibres are automatically small for any notion of smallness it might be possible to reconstruct their classifier from the collection of object classifiers. But I guess what this special property of theirs actually does is just make it possible for this axiom to be required at all.
The point about pretopoi is very interesting; -truncated morphisms do seem to keep a weirdly important role in -categories, as for example in Germán Stefanich's recent work on regular -categories — or even Lurie's -pretopoi in SAG.
David Kern (Feb 02 2024 at 02:04):In my mind, the existence of object classifiers is a kind of "solution set condition" on a regular category E (since I like the "allegorical" characterisation as providing a right-adjoint to the inclusion of E into its -truncated spans); is it right then to think of this axiom as being a size constraint on a pretopos?
David Kern (Feb 02 2024 at 02:45):David Kern said:
In my mind, the existence of object classifiers is a kind of "solution set condition" on a regular category E (since I like the "allegorical" characterisation as providing a right-adjoint to the inclusion of E into its -truncated spans); is it right then to think of this axiom as being a size constraint on a pretopos?
I should maybe be less sloppy, and specify that this only really makes sense for -topoi (which are the ones where all colimits are Van Kampen) and not so much for -topoi.
Mike Shulman (Feb 02 2024 at 12:12):I don't really think of it as a "size constraint", but I guess it can be used for some of the same things that a size condition like accessibility can be.