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John Onstead (Nov 07 2024 at 19:21):I'm still thinking about a few things, but in the meantime the above discussion inspired a sort of side question about this observation:
Mike Shulman said:
The first is categorical dimension: is a 2-category, not a 1-category, so its "nerve" doesn't land in presheaves of sets but rather presheaves of categories. Similarly, the nerve for -topoi lands in presheaves of -categories. But we can take this seriously, and for any and any -topos , get a hoped-for equivalence between the -topos induced by (consisting of sheaves of -categories on ) and the local homeomorphisms of -topoi over . At we can hope for it to stabilize.
I noticed there's a lot of times when we have some functor between a 1-category and a 2-category. When this happens, most of the time we work to try and "upgrade" the 1-category into a 2-category so the dimensions match. But this can be messy and it doesn't always work out. Why don't we instead just "downgrade" the 2-category? Unlike trying to upgrade the 1-category, which you might not be able to do uniquely or at all, there's always a canonical way to "downgrade" a 2-category back into a 1-category: by throwing out all the 2-morphisms. It's even done with Cat occasionally; there's some properties that the 1-category Cat has that the 2-category doesn't.
Notification Bot (Nov 07 2024 at 19:28):A message was moved here from #learning: questions > Motivating Sheaves with Locality by Mike Shulman.
Mike Shulman (Nov 07 2024 at 19:28):Of course, for a functor from a 1-category to a 2-category, there is no problem: you can just treat the 1-category as a 2-category with only identity 2-morphisms.
Mike Shulman (Nov 07 2024 at 19:29):There are two problems with throwing away the 2-morphisms in a 2-category.
Mike Shulman (Nov 07 2024 at 19:29):First, if the 2-category is not a strict 2-category, you can't even do it, as composition of 1-morphisms is not strictly associative.
Mike Shulman (Nov 07 2024 at 19:30):Second, it violates the [[principle of equivalence]], since a functor out of such a "downgraded" 1-category can send isomorphic 1-morphisms to different things, or equivalent objects to non-isomorphic things. Similarly, it's not invariant under equivalence of 2-categories: two equivalent 2-categories can have inequivalent underlying 1-categories.
Mike Shulman (Nov 07 2024 at 19:31):That doesn't mean we don't ever do it; as you pointed out, sometimes people do talk about the 1-category of categories. But it's "fraught" and not usually to be a first choice.
Mike Shulman (Nov 07 2024 at 19:36):I would say a good rule of thumb for when we might consider it is to think about whether the underlying 1-category of the 2-category in question is "meaningful" or "well-defined" on its own. For example, in the case of Cat, we can certainly define categories and functors without talking about natural transformations, and most everyone would agree that this is at least a meaningful and well-defined thing.
But in the case of Topos, I would argue that the 2-category Topos is not even well-defined more precisely than up to equivalence of 2-categories. E.g. is a geometric morphism a pair of adjoint functors where the left adjoint is left exact? Or is it a functor for which there exists a left exact left adjoint? Or is it a functor in the opposite direction that is left exact and for which there exists a right adjoint? When working in the 2-category up to equivalence, we don't need to take a stand on questions like that since they all give equivalent definitions; but if we wanted to talk about the underlying 1-category of Topos we'd need to make a choice, since they would give inequivalent 1-categories.