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I’ve had reason to think about locally internal categories/locally small fibrations over a base topos lately, and I was asked to what extent one can view these as categories of families of objects of a “locally small category” in a structural axiomatic set theory. To me it seems like one should take the fibration to be a stack, since given compatible families of objects on some cover, then one should definitely be able to glue them. Maybe I’m looking in the wrong places, but I don’t see any statements to this effect in the various papers on locally internal categories (in all their various guises and names), by Penon, Benabou, Paré–Schumacher the Baby Elephant, and The Elephant. I didn’t read them thoroughly, but I also didn’t see it in Mike’s Sets for category theory or Enriched indexed categories.
Does anyone else concur, or know of a result in the literature close to this?
Or maybe the more pointed question to ask is: why should I think of arbitrary (locally internal) fibred categories as encoding “families of objects of a large category” if the fibred category is not a stack? This is one of the motivations given for fibred/indexed categories, but they are much more general to that. I’m not saying the non-stack fibred categories are not important, just the intuition seems slightly mismatched.
My current opinion is that the best context in which to answer this question is HoTT. For any site $C$ we have a 2- (indeed an $\infty$-) topos of stacks (of groupoids) on $C$, which therefore models HoTT (perhaps 2-truncated). Internal to that we can therefore define univalent categories, which interpret to stacks of categories on $C$. If the self-indexing of $C$ is a stack over itself, say $S_C$, it provides an extra universe in that model of HoTT with respect to which we can define (still internally) a notion of “locally small category”, meaning a univalent category $D$ whose hom-sets lie in the universe $S_C$, which will interpret semantically as a locally internal category that’s a stack. The “$I$-indexed families” of an indexed category can then be identified with families $I\to D$ in the usual sense (where $I$ is identified with its representable sheaf).
To your more pointed question, I would say that you can just choose the topology on $C$ as you like. Even if $C$ is a topos, you don’t have to equip it with its coherent or canonical topology; you can give it the trivial topology instead.
One day I’m going to write all of this up…
OK, thanks.
To your more pointed question, I would say that you can just choose the topology on $C$ as you like.
well, alright. I guess the more flexible way to say this would be to talk about the base category as a site, rather than as a bare category, and not presuppose the trivial topology.
One day I’m going to write all of this up…
I know the feeling (-:
It is different, but maybe it is relevant: Durov had in his thesis a chapter on homotopical algebra in sheaf setup. He is interested in cohomology of quasicoherent sheaves over generalized schemes, where rings are replaced by finitary monads. Because it is nonabelian he needs to resort to homotopical algebra. But regarding the local nature he needs to work not with model categories but with model stacks where he finds that he must weaken the analogues of axioms of Quillen for model categories when defining model stacks.
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