Locally internal stacks as categories of families?

[David Roberts <droberts.65537@gmail.com>]

Dear all

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, Bénabou, Paré–Schumacher the Baby Elephant, and The
Elephant (to name a few). I didn’t read them thoroughly, but I also
didn’t see it in Mike Shulman’s Sets for category theory or  Enriched
indexed categories.

Does anyone else concur, or know of a result in the literature close to this?

Regards,
David

David Roberts
Webpage: https://ncatlab.org/nlab/show/David+Roberts
Blog: https://thehighergeometer.wordpress.com


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