CAUTION: The Sender of this email is not from within Dalhousie. 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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]