CAUTION: The Sender of this email is not from within Dalhousie. Dear all, We know from work of Burroni, Lambek, Macdonald–Stone and Dubuc–Kelley that toposes are monadic over Graph and over Cat (the 1-category), and even (and I don't know to whom this is due) 2-monadic over Cat (the 2-category). I was wondering recently if there is a sensible notion of a *topos object* in a 2- or bicategory K. One would need suitable structure on K to support this, I presume finite limits (of the appropriate weakness) at minimum. I would imagine possibly also an involution akin to (-)^op, for the following reason. In the work of the above authors, the definition of topos is taken to be that of a cartesian closed category with subobject classifier. However, one could take the terminal object + pullbacks + power objects definition instead, provided one gave these as functors satisfying certain conditions. And here I'm not sure if one would take the covariant or contravariant power object functor. If the latter, we clearly need that involution. Alternatively, one could take the approach that the relation \in appearing in the definition of power object is a universal relation, and one could potentially think of Rel(E) as a suitable completion in Cat of the putative topos E, and abstract this. It seems a fair bit more overhead though, and probably more complicated than it's worth (modulo the fact this whole exercise might also be so!) My motivation for this, such as it is, is that if one could define topos objects in a suitable bicategory K, one could take, for instance, K to be fibrations over a suitable base, or categories and anafunctors or a combination of these. Maybe it's a sledgehammer approach, but it seems curious and maybe interesting in its own right. Best regards, 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/ ]