From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/10273 Path: news.gmane.io!.POSTED.blaine.gmane.org!not-for-mail From: David Roberts Newsgroups: gmane.science.mathematics.categories Subject: Topos objects Date: Mon, 31 Aug 2020 09:43:06 +0930 Message-ID: Reply-To: David Roberts Mime-Version: 1.0 Content-Type: text/plain; charset="UTF-8" Content-Transfer-Encoding: quoted-printable Injection-Info: ciao.gmane.io; posting-host="blaine.gmane.org:116.202.254.214"; logging-data="30462"; mail-complaints-to="usenet@ciao.gmane.io" To: "categories@mta.ca list" Original-X-From: majordomo@rr.mta.ca Tue Sep 01 04:23:02 2020 Return-path: Envelope-to: gsmc-categories@m.gmane-mx.org Original-Received: from smtp2.mta.ca ([198.164.44.55]) by ciao.gmane.io with esmtps (TLS1.2:ECDHE_RSA_AES_256_GCM_SHA384:256) (Exim 4.92) (envelope-from ) id 1kCvwr-0007lI-CA for gsmc-categories@m.gmane-mx.org; Tue, 01 Sep 2020 04:23:01 +0200 Original-Received: from rr.mta.ca ([198.164.44.159]:35990) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1kCvwU-00058t-69; Mon, 31 Aug 2020 23:22:38 -0300 Original-Received: from majordomo by rr.mta.ca with local (Exim 4.92.1) (envelope-from ) id 1kCvuU-0004eL-0k for categories-list@rr.mta.ca; Mon, 31 Aug 2020 23:20:34 -0300 Precedence: bulk Xref: news.gmane.io gmane.science.mathematics.categories:10273 Archived-At: Dear all, We know from work of Burroni, Lambek, Macdonald=E2=80=93Stone and Dubuc=E2= =80=93Kelley 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/ ]