From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/10276 Path: news.gmane.io!.POSTED.blaine.gmane.org!not-for-mail From: Steve Vickers Newsgroups: gmane.science.mathematics.categories Subject: Re: Topos objects Date: Wed, 2 Sep 2020 11:45:42 +0100 Message-ID: References: Reply-To: Steve Vickers Mime-Version: 1.0 (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="34971"; mail-complaints-to="usenet@ciao.gmane.io" Cc: "categories@mta.ca list" To: David Roberts Original-X-From: majordomo@rr.mta.ca Fri Sep 04 02:55:05 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 1kE00O-0008s1-3i for gsmc-categories@m.gmane-mx.org; Fri, 04 Sep 2020 02:55:04 +0200 Original-Received: from rr.mta.ca ([198.164.44.159]:36248) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1kDzzz-0002xI-Is; Thu, 03 Sep 2020 21:54:39 -0300 Original-Received: from majordomo by rr.mta.ca with local (Exim 4.92.1) (envelope-from ) id 1kDzxw-0003Oh-FC for categories-list@rr.mta.ca; Thu, 03 Sep 2020 21:52:32 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.io gmane.science.mathematics.categories:10276 Archived-At: Dear David, My first reaction when I read your post was that there was an obvious answer= of Yes. But the more I reread it, the more I wondered whether I truly under= stood what the question was. What did you mean by =E2=80=9Ctopos=E2=80=9D? A= nd what would a =E2=80=9Ctopos object=E2=80=9D be? Here=E2=80=99s my initial obvious answer. If =E2=80=9Ctopos=E2=80=9D means elementary topos (optionally with nno), the= n the theory of toposes is cartesian (aka essentially algebraic), so if =E2=80= =9Ctopos object=E2=80=9D means =E2=80=9Cinternal topos=E2=80=9D (just as eg =E2= =80=9Cgroup object=E2=80=9D means =E2=80=9Cinternal group=E2=80=9D), then th= ere is a standard notion of topos object, model of that cartesian theory, in= any 1-category with finite limits. At least, if you take the topos structur= e to be canonically given. It seemed too easy. Perhaps you meant something deeper. First I started to wonder why you stressed the 2-categories. Was it to enabl= e some up-to-isomorphism laxity? Next I wondered if I was interpreting =E2=80=9Ctopos object=E2=80=9D correct= ly. After all, the theory of toposes has two sorts, for objects and arrows, a= nd an internal topos is carried by two objects. Were you instead thinking of= a single object, with its category structure implied by the ambient 2-cells= ? For instance, in a 2-category with finite products, a =E2=80=9Cfinite prod= uct object=E2=80=9D X could be one for which the diagonals X -> 1 and X -> X= x X have right adjoints. Finally, there is the question of what a =E2=80=9Ctopos=E2=80=9D is. If it i= s akin to a Grothendieck topos, a category of sheaves for a generalized spac= e, then then the relevant structure is that not of elementary toposes, but o= f =E2=80=9Call=E2=80=9D colimits and finite limits as in Giraud=E2=80=99s th= eorem. I find it a very interesting question when an object in a 2-category m= ight have topos structure in that sense. For example, suppose your 2-category C is that of Grothendieck toposes and g= eometric morphisms (maps). Then the object classifier O surely is a =E2=80=9C= topos object=E2=80=9D in C, and so also would be O^X for any exponentiable t= opos X. The points of O^X are the maps X -> O, ie the objects of X, so it is= reasonable to imagine that the topos structure of X might be reflected in t= he C-internal structure on O^X. Finitary products and coproducts for O, given by maps O^n -> O, are easy. Eg= for binary products, n=3D2, map (U, V) |-> UxV. (U, V are points of O, ie s= ets. The construction is geometric and so does give a geometric morphism.) T= he infinitary colimits are harder. I conjecture that they are given by M-alg= ebra structure for O, where M is the symmetric topos monad on C. That should= all lift to O^X. The question with C =3D Grothendieck toposes generalizes to Grothendieck top= oses over S (bounded S-toposes) and - a particular interest of my own - a re= lated 2-category based on arithmetic universes. All the best, Steve. > On 1 Sep 2020, at 03:23, droberts.65537@gmail.com wrote: >=20 > =EF=BB=BFDear all, >=20 > 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). >=20 > 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. >=20 > 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!) >=20 > 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. >=20 > Best regards, >=20 > David Roberts > Webpage: https://ncatlab.org/nlab/show/David+Roberts > Blog: https://thehighergeometer.wordpress.com >=20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]