From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/10278 Path: news.gmane.io!.POSTED.blaine.gmane.org!not-for-mail From: David Roberts Newsgroups: gmane.science.mathematics.categories Subject: Re: Topos objects Date: Thu, 3 Sep 2020 12:46:02 +0930 Message-ID: References: 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="12279"; mail-complaints-to="usenet@ciao.gmane.io" Cc: "categories@mta.ca list" To: Steve Vickers Original-X-From: majordomo@rr.mta.ca Fri Sep 04 02:58:45 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 1kE03x-00032R-1S for gsmc-categories@m.gmane-mx.org; Fri, 04 Sep 2020 02:58:45 +0200 Original-Received: from rr.mta.ca ([198.164.44.159]:36288) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1kE03r-0003XM-AT; Thu, 03 Sep 2020 21:58:39 -0300 Original-Received: from majordomo by rr.mta.ca with local (Exim 4.92.1) (envelope-from ) id 1kE01n-0003UH-PN for categories-list@rr.mta.ca; Thu, 03 Sep 2020 21:56:31 -0300 In-Reply-To: <9583BDE6-9E93-4F84-9711-6A29DC8C7B23@cs.bham.ac.uk> Precedence: bulk Xref: news.gmane.io gmane.science.mathematics.categories:10278 Archived-At: Dear Steve, thank you for the thoughtful reply. You are correct in that I did not mean an internal topos, and I should have specified that at the time of writing I had in mind an elementary topos. > 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 product object=E2=80=9D X could be one for wh= ich the diagonals > X -> 1 and X -> X x X have right adjoints. Yes, this is the concept I had in mind. It is less clear to me if, in addition to finite limits, it would be easier to ask for cartesian closedness+subobject classifier or for power objects. The latter seems to me to be simpler, at least in the case when the 2-category admits a construction analogous to E |---> Rel(E) (not that I know how to do it, but it seems a reasonable first line of attack). There should be some 1-arrow in the/an ambient 2-category with an adjoint giving the power object and the universal relation \in. One result that would indicate this is on the right track is that a topos object in Cat should just be an elementary topos (with specified finite limits, power objects), a topos object in Cat_ana (anafunctors instead of functors) should be a topos (with *unspecified* finite limits and power objects), a topos object in Fib(B) should be a fibred topos, and a topos object in Cat_cocomp should be a cocomplete topos. I'd be interested to hear (even if off-list) your ideas relating to arithmetic universes (AUs). I could imagine asking for a topos object in categories over an AU. Or perhaps in some kind of syntactic 2-category. For an even more exotic example, imagine taking the 2-category of be of the sort Riehl and Verity use for their synthetic \infty-cateogry work. Then one might argue that a topos object in such a 2-category is a candidate for an elementary (\infty,1)-topos, since these 2-categories are meant to have as objects (\infty,1)-categories. The question of what morphisms to take is then an even more interesting question. The easiest answer is that one could ask for logical morphisms. But clearly this is not sufficient for all purposes. If looking for geometric morphisms I would myself aim for something like logos morphisms (=C3=A0 la Anel=E2=80=93Joyal), namely left = exact cocontinuous arrows, considering these as morphisms in the opposite category. Best regards, David Roberts Webpage: https://ncatlab.org/nlab/show/David+Roberts Blog: https://thehighergeometer.wordpress.com On Wed, 2 Sep 2020 at 20:15, Steve Vickers wrot= e: > > Dear David, > > My first reaction when I read your post was that there was an obvious ans= wer of Yes. But the more I reread it, the more I wondered whether I truly u= nderstood what the question was. What did you mean by =E2=80=9Ctopos=E2=80= =9D? And 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), = then 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 there is a standard notion of topos object, model of that cartes= ian theory, in any 1-category with finite limits. At least, if you take the= topos structure 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 en= able some up-to-isomorphism laxity? > > Next I wondered if I was interpreting =E2=80=9Ctopos object=E2=80=9D corr= ectly. After all, the theory of toposes has two sorts, for objects and arro= ws, and an internal topos is carried by two objects. Were you instead think= ing 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=9Cfin= ite product object=E2=80=9D X could be one for which the diagonals X -> 1 a= nd X -> X x X have right adjoints. > > Finally, there is the question of what a =E2=80=9Ctopos=E2=80=9D is. If i= t is akin to a Grothendieck topos, a category of sheaves for a generalized = space, then then the relevant structure is that not of elementary toposes, = but of =E2=80=9Call=E2=80=9D colimits and finite limits as in Giraud=E2=80= =99s theorem. I find it a very interesting question when an object in a 2-c= ategory might have topos structure in that sense. > > For example, suppose your 2-category C is that of Grothendieck toposes an= d geometric morphisms (maps). Then the object classifier O surely is a =E2= =80=9Ctopos object=E2=80=9D in C, and so also would be O^X for any exponent= iable topos 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 refle= cted in the 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 sets. The construction is geometric and so does give a geometric morphis= m.) The infinitary colimits are harder. I conjecture that they are given by= M-algebra 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 = toposes over S (bounded S-toposes) and - a particular interest of my own - = a related 2-category based on arithmetic universes. > > All the best, > > Steve. > >> On 1 Sep 2020, at 03:23, droberts.65537@gmail.com wrote: >> >> =EF=BB=BFDear 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/ ]