From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/10280 Path: news.gmane.io!.POSTED.blaine.gmane.org!not-for-mail From: Alexander Gietelink Oldenziel Newsgroups: gmane.science.mathematics.categories Subject: Re: Topos objects Date: Fri, 4 Sep 2020 13:52:57 +0200 Message-ID: References: Reply-To: Alexander Gietelink Oldenziel 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="16386"; mail-complaints-to="usenet@ciao.gmane.io" Cc: David Roberts , "categories@mta.ca list" To: Steve Vickers Original-X-From: majordomo@rr.mta.ca Mon Sep 07 23:49:29 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 1kFP0z-00048E-7l for gsmc-categories@m.gmane-mx.org; Mon, 07 Sep 2020 23:49:29 +0200 Original-Received: from rr.mta.ca ([198.164.44.159]:36598) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1kFP0a-0008Ft-Hq; Mon, 07 Sep 2020 18:49:04 -0300 Original-Received: from majordomo by rr.mta.ca with local (Exim 4.92.1) (envelope-from ) id 1kFOxs-0006LG-BK for categories-list@rr.mta.ca; Mon, 07 Sep 2020 18:46:16 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.io gmane.science.mathematics.categories:10280 Archived-At: Dear Steve, I believe David is indeed talking about elementary topoi internal to a suitable bicategory. Your remarks about the object classifier, exponentiable topoi [image: S[\mathbb{O}], S[\mathbb{O}]^X] in [image: GTopos/S] are well-taken. I believe it is sometimes preferred to say they are ' logos' objects inside the category of topoi, i.e. formal duals to topoi rather than a topos object. For simplicity, let us consider just [image: S[\mathbb{O}]]. It has finite limits in the sense that for any finite diagram [image: J] we have a geometric morphism [image: \varprojlim: S[\mathbb{O}]^J \to S[\mathbb{O}]] which is right adjoint to the diagonal [image: \Delta: S[\mathbb{O}] \to S[\mathbb{O}]]. Indeed, we may define this simply on points by [image: \{F(j)\}_{j\in J} \mapsto \varprojlim F(j)], noting that geometric morphisms preserve finite limits. Similarly, for infinite colimits we have a morphism [image: \varinjlim: S[\mathbb{O}]^J \to S[\mathbb{O}]], defined similarly. To say [image: S[\mathbb{O}]] is a logos object, we should probably say it satisfies some version of the Giraud axioms. I am not completely confident how this would work precisely. I do understand the locale case. Here the Sierpinski locale [image: \mathbb{S}] is an internal frame in [image: Locale] which simply means we have [image: \wedge: \mathbb{S}^2 \to \mathbb{S}, \vee_I: \mathbb{S}^I \to \mathbb{S}] which distribute. The Giraud axioms can also be seen as a sort of distributivity axioms, but the details elude me. You mention something about using the M-algebra structure of [image: S[\mathbb{O}]] where [image: M] is the symmetric topos construction. I am not sure how the symmetric topos monad will help, could you say more about this? best, Alexander On Fri, 4 Sep 2020 at 02:54, Steve Vickers wrote: > 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 understood what the question was. What did you mean by =E2=80=9Ctop= os=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 cartesian > 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 > enable 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 arrows, and 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 pro= duct > 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 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 ve= ry > interesting question when an object in a 2-category 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 exponentiable topo= s 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 > 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 morphism= .) > 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. Tha= t > 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. > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]