From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9548 Path: news.gmane.org!.POSTED!not-for-mail From: Marta Bunge Newsgroups: gmane.science.mathematics.categories Subject: A la recherche de N Date: Tue, 13 Feb 2018 08:23:32 -0500 (EST) Message-ID: Reply-To: Marta Bunge NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=utf-8 Content-Transfer-Encoding: quoted-printable X-Trace: blaine.gmane.org 1518563050 5854 195.159.176.226 (13 Feb 2018 23:04:10 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Tue, 13 Feb 2018 23:04:10 +0000 (UTC) Cc: Steve Vickers , cft71@hotmail.com To: categories@mta.ca Original-X-From: majordomo@mlist.mta.ca Wed Feb 14 00:04:06 2018 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp2.mta.ca ([198.164.44.40]) by blaine.gmane.org with esmtp (Exim 4.84_2) (envelope-from ) id 1eljbv-00081Y-Ul for gsmc-categories@m.gmane.org; Wed, 14 Feb 2018 00:03:40 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:41835) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1eljdU-0004N5-Mv; Tue, 13 Feb 2018 19:05:16 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1eljcV-00088r-Bl for categories-list@mlist.mta.ca; Tue, 13 Feb 2018 19:04:15 -0400 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9548 Archived-At: Dear fellow categorists,=20 Preamble.=20 I have a question that is the key to getting the (co)KZ-monad Tau on the op= posite R of the 2-category bounded S-toposes =E2=80=9Cjust as=E2=80=9D Sigm= a was defined in Bunge-Carboni =E2=80=9CThe symmetric topos=E2=80=9D in 199= 5. In turn, just as taking opposites to Sigma gives the KZ-monad M on BTop_= S, pursued in several papers including the Bunge-Funk 2006 book, Singular C= overings of Toposes. In my =E2=80=9CPitts monads and a lax descent theorem= =E2=80=9D (2015) I used M to derive that S-essental surjections of toposes= are of effective descent, (a result due to Pitts (1986), along with other = known and unknown (lax)descent theorems, moreover in a unified manner. The = goal here is to do likewise by taking opposites to Tau, thus giving a (co)-= KZ monad N on BTop_S, to be used among other things, to derive that relativ= ely tidy surjections of toposes are of effective lax descent, a result due = to Moerdijk and Vermeulen (2000). In the Bunge-Carboni paper, Sigma was ob= tained as the left adjoint to the forgetful R=E2=80=94> A, where A was the = 2-category of locally presentable categories and cocontinuous functors as m= orphisms. It involved a lex completion at the level of the sites. The key t= hen was simply to know that coinverters in R and in A both existed and that= those in R could be calculated in A. We took the word of Max Kelly and And= y Pitts that this was indeed the case. Dually, Tau (if it existed) should b= e the left adjoint to the forgetful R=E2=80=94> B, where B now is the 2-cat= egory of locally presentable categories with lex and filtered colimit prese= rving functors as morphisms.=20 The Question.=20 Let B be the 2-category of locally presentable categories with lex and filt= ered colimit preserving functors as morphisms. Let R be the opposite of the= 2-category BTop_S of bounded S-toposes, geometric morphisms and 2-cells. D= o coinverters in B exist? If so, are those in R calculated just as in B? I= f not, is there a known class of toposes, strictly larger than that of the = coherent toposes, for which this is the case? Further comments. The locales version of an answer to my question is in the Johnstone-Vickers= paper =E2=80=9CPreframe presentations present=E2=80=99 (1991) but, just as= with Carboni we did not look at suplattices for an inspiration of how to g= et Sigma, I do not want to do likewise for Tau looking at locale preframes,= as the transition from locales to toposes is not perfect. Let me point out= howevet that the case of N restricted to coherent toposes is easy in that= setting, as the sites of coherent toposes are categories with finite limit= s already, so that the frame completion involves only adding finite colimit= s, unlike the frame completion of a preframe which involves adding both fin= ite limits and filtered colimits (plus the distributive aspect). I recentl= y learnt that also Steve Vickers and Christopher Townsend have been looking= for this N for a long time and after learning about the symmetric monad (S= igma and M). Steve, who wrote to categories about it last week, is actually= looking for the geometric counterpart of (the points of) an N(E) for E a = topos, just as the complete spreads with a locally connected domain are the= counterpart of the Lawvere distributions on toposes, that is, of (the poi= nts) of M(E).=20 Conclusion. Any concrete pointers to an answer will be appreciated. I am aware that the= re would be other ways to tackle this completion-cocompletion matter (bicom= pletions, distributive laws) if it were just a matter of getting N, but not= so for my current purpose as I explained above.=20 Best wishes, Marta [For admin and other information see: http://www.mta.ca/~cat-dist/ ]