From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9412 Path: news.gmane.org!.POSTED!not-for-mail From: ptj@maths.cam.ac.uk Newsgroups: gmane.science.mathematics.categories Subject: Re: Some questions about different notions of "theory" Date: 07 Nov 2017 23:57:07 +0000 Message-ID: References: Reply-To: ptj@maths.cam.ac.uk NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 Content-Type: text/plain; format=flowed; charset=ISO-8859-1 X-Trace: blaine.gmane.org 1510186235 26471 195.159.176.226 (9 Nov 2017 00:10:35 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Thu, 9 Nov 2017 00:10:35 +0000 (UTC) Cc: categories To: Mike Stay Original-X-From: majordomo@mlist.mta.ca Thu Nov 09 01:10:31 2017 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 1eCaQP-0006ZF-CM for gsmc-categories@m.gmane.org; Thu, 09 Nov 2017 01:10:29 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:59071) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1eCaQj-00019n-Qx; Wed, 08 Nov 2017 20:10:49 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1eCaPJ-0005oh-24 for categories-list@mlist.mta.ca; Wed, 08 Nov 2017 20:09:21 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9412 Archived-At: 1) I'm not sure what Mike means by `those monads that correspond to toposes' since most toposes don't correspond to monads on anything. I did investigate those toposes which are monadic over Set, or a power of Set, in my papers `When is a variety a topos?', Algebra Universalis 21 (1985), 198--212, and `Collapsed toposes and cartesian closed varieties', J. Algebra 129 (1990), 446--480. 2) A possible answer to this question is that the (2-)category of finitely presented minimal toposes (and logical functors) is equivalent to the dual of the free topos (on no generators), where a topos is said to be minimal if it has no proper full logical subtoposes. This is a result of Peter Freyd, but I don't know whether he ever published it. Peter Johnstone On Nov 7 2017, Mike Stay wrote: >1) Finitary monads correspond to Lawvere theories. Is there a name >for those monads that correspond to toposes? > >2) In topos theory is there any analogous result to Lawvere's theorem >that the opposite of the category of free finitely generated gadgets >is equivalent to the Lawvere theory of gadgets? Something like "the >opposite of the category of fooable gadgets is equivalent to the topos >of gadgets"? > >3) nLab says a sketch is a small category T equipped with subsets >(L,C) of its limit cones and colimit cocones. A model of a sketch is >a Set-valued functor preserving the specified limits and colimits. Is >preserving limits and colimits like a ring homomorphism? Preserving >both limits and colimits sounds like it ought to involve profunctors, >but maybe I'm level slipping. > > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]