From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9420 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: 10 Nov 2017 00:00:35 +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 1510356575 5127 195.159.176.226 (10 Nov 2017 23:29:35 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Fri, 10 Nov 2017 23:29:35 +0000 (UTC) Cc: categories To: Mike Stay Original-X-From: majordomo@mlist.mta.ca Sat Nov 11 00:29:29 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 1eDIjp-00012g-2J for gsmc-categories@m.gmane.org; Sat, 11 Nov 2017 00:29:29 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:59666) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1eDIkz-0004LS-Lz; Fri, 10 Nov 2017 19:30:41 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1eDIjY-00008v-JH for categories-list@mlist.mta.ca; Fri, 10 Nov 2017 19:29:12 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9420 Archived-At: Dear Mike, Thanks for the clarification, but the same comment applies: most higher- order theories don't correspond to anything as simply described as a monad (just as most first-order theories don't, but only the very special class of algebraic theories). For first-order theories (including infinitary ones) you have the notion of sketch, but that (unlike a monad) is still presentation-dependent (i.e. it varies according to which parts of the structure you're modelling you regard as primitive); if you want something that doesn't depend on the presentation, you need to go all the way to the syntactic category, which essentially consists of `everything that has to be present in a category containing a model of the theory under consideration' (just as a Lawvere theory contains all the operations which have to be present in a model of an algebraic theory). Syntactic categories can also be constructed for higher-order theories, and they are in fact toposes; but they tend to be insanely complicated (even the free topos, which corresponds to the empty theory, has an incredibly rich structure). Peter Johnstone On Nov 9 2017, Mike Stay wrote: >On Tue, Nov 7, 2017 at 4:57 PM, wrote: >> 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'm sorry, I meant "those monads that correspond to higher-order >theories", where a higher-order theory gets interpreted in a topos in >the same way that a Lawvere theory gets interpreted in a category with >finite products. > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]