From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9253 Path: news.gmane.org!.POSTED!not-for-mail From: David Roberts Newsgroups: gmane.science.mathematics.categories Subject: Models of finite-limit sketches in internal logic of a (pre)topos Date: Tue, 11 Jul 2017 08:44:44 +0930 Message-ID: Reply-To: David Roberts NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="UTF-8" X-Trace: blaine.gmane.org 1499803560 5326 195.159.176.226 (11 Jul 2017 20:06:00 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Tue, 11 Jul 2017 20:06:00 +0000 (UTC) To: "categories@mta.ca list" Original-X-From: majordomo@mlist.mta.ca Tue Jul 11 22:05:55 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 1dV1Pp-0000ik-MG for gsmc-categories@m.gmane.org; Tue, 11 Jul 2017 22:05:49 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:51924) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1dV1QQ-0003ca-U5; Tue, 11 Jul 2017 17:06:26 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1dV1Oq-00072N-C8 for categories-list@mlist.mta.ca; Tue, 11 Jul 2017 17:04:48 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9253 Archived-At: Hi all, I believe that if one has some finite limit sketch S, then models of S in the internal logic of a topos E should be equivalent to external models. I'm thinking here about forcing from the sheaf-theoretic viewpoint, so that some algebraic gizmo in the forced model(=in internal logic of the topos) is none other than that algebraic gizmo internal to the category from the external perspective. Or, that a model in some filterquotient E/~ of a topos E is equivalent to a model in E. Is there a reference I could point to? Or is it obvious because a finite-limit sketch uses no quantifiers etc? I would guess such reasoning to hold in a much more general setting than a topos, for instance pretoposes or regular categories. A second question, that I do not know the answer to: how far can one generalise theories (from finite-limit etc) and still get {models in internal logic} ~ {models in the category}? Here "the category" has sufficient structure to interpret the theory. Thanks, David -- David Roberts http://ncatlab.org/nlab/show/David+Roberts [For admin and other information see: http://www.mta.ca/~cat-dist/ ]