From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/10261 Path: news.gmane.io!.POSTED.blaine.gmane.org!not-for-mail From: David Roberts Newsgroups: gmane.science.mathematics.categories Subject: Object of connected components for internal categories Date: Sun, 26 Jul 2020 12:39:42 +0930 Message-ID: Reply-To: David Roberts Mime-Version: 1.0 Content-Type: text/plain; charset="UTF-8" Injection-Info: ciao.gmane.io; posting-host="blaine.gmane.org:116.202.254.214"; logging-data="8790"; mail-complaints-to="usenet@ciao.gmane.io" To: "categories@mta.ca list" Original-X-From: majordomo@rr.mta.ca Sun Jul 26 19:32:51 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 1jzkW2-000283-SC for gsmc-categories@m.gmane-mx.org; Sun, 26 Jul 2020 19:32:50 +0200 Original-Received: from rr.mta.ca ([198.164.44.159]:32970) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1jzkVf-0003II-3N; Sun, 26 Jul 2020 14:32:27 -0300 Original-Received: from majordomo by rr.mta.ca with local (Exim 4.92.1) (envelope-from ) id 1jzkVL-0003gi-BP for categories-list@rr.mta.ca; Sun, 26 Jul 2020 14:32:07 -0300 Precedence: bulk Xref: news.gmane.io gmane.science.mathematics.categories:10261 Archived-At: Dear all, I was wondering if there is any place in the literature that people consider a hypothesis on a category E of the following form: (*) For an internal category C in E, there is an object of connected components pi_0(C). This is equivalent to saying that s,t: Mor(C) ==> Obj(C) has a coequaliser for any C. One setup I know that is *sufficient* is if C is a sigma-pretopos (in the language of the Elephant) aka an aleph_1-ary pretopos (in the language of Shulman's exact completions article). This is a pretopos that additionally has countable disjoint sums (rather than just finite). This is sufficient because then we can take the underlying internal graph of C, then form the corresponding relation, freely form the associated symmetric, reflexive relation (this part so far only requires the finitary structure) and then form the transitive closure (this requires the countable sums). However, given an internal category C I already have a transitive (and reflexive) relation, namely the preorder reflection of C (which only requires finitary operations). So while I don't necessarily expect any other construction that doesn't require the countable sums, I was wondering if people had considered this condition (*) directly and in isolation. My best guess is it would be in some cohesion-type setup, but I'm not interested in the rest of the cohesion axioms at this time. Many thanks, David David Roberts Webpage: https://ncatlab.org/nlab/show/David+Roberts Blog: https://thehighergeometer.wordpress.com [For admin and other information see: http://www.mta.ca/~cat-dist/ ]