From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/10228 Path: news.gmane.io!.POSTED.ciao.gmane.io!not-for-mail From: Alexander Campbell Newsgroups: gmane.science.mathematics.categories Subject: Adjoining companions Date: Thu, 4 Jun 2020 19:21:39 +1000 Message-ID: Reply-To: Alexander Campbell Mime-Version: 1.0 Content-Type: text/plain; charset="UTF-8" Injection-Info: ciao.gmane.io; posting-host="ciao.gmane.io:159.69.161.202"; logging-data="6872"; mail-complaints-to="usenet@ciao.gmane.io" To: categories@mta.ca Original-X-From: majordomo@rr.mta.ca Thu Jun 04 16:12:35 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 1jgqbi-0001fV-3L for gsmc-categories@m.gmane-mx.org; Thu, 04 Jun 2020 16:12:34 +0200 Original-Received: from rr.mta.ca ([198.164.44.159]:55498) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1jgqZl-0006Gb-A5; Thu, 04 Jun 2020 11:10:33 -0300 Original-Received: from majordomo by rr.mta.ca with local (Exim 4.92.1) (envelope-from ) id 1jgqWZ-0002MB-Fj for categories-list@rr.mta.ca; Thu, 04 Jun 2020 11:07:15 -0300 Precedence: bulk Xref: news.gmane.io gmane.science.mathematics.categories:10228 Archived-At: Dear categorists, I have encountered the following pair of thorny questions in the course of an investigation into double categories, to which I would like to know the answers. I would be grateful for any help you may have to offer. Given a horizontal morphism f in a double category A, we may construct a new double category A[f_*] by freely adjoining a vertical companion for f. This new double category comes equipped with a double functor A --> A[f_*], which is the pushout of the inclusion H(2) --> Sq(2) along the double functor H(2) --> A that picks out the horizontal morphism f. (Here H(2) and Sq(2) denote the "free-living horizontal morphism" and the "free-living companion pair" respectively.) Question 1: Is the double functor A --> A[f_*] fully faithful on squares? A related question concerns the "double category of squares" functor Sq : 2-Cat --> DblCat and its left adjoint St : DblCat --> 2-Cat. Question 2: Is the unit double functor A --> Sq(St(A)) fully faithful on squares? Best regards, Alexander [For admin and other information see: http://www.mta.ca/~cat-dist/ ]