From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/10044 Path: news.gmane.org!.POSTED.blaine.gmane.org!not-for-mail From: Joseph Collins Newsgroups: gmane.science.mathematics.categories Subject: Formally adding morphisms Date: Tue, 12 Nov 2019 18:04:31 +0000 Message-ID: Reply-To: Joseph Collins Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable Injection-Info: blaine.gmane.org; posting-host="blaine.gmane.org:195.159.176.226"; logging-data="102471"; mail-complaints-to="usenet@blaine.gmane.org" To: "categories@mta.ca" Original-X-From: majordomo@rr.mta.ca Thu Nov 14 15:56:34 2019 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp2.mta.ca ([198.164.44.55]) by blaine.gmane.org with esmtps (TLS1.2:ECDHE_RSA_AES_256_GCM_SHA384:256) (Exim 4.89) (envelope-from ) id 1iVGXt-000QON-6R for gsmc-categories@m.gmane.org; Thu, 14 Nov 2019 15:56:29 +0100 Original-Received: from rr.mta.ca ([198.164.44.159]:35270) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1iVGWD-0003Me-MR; Thu, 14 Nov 2019 10:54:45 -0400 Original-Received: from majordomo by rr.mta.ca with local (Exim 4.92.1) (envelope-from ) id 1iVGVj-0003Zl-Ob for categories-list@rr.mta.ca; Thu, 14 Nov 2019 10:54:15 -0400 Accept-Language: en-GB, en-US Content-Language: en-GB Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:10044 Archived-At: Hey all Suppose that we have a category A. If we want to formally add a single morp= hism, say f:X -> Y, where X,Y are in A, but f is not in A, we can do the fo= llowing: we look at the discrete category containing only X and Y - let us= denote that as (X Y) - and the category with two objects and only a sing= le morphism between them. Let's call this one (X -> Y). There are natural embeddings (X Y) -> A and (X Y) -> (X -> Y). We take= the pushout of these functors, and as one might expect, we get the union o= f A and (X -> Y). This is basically A, but with an extra morphism formally = added in. Let's call this new morphism f and the new category A_f. This cat= egory is not particularly interesting, but I can then quotient it by some e= quations involving f and it becomes more interesting. I don't think that I am doing anything particularly modern, and I expect th= at someone else will have done something similar in the past, but my search= has not been very fruitful. Does anyone have any references that they can = throw my way? Thanks Joe [For admin and other information see: http://www.mta.ca/~cat-dist/ ]