From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/10053 Path: news.gmane.org!.POSTED.blaine.gmane.org!not-for-mail From: Fernando Lucatelli Nunes Newsgroups: gmane.science.mathematics.categories Subject: Re: Formally adding morphisms Date: Sun, 17 Nov 2019 06:58:59 +0200 Message-ID: References: Reply-To: Fernando Lucatelli Nunes Mime-Version: 1.0 Content-Type: text/plain; charset="UTF-8" Injection-Info: blaine.gmane.org; posting-host="blaine.gmane.org:195.159.176.226"; logging-data="217491"; mail-complaints-to="usenet@blaine.gmane.org" Cc: To: Joseph Collins Original-X-From: majordomo@rr.mta.ca Tue Nov 19 23:44:33 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 1iXCEa-000uOG-V5 for gsmc-categories@m.gmane.org; Tue, 19 Nov 2019 23:44:33 +0100 Original-Received: from rr.mta.ca ([198.164.44.159]:35844) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1iXCDc-00021d-DL; Tue, 19 Nov 2019 18:43:32 -0400 Original-Received: from majordomo by rr.mta.ca with local (Exim 4.92.1) (envelope-from ) id 1iXCD2-0000eb-5D for categories-list@rr.mta.ca; Tue, 19 Nov 2019 18:42:56 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:10053 Archived-At: Dear Joseph, The objects of a category A is in bijective relation with the functors 1\to A, in which 1 is the terminal category. The free addition of a morphism in a category is actually a very particular case of a Cat-weighted colimit (see Limits indexed by category-valued 2-functors, Street), called coinserter (see pag. 307 of Elementary observations on 2-categorical limits, Kelly), in Cat. Given a pair of objects x: 1\to A, y: 1\to A, the coinserter of the pair x: 1\to A and y: 1\to A is the category you are looking for (and, as any Cat-enriched colimit of them Cat-category Cat, it can be constructed out of the coequalizers, coproducts and products). Fernando On Thu, Nov 14, 2019 at 2:56 PM Joseph Collins wrote: > Hey all > > Suppose that we have a category A. If we want to formally add a single > morphism, say f:X -> Y, where X,Y are in A, but f is not in A, we can do > the following: 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 single 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 of 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 category is not particularly interesting, but I can then quotient it > by some equations involving f and it becomes more interesting. > > I don't think that I am doing anything particularly modern, and I expect > that 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/ ]