From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8800 Path: news.gmane.org!not-for-mail From: Richard Garner Newsgroups: gmane.science.mathematics.categories Subject: Re: two categories of interest Date: Tue, 19 Jan 2016 13:51:33 +1100 Message-ID: References: Reply-To: Richard Garner NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=utf-8 Content-Transfer-Encoding: 8bit X-Trace: ger.gmane.org 1453297897 27176 80.91.229.3 (20 Jan 2016 13:51:37 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 20 Jan 2016 13:51:37 +0000 (UTC) Cc: "categories\@mta.ca" To: Paul B Levy Original-X-From: majordomo@mlist.mta.ca Wed Jan 20 14:51:30 2016 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.7.22]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1aLtAU-0004F1-NV for gsmc-categories@m.gmane.org; Wed, 20 Jan 2016 14:51:26 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:47747) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1aLt9s-0005Hs-Rg; Wed, 20 Jan 2016 09:50:48 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1aLt9n-0001I0-OI for categories-list@mlist.mta.ca; Wed, 20 Jan 2016 09:50:43 -0400 In-reply-to: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8800 Archived-At: Dear Paul, > Have either of the following categories been studied before? > > 1. A "set with loners" is a set A with a subset U, whose elements are > called "loners". A "loner-respecting function" (A,U) --> (B,V) is a > function A --> B such that for any x in U, f(x) is in V and its only > f-preimage is x. Let SWL be the category of sets with loners and > loner-respecting functions, and Inj the category of sets and injections. > Both Set and Inj are isomorphic to full subcategories of SWL. I don't know if this category has been studied, but it looks like you can also describe it as follows. Take the category Inj x Set. On here there is a monad defined by T(A,B) = (A,A+B). The Kleisli category of this monad appears to be SWL. That presentation seems to make it look dual to Dialectica-type stuff. > 2. For sets A and B, a "sum preorder" from A to B is a preorder on A+B. > Example: A is the set of men, B is the set of women, take the preorder > "younger than or the same age as". An equivalence relation on A+B is > called a "corelation" from A to B. Given sum preorders R : A --> B and > S : B --> C, obtain the composite by taking the least preorder on A+B+C > that contains R and S, and then restricting to A+C. Let SumPreord be > the category of sets and sum preorders, Rel the category of sets and > relations, and Corel the category of sets and corelations. Both Rel and > Corel are isomorphic to lluf subcategories of SumPreord. Yes: Dosen, Petric, "Syntax for split preorders", Annals of Pure and Applied Logic 164 (2013) 443???481 I think Danos and Regnier might also talk about related things somewhere, but I can't exactly tell you where (or if I am remembering correctly). Richard [For admin and other information see: http://www.mta.ca/~cat-dist/ ]