From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7685 Path: news.gmane.org!not-for-mail From: Peter May Newsgroups: gmane.science.mathematics.categories Subject: Re: Terminology Date: Mon, 29 Apr 2013 19:58:05 -0500 Message-ID: References: <20130429200548.GA21933@ugcs.caltech.edu> Reply-To: Peter May NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1367326513 23948 80.91.229.3 (30 Apr 2013 12:55:13 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Tue, 30 Apr 2013 12:55:13 +0000 (UTC) Cc: Categories To: Toby Bartels Original-X-From: majordomo@mlist.mta.ca Tue Apr 30 14:55:14 2013 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.1.186]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1UXA5Q-00052Y-Pj for gsmc-categories@m.gmane.org; Tue, 30 Apr 2013 14:55:12 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:42894) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1UXA3X-0002ej-S8; Tue, 30 Apr 2013 09:53:15 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1UXA3X-0006Hu-C9 for categories-list@mlist.mta.ca; Tue, 30 Apr 2013 09:53:15 -0300 In-Reply-To: <20130429200548.GA21933@ugcs.caltech.edu> Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7685 Archived-At: I apologize for poor taste, but I do like chaotic: one might substitute indiscrete in the title ``Chaotic categories and equivariant classifying spaces'' (posted at http://front.math.ucdavis.edu/1201.5178), but surely not essentially subterminal. The comment I'd really like to make is that such categories can be seriously interesting. Peter On 4/29/13 3:05 PM, Toby Bartels wrote: > Thomas Streicher wrote: > >> Jean B?nabou wrote: >>> What would you call a category X such that the functor X --> 1 is >>> full and faithful? Please don't tell me what they are, I know that. >> Sticking to the pattern I suggested I'd call it "essentially subterminal". > I learnt to call that an "indiscrete category", so I probably would. > (Another term that I've heard is "chaotic category", which I never liked.) > Of course, I could also call it a "truth value", > but only in a context where I would expect this to be understood > (and being "non-evil", that is working up to equivalence, > is not actually sufficient for that). Thus the nLab has > http://ncatlab.org/nlab/show/indiscrete+category as its own page. > >>> Non evil is essentially evil. >>> I rather like this conclusion, don't you? > It is beautiful, but is it accurate? > >> I'd expect the people abhoring evilness would >> say that full and faithful and essentially surjective is an "evil" notion >> of equivalence as opposed to the "good" one of adjoint pair where unit and >> counit are isos. The latter makes sense in any 2-category whereas the former >> doesn't. However, often you just get the "evil" version when not having >> a strong form of AC (for classes) available. > On the contrary, an ff and eso functor between two categories > is enough for the people who abhor evil, as far as I know, > to decide that the categories are equivalent (and so essentially the same). > Yet at the same time, these people tend to abhor AC! How can this be? > It works if one works in a 2-category whose 1-morphisms are anafunctors. > Then it is a theorem requiring no choice (and true internal to any topos) > that any ff and eso functor can be enriched to an adjoint equivalence > (and in an essentially unique way). > > Of course, "abhor" here should really be read as "consider optional". > It is possible to work with strict categories, or to work with AC, > but the main principles and results of category theory do not require either. > > > --Toby [For admin and other information see: http://www.mta.ca/~cat-dist/ ]