From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7677 Path: news.gmane.org!not-for-mail From: =?iso-8859-1?Q?Jean_B=E9nabou?= Newsgroups: gmane.science.mathematics.categories Subject: Re: Terminology Date: Sun, 28 Apr 2013 05:49:16 +0200 Message-ID: References: <20130427130857.GC16801@mathematik.tu-darmstadt.de> Reply-To: =?iso-8859-1?Q?Jean_B=E9nabou?= NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 (Apple Message framework v1283) Content-Type: text/plain; charset=iso-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1367187424 18554 80.91.229.3 (28 Apr 2013 22:17:04 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sun, 28 Apr 2013 22:17:04 +0000 (UTC) Cc: Categories To: Thomas Streicher Original-X-From: majordomo@mlist.mta.ca Mon Apr 29 00:17:08 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 1UWZu7-00012I-SD for gsmc-categories@m.gmane.org; Mon, 29 Apr 2013 00:17:07 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:41123) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1UWZsB-0008Dm-ID; Sun, 28 Apr 2013 19:15:07 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1UWZsC-0006uE-GQ for categories-list@mlist.mta.ca; Sun, 28 Apr 2013 19:15:08 -0300 In-Reply-To: <20130427130857.GC16801@mathematik.tu-darmstadt.de> Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7677 Archived-At: Dear Thomas, Dear all, My definition of poset is: "preordered set". I don't know if there is a = general agreement, since some answers seemed to suppose that I meant = "partially ordered set". It is because I feared this confusion that i = specified by adding: equivalent to a discrete category.=20 Of course I knew that they were "equivalence relations", and had also = many other simple characterizations. One which I like and need is: X is = equivalent to a discrete category iff the functor X --> 1 is faithful = and conservative (i.e. reflects isos) because it has the following = generalization: Let P: X --> S be a prefibration. The following are equivalent: (i) P is equivalent to a discrete fibration. (ii) P is faithful and consevative. (iii) each fiber of P is equivalent to a discrete category. Thus my question was not: what are such categories, for which I knew = perfectly many answers, but : is there a well established name for them?=20= Suggestions such as "setoids" or "essentially discrete" show that this = is not the case. I don't like very much "setoids", and I am very tempted by "essentially = discrete" as Thomas suggested.=20 But I shall make my question a bit more difficult.=20 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. I'm not = even asking if there is a we'll established name for them. I don't think = there is one. What I ask is: Could you suggest one? Preferably a name = which would be suitable when we work with categories internal to a Topos = E where supports don't split. As a side remark, let me say that I don't care very much for the = distinction between "evil" and "non evil". Apart from obvious moral or = philosophical reasons, for the following purely mathematical one: = Non-evilness depends on the notion of equivalence of categories. And = this in turn may heavily depend on which notion of equivalence you = chose. And some of these notions depend on the axiom of choice, which I = might be tempted to call "evil". Thus we'd reach the following = conclusion:=20 Non evil is essentially evil.=20 I rather like this conclusion, don't you? Best regards, Jean Le 27 avr. 2013 =E0 15:08, Thomas Streicher a =E9crit : > Dear Jean, >=20 a >=20 >> As many of you I presume, I have for ages, and very often, had to = deal with categories which are both groupo=EFds and posets, or again = which are equivalent to a discrete category. Is there a well established = name for them? >=20 > What about "essentially discrete" like in "essentially small" or > "essentially surjective". Generally, for any property P of categories > I would say a category is "essentially P" if it is equivalent to a > category with property P. > So "essentially" is a kind of magic word transforming "evil" = properties > into "non-evil" ones. (I don't think one should always do this!) >=20 > Thomas [For admin and other information see: http://www.mta.ca/~cat-dist/ ]