From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2934 Path: news.gmane.org!not-for-mail From: Marco Grandis Newsgroups: gmane.science.mathematics.categories Subject: Re: Name for a concept Date: Mon, 5 Dec 2005 15:44:26 +0100 Message-ID: <41D4B4D6-BBF1-4371-9339-046EA2ADACF2@dima.unige.it> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 (Apple Message framework v733) Content-Type: text/plain; charset=ISO-8859-1; delsp=yes; format=flowed Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1241018992 6414 80.91.229.2 (29 Apr 2009 15:29:52 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:29:52 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Tue Dec 6 19:53:52 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 06 Dec 2005 19:53:52 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EjmV4-0000KP-UI for categories-list@mta.ca; Tue, 06 Dec 2005 19:45:34 -0400 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 9 Original-Lines: 89 Xref: news.gmane.org gmane.science.mathematics.categories:2934 Archived-At: In reply to M. Barr's posting. Richard Wood tells me that my posting on this subject, dated 2 =20 December 2005, was unreadable with 'elm' and nearly so with 'pine', =20 due to rich-text marks. I am reposting it in plain text (hopefully), with a few small =20 additions - and apologies MG _____ I think that such squares should be called "exact" or =20 "semicartesian" (where cartesian square =3D pb, cocartesian =3D po). They should be viewed as the natural self-dual generalisation of =20 pullback and pushout (and their name should be "self-dual", in some =20 way). They appear whenever one studies categories of relations. 1. In an abelian category (where they are chracterised by the exact =20 sequence you have mentioned), I would prefer "exact", or "Hilton-exact". Hilton considered such squares (for abelian categories), and proved =20 that an equivalent condition is that this square (of proper =20 morphisms) is "bicommutative" in the category of relations (i.e. it =20 commutes and stays commutative when you reverse two "parallel" arrows =20= - as relations). Plainly: bicartesian square =3D> pullback =3D> exact; and dually. REFERENCE: P. Hilton, Correspondences and exact squares, in: Proc. Conf. on =20 Categorical Algebra, La Jolla 1965, Springer, pp. 254-271. 2. Studying more general categories of relations, I considered =20 "semicartesian squares" (f,g, h,k), defined - in any category - as =20 the commutative squares satisfying the following self-dual property: Whenever (f',g', h,k) and (f,g, h',k') commute, also the outer =20 square (f',g', h',k') commutes B f' f h h' A' A D D' g' g k k' C (add slanting arrows f': A' --> B, g': A --> C, f: A --> B, etc). - Again: bicartesian square =3D> pullback =3D> semicartesian, and =20= dually. - If pb's and/or po's exist, there are a lot of equivalent =20 properties; eg: -- (f,g) and the pb of (h,k) have the same po (or the same =20 commutative squares out of them). - In an abelian category, semicartesian amounts to the previous notion. - In Set, it characterises again those squares which are =20 bicommutative in Rel. REFERENCE: M. Grandis, Sym=E9trisations de categories et factorisations =20 quaternaires, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. =20 14 sez. 1 (1977), 133-207. 3. A 2-dimensional version of this property (actually a STRUCTURE on =20 2-cells), was introduced by Guitart, and called "H-exact", if I =20 remember well (H for Hilton) REFERENCES: - R. Guitart, Carr=E9s exacts et carr=E9s deductifs, Diagrammes 6 = (1981), =20 G1-G17. - R. Guitart and L. Van den Bril, Calcul des satellites et =20 pr=E9sentations des bimodules =E0 l'aide des carr=E9s exacts, Cahiers =20= Topologie G=E9om. Diff=E9rentielle 24 (1983), no. 3, 299-330. (and some other papers by the same authors). Best regards Marco Grandis