From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2930 Path: news.gmane.org!not-for-mail From: Marco Grandis Newsgroups: gmane.science.mathematics.categories Subject: Re: Name for a concept Date: Fri, 2 Dec 2005 14:51:38 +0100 Message-ID: <15DD9EBD-B91E-4386-A143-DD3A4348F0DB@dima.unige.it> References: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 (Apple Message framework v733) Content-Type: multipart/alternative; boundary=Apple-Mail-5-605476314 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1241018989 6399 80.91.229.2 (29 Apr 2009 15:29:49 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:29:49 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Fri Dec 2 14:02:54 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 02 Dec 2005 14:02:54 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EiFDV-0006NY-Qm for categories-list@mta.ca; Fri, 02 Dec 2005 14:01:05 -0400 In-Reply-To: Original-Content-Type: text/plain;charset=ISO-8859-1;delsp=yes;format=flowed Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 5 Original-Lines: 313 Xref: news.gmane.org gmane.science.mathematics.categories:2930 Archived-At: I think that such squares should be called "exact" or =20 "semicartesian" (where cartesian square =3D pb). They should be viewed as the natural self-dual generalisation of =20 pullback and pushout. They appear whenever one studies categories of =20 relations. 1. In an abelian category, 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 - by =20 the following self-dual property (after being commutative, of course): Whenever (f',g', h,k) and (f,g, h',k') commute, also the =20 external 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, f: A --> B, etc). - Again: bicartesian square =3D> pullback =3D> semicartesian, and =20= dually. - If pb's and/or po's exist, one can give 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 --Apple-Mail-5-605476314 Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset=ISO-8859-1
I think that such squares = should be called "exact" or "semicartesian" (where cartesian square =3D = pb).
They should be viewed as the=A0natural self-dual = generalisation of pullback and pushout. They appear whenever one studies = categories of relations.

1.= In an abelian category, I would prefer "exact", or = "Hilton-exact".

Hilton considered such = squares (for abelian categories), and proved that an equivalent = condition is that this square (of proper morphisms) is "bicommutative" = in the category of relations (i.e. it commutes and stays commutative = when you reverse two "parallel" arrows - as relations).

Plainly: =A0 bicartesian square=A0 =3D>=A0 pullback=A0 =3D>=A0 exact;=A0 and = dually.

REFERENCE:
P. Hilton, Correspondences and = exact squares, in: Proc. Conf. on Categorical Algebra, La Jolla = 1965, Springer, pp. 254-271.

2.= Studying more general categories of relations, I considered = "semicartesian squares"=A0 (f,g, h,k),=A0 defined - in any category - by = the following self-dual property (after being commutative, of = course):

=A0Whenever=A0 (f',g', h,k)=A0 and=A0 (f,g, h',k')=A0 commute, also the external square=A0 (f',g', h',k')=A0 commutes

=A0=A0 =A0=A0 =A0=A0 =A0=A0 = =A0=A0 =A0=A0 =A0=A0 =A0=A0 =A0=A0 =A0B =A0
=A0= =A0 =A0=A0 =A0=A0 =A0f'=A0 =A0=A0 =A0=A0 =A0=A0 =A0f=A0 =A0 =A0=A0 = =A0h = =A0 =A0=A0 =A0=A0= =A0h'
=A0 A'=A0=A0 =A0=A0 =A0=A0 =A0=A0 = =A0A = =A0 =A0=A0 =A0=A0= =A0=A0 =A0=A0 =A0D =A0 =A0=A0 =A0=A0 =A0=A0 =A0D'
=A0= =A0 =A0=A0 =A0=A0 =A0g'=A0=A0 =A0=A0 =A0=A0 =A0g=A0=A0 =A0=A0 = =A0k = =A0=A0 =A0=A0 = =A0=A0 =A0k'
=A0 =A0=A0 =A0=A0 =A0=A0 = =A0=A0 =A0=A0 =A0=A0 =A0=A0 =A0=A0 =A0C

(add slanting arrows=A0 f': A' --> B, =A0f: A = --> B,=A0 =A0etc).

- Again: bicartesian = square=A0 = =3D>=A0 pullback=A0 =3D>=A0 semicartesian,=A0 and dually.

- If pb's=A0 and/or=A0 = po's exist, one can give a lot of equivalent properties; = eg:

--=A0 (f,g)=A0 and=A0the pb of=A0 (h,k) have the same = po (or the same commutative squares out of them).

- = In an abelian category, semicartesian amounts to the previous = notion.
- In Set, it characterises again those squares which = are bicommutative in Rel.

REFERENCE:
M. Grandis, Sym=E9trisations de = categories et factorisations quaternaires, Atti Accad. Naz. Lincei = Mem. Cl. Sci. Fis. Mat. Natur. 14 sez. 1 (1977), 133-207.

3. A 2-dimensional version = of this property (actually a STRUCTURE on 2-cells), was introduced by = Guitart, and called "H-exact", if I remember well (H for = Hilton)

REFERENCES:

- R. Guitart, Carr=E9s = exacts et carr=E9s deductifs, Diagrammes 6 (1981), = G1-G17.
- R. Guitart and L. Van den Bril, Calcul des = satellites et pr=E9sentations des bimodules =E0 l'aide des carr=E9s = exacts, Cahiers Topologie G=E9om. Diff=E9rentielle 24 (1983), no. 3, = 299-330.
(and some other papers by the same = authors).

Best regards

Marco = Grandis

= --Apple-Mail-5-605476314--