From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2917 Path: news.gmane.org!not-for-mail From: Francis Borceux Newsgroups: gmane.science.mathematics.categories Subject: semi-categories Date: Tue, 29 Nov 2005 14:55:58 +0100 Message-ID: NNTP-Posting-Host: main.gmane.org Content-Type: text/plain; charset="iso-8859-1" ; format="flowed" Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1241018982 6350 80.91.229.2 (29 Apr 2009 15:29:42 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:29:42 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Wed Nov 30 15:04:50 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 30 Nov 2005 15:04:50 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EhX83-00022B-VB for categories-list@mta.ca; Wed, 30 Nov 2005 14:56:31 -0400 X-Mailer: Eudora F6.0.2 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 62 Original-Lines: 124 Xref: news.gmane.org gmane.science.mathematics.categories:2917 Archived-At: There has been a long discussion on the list=20 about "categories without identities", whatever=20 you decide to call them. And the attention has=20 been brought to axioms which could -- in this=20 more general context -- replace the identity=20 axiom. I would like to focus on a very striking categorical aspect of this problem. A (right) module M on a ring R with unit must satisfy the axiom m1=3Dm =2E.. but what about the case when R does not have a unit ? Simply dropping the axiom m1=3Dm leaves you with=20 the unpleasant situation where you have two=20 different notions of module, in the case where R=20 has a unit. Therefore people working in linear algebra have considered the axiom the scalar multiplication M@R ---> M is an isomorphism (@=3Dtensor product sign) which is equivalent to the axiom m1=3Dm, when the=20 ring has a unit ... but makes perfect sense when=20 the ring does not have a unit. Such modules are=20 generally called "Taylor regular". A ring R with unit is simply a one-objet additive=20 category and a right module M on R is simply an=20 additive presheaf M ---> Ab (=3Dthe category of=20 abelian groups). A ring without unit is thus a "one-object=20 additive category without identity", again=20 whatever you decide to call this. But what is the analogue of the axiom M@R ---> M is an isomorphism when R is now an arbitrary small (enriched)=20 "category without identities" and M is an=20 arbitrary (enriched) presheaf on it ? All of us know that to define a (co)limit, we do=20 not need at all to start with an indexing=20 category: an arbitrary graph with arbitrary=20 commutativity conditions works perfectly well. In=20 particular, a "category without identities" is=20 all right. And the same holds in the enriched=20 case, with (co)limits replaced by "weighted=20 (co)limits". Now every presheaf on a small category is=20 canonically a colimit of representable ones ...=20 but this result depends heavily on the existence=20 of identities ! When you work with a presheaf M=20 on a "category R without identities", you still=20 have a canonical morphism canonical colimit of representables ---> R and you can call M "Taylor regular" when this is=20 an isomorphism. Again in the enriched case,=20 "colimit" means "weighted colimit". This=20 recaptures exactly the case of "Taylor regular=20 modules", when working with Ab-enriched=20 categories. A sensible axiom to put on a "category R without=20 identities" is the fact that the representable=20 functors are "Taylor regular". (We should=20 certainly call this something else than "Taylor=20 regular", but let me keep this terminology in=20 this message.) And when R is a "Taylor regular category without identities", the constructi= on presheaf on R |---> corresponding canonical colimit of representables yields a reflection for the inclusion of Taylor=20 regular presheaves in all presheaves. A very striking property is the existence of a=20 further (necessarily full and faithful) left=20 adjoint to this reflection. This second inclusion=20 provides in fact an equivalence with the full=20 subcategory of those presheaves which satisfy the=20 Yoneda isomorphism. This yields thus a nice example of what Bill=20 Lawvere calls the "unity of opposites": the two=20 inclusions identify the category of Taylor=20 regular presheaves with * on one side, those presheaves which are colimits of representables; * on the other side, those presheaves which satisfy the Yoneda lemma. This underlines the pertinence of these "Taylor=20 regular categories without identities". To my knowledge, the best treatment of these=20 questions is to be found in various papers by=20 Marie-Anne Moens and by Isar Stubbe, in=20 particular in the "Cahiers" and in "TAC". And very interesting examples occur in functional=20 analysis (the identity on a Hilbert space is a=20 compact operator ... if and only if the space is=20 finite dimensional) and also in the theory of=20 quantales. =46rancis Borceux -- =46rancis BORCEUX D=E9partement de Math=E9matique Universit=E9 Catholique de Louvain 2 chemin du Cyclotron 1348 Louvain-la-Neuve (Belgique) t=E9l. +32(0)10473170, fax. +32(0)10472530 borceux@math.ucl.ac.be