From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3635 Path: news.gmane.org!not-for-mail From: "Jamie Vicary" Newsgroups: gmane.science.mathematics.categories Subject: Re: Equalisers and coequalisers in categories with a \dag-involution Date: Sat, 17 Feb 2007 17:39:16 +0000 Message-ID: NNTP-Posting-Host: main.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 1241019425 9574 80.91.229.2 (29 Apr 2009 15:37:05 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:37:05 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Sun Feb 18 10:33:40 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 18 Feb 2007 10:33:40 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1HImvQ-0003DH-Bg for categories-list@mta.ca; Sun, 18 Feb 2007 10:22:00 -0400 Content-Disposition: inline Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 28 Original-Lines: 37 Xref: news.gmane.org gmane.science.mathematics.categories:3635 Archived-At: > Could we make the following definition: a dagger-category has > 'finite bilimits' if any finite diagram D in the category has an > 'isometric cone', a cone for which all the associated morphisms to the > objects of D are isometries, along with some sort of condition that > the isometries are orthogonal in the correct way. It is interesting to > consider this in the case of products and equalisers: for products > AxB, so it seems, the isometries to A and B will generally be > _projectors_, but for equalisers E-e->A=f,g=>B, the isometry e will > generally be an _injector_! So we cannot ask for the cone morphisms to > be isometric projectors, or isometric injectors. But perhaps this is > OK, and we can just require them to be isometries. This definition of > bilimit has the 'local flavour' of the definition of biproducts, but > cooking up a generally-applicable orthogonality condition on the > isometries seems tricky. Fred Linton has pointed out to me that my terminology here is not standard. By "isometric injector", I mean a morphism which is unitary on its range, i.e., one-to-one and norm-preserving in the case of Hilbert spaces; I believe this is usually simply referred to as an isometry. By "isometric projector", I mean a morphism which is unitary on the complement of its kernel; some people prefer to call this a "partial isometry". I was then using the terms "isometric" and "isometry" to mean "isometric projector or isometric injector". Anyway, the simple prescription I give for a bilimit cannot work, as it is easy to find diagrams in the category of finite-dimensional Hilbert spaces, our canonical example of a strongly compact-closed category with biproducts, for which the colimit and limit are not isomorphic. A diagram f:A-->B for non-iso A and B is the simplest example. However, if we restrict to diagrams F:D-->FdHilb such that D admits a dagger-operation compatible with the dagger on FdHilb, then I believe the conjecture becomes plausible. Regards, Jamie Vicary.