From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3632 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: Fri, 16 Feb 2007 10:14:46 +0000 Message-ID: <131dedfb0702160214u62adb9fva396dab7be1678b7@mail.gmail.com> 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 1241019422 9557 80.91.229.2 (29 Apr 2009 15:37:02 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:37:02 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Fri Feb 16 12:25:31 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 16 Feb 2007 12:25:31 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1HI5p9-0003I9-DJ for categories-list@mta.ca; Fri, 16 Feb 2007 12:20:39 -0400 Content-Disposition: inline Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 25 Original-Lines: 34 Xref: news.gmane.org gmane.science.mathematics.categories:3632 Archived-At: Peter, Thank you for that detailed response! So it seems that if these dagger-subobjects do exist, then then will have good properties. But existence is tricky; in particular, there does not seem to be an elegant property (analagous to having finite limits and colimits) that will guarantee that all of this works. 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. Of course, in the light of http://www.arxiv.org/abs/math.CT/0604542 , perhaps we only need require that our dagger-category has products and equalizers in order for it to have 'finite bilimits'! In remark 2.6 of [2] cited in your email below, you write that if a dagger-category has products then it must of course have coproducts, but it need not have biproducts. Presumably, math.CT/0604542 proves you wrong here? Jamie.