From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5431 Path: news.gmane.org!not-for-mail From: "Reinhard Boerger" Newsgroups: gmane.science.mathematics.categories Subject: Re: A well kept secret Date: Mon, 28 Dec 2009 11:07:29 +0100 Message-ID: References: Reply-To: "Reinhard Boerger" NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1262041917 29703 80.91.229.12 (28 Dec 2009 23:11:57 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Mon, 28 Dec 2009 23:11:57 +0000 (UTC) To: Original-X-From: categories@mta.ca Tue Dec 29 00:11:50 2009 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.50) id 1NPOkg-0005Q6-6f for gsmc-categories@m.gmane.org; Tue, 29 Dec 2009 00:11:50 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1NPOJ8-0007Ek-EE for categories-list@mta.ca; Mon, 28 Dec 2009 18:43:22 -0400 In-Reply-To: Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5431 Archived-At: Hello, I still like to add some remarks. Category theory is one part of mathematics, and it should be treated not better, but not worse than = others. It looks more important to me that categorical thinking becomes popular = in other areas in mathematics. Some years ago, a functional analyst needed = half an our to prove that homeomorphic Banach spaces have homemorphic duals, = a simple consequence of the fact that all functors preserve isomorphisms.=20 Another example from my own experience: People, who worked about orthomodular lattices noticed that they have no tensor product. So they tried to weaken the notions and ended up with effect algebras, but unfortunately they did not admit a tensor product either. Su people = looked for other notions. But they had already shown that a tensor product of effect algebras exists if one admits 0=3D1; i.e. the tensor product my collapse. But because they did not admit this, they had to formulate = their result more complicated. Later I saw that tensor products of orthomodular posets exist if one = admits 0=3D1; the easy proof uses the Adjoint Functor Theorem and does not give = much insight into the structure. It also seems to work for orthomodular = lattices. My preference for orthomodular posets rather than lattices is also = inspired by categorical thinking. The idempotents of an arbitrary ring with 1 = form an orthomodular poset, and this construction yields a functor. This is the non-commutative analogue to the Boolean algebra of idempotents of an arbitrary ring. But most people were inspired by quantum dynamics and = were looking for an abstraction for the set of projections of a Hilbert = space. Here joins and meets exist (somehow accidentially) because projections correspond to closed subspaces. But they are not continuous and have no physical meaning in general. I think it is often better to look for functorial notions than to use ad-hoc-abtractions. Greetings Reinhard [For admin and other information see: http://www.mta.ca/~cat-dist/ ]