From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7087 Path: news.gmane.org!not-for-mail From: David Roberts Newsgroups: gmane.science.mathematics.categories Subject: Re: Dualities arising via pairs of schizophrenic objects Date: Wed, 30 Nov 2011 10:07:15 +1030 Message-ID: References: Reply-To: David Roberts NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8 X-Trace: dough.gmane.org 1322832755 11822 80.91.229.12 (2 Dec 2011 13:32:35 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Fri, 2 Dec 2011 13:32:35 +0000 (UTC) To: Categories Original-X-From: majordomo@mlist.mta.ca Fri Dec 02 14:32:31 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.4]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1RWTE7-0007go-CJ for gsmc-categories@m.gmane.org; Fri, 02 Dec 2011 14:32:31 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:32980) by smtpx.mta.ca with esmtp (Exim 4.76) (envelope-from ) id 1RWTCc-00042V-6p; Fri, 02 Dec 2011 09:30:58 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1RWTCa-00033M-LJ for categories-list@mlist.mta.ca; Fri, 02 Dec 2011 09:30:56 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7087 Archived-At: Vaughan wrote: > What's odd is that the term seems to be used precisely when it is > mathematically inappropriate in the above sense (quite apart from > medical or sensitivity issues). > > The real manifestation of the "same" entity is not with objects at all > but with homsets, namely the homset C(D'(d'), d) in C and the homset > C'(D(d), d') in C', which *do* have the same number of morphisms. > > If anything deserves the epithet in question it is that homset in each > category. The two homsets are in bijection, but their targets don't > correspond, having only in common that they are the dualizers in the > respective categories. Yes, I always thought it odd that even when one wants to accept category-theoretic foundations (e.g. ETCS or similar), then suddenly something like this comes along, where people start saying there is a thing which is an object of two different categories. Such a property isn't even expressible in type theory-style foundations, where even elements of two different sets aren't comparable... But since both of the categories in each pair Vaughan mentioned are enriched over Set, we *are* allowed to compare hom-sets, at least using some sort of roughly canonical isomorphism. Using the formalism suggested (hom-sets), it seems much easier to set down a definition of these slippery objects. And (more whimsically) regarding terminology: if someone wanted to use the analogy of a door, why not a window? We can have fenestral objects, by which one can 'see' from one category to another. And unfortunately, 'liminal' gives rise to subliminal, which might be a natural prefix extension mathematically but is even more confusing than the existing inaccurate term. :-) David [For admin and other information see: http://www.mta.ca/~cat-dist/ ]