From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5651 Path: news.gmane.org!not-for-mail From: Toby Bartels Newsgroups: gmane.science.mathematics.categories Subject: Re: equality is beautiful Date: Sun, 21 Mar 2010 12:36:32 -0700 Message-ID: References: Reply-To: Toby Bartels NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii X-Trace: dough.gmane.org 1269270933 1126 80.91.229.12 (22 Mar 2010 15:15:33 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Mon, 22 Mar 2010 15:15:33 +0000 (UTC) To: categories@mta.ca Original-X-From: categories@mta.ca Mon Mar 22 16:15:27 2010 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 4.69) (envelope-from ) id 1NtjLh-0005y6-OI for gsmc-categories@m.gmane.org; Mon, 22 Mar 2010 16:15:25 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Ntiol-0006Ls-1G for categories-list@mta.ca; Mon, 22 Mar 2010 11:41:23 -0300 Content-Disposition: inline In-Reply-To: Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5651 Archived-At: Richard Garner wrote: >however, if one wishes this notion of product >to become a special case of the notion of limit (a demand >which seems not unreasonable) then it is enough to ask your >type theory to have identity types: for then any preset A can >be made into a category A# whose hom-setoids are the identity >types Id_A(x,y) equipped with their propositional equality. And as well, any preset is thus made into a set(oid). (Categorially, we form the free set on a given preset.) And so any category is strict. Although this is a feature of most type theories in practice, I've always found it rather artificial. Bishop's insight is that you have to *define* equality, and while it's a step up to say that you *can* define equality if you wish to, it's unsatisfying to fall back and say you don't *have* to. Not that identity types can't have their own interesting structure. The elements of identity types have their own identity types, etc, so every type becomes not only a set but an infinity-groupoid; see Awodey & Warren's paper at http://arxiv.org/abs/0709.0248 and Michael Warren's PhD thesis at http://aix1.uottawa.ca/~mwarren/Papers/ But in the philosophical mode where I avoid evil in category theory, I don't see the justification for identity types in general. --Toby [For admin and other information see: http://www.mta.ca/~cat-dist/ ]