From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6219 Path: news.gmane.org!not-for-mail From: Toby Bartels Newsgroups: gmane.science.mathematics.categories Subject: Re: Freyd Categories Date: Sat, 25 Sep 2010 18:48:03 -0300 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 1285451368 14815 80.91.229.12 (25 Sep 2010 21:49:28 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 25 Sep 2010 21:49:28 +0000 (UTC) Cc: David Leduc To: categories Original-X-From: majordomo@mlist.mta.ca Sat Sep 25 23:49:23 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.138]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1OzccU-00017g-WB for gsmc-categories@m.gmane.org; Sat, 25 Sep 2010 23:49:23 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:48420) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1OzcbI-0004qW-UD; Sat, 25 Sep 2010 18:48:08 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1OzcbD-0000BQ-Ru for categories-list@mlist.mta.ca; Sat, 25 Sep 2010 18:48:04 -0300 Content-Disposition: inline In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6219 Archived-At: David Leduc wrote: >A Freyd category is essentially a functor. >So why is it called a category?! As I understand it, one says a Freyd category on C, where C is the source/domain of the functor. So a Freyd category on C is a category K (with certain structure), together with a functor C -> K (with certain properties). More simply, a Freyd category on C is a category K together with certain extra stuff. Calling that whole business (a category together with ...) a "category" is an abuse of language akin to terms like "partially ordered set"; a partially ordered set is really a set together with certain structure. There may be a better reason why Power and Thielecke used this terminology, but if so, they don't explain it in the papers that I have found (but I have not been able to read their first paper on the subject, Environments, Continuation Semantics and Indexed Categories, so maybe there is an explanation there). >More generally, is there a way to see a functor as being a category? More generally, a functor is a pair of categories equipped with extra stuff, which is trivial, but I don't think that there's anything deeper than that. You could do something like the trick that encodes a function as a set (as everything must be) in the foundation of material set theory: a function f: X -> Y is the set {(a,b) in X x Y | f(a) = b}. But this set is isomorphic, in the cateory of sets, to X itself, so really we need this set together with maps from it to X and Y. Similarly, think of a functor F: X -> Y as the category with {(a,b) in Ob X x Ob Y | F(a) = b} as set of objects and Hom((a,b), (c,d)) := {(f,g) in X(a,c) x Y(b,d) | F(f) = g}, with the obvious composition; but again, this is isomorphic to X, so we really need it together with functors from it to X and Y. So I don't think that this really helps anything. --Toby [For admin and other information see: http://www.mta.ca/~cat-dist/ ]