From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6186 Path: news.gmane.org!not-for-mail From: Toby Bartels Newsgroups: gmane.science.mathematics.categories Subject: Re: are fibrations evil? Date: Fri, 17 Sep 2010 00:44:18 -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 1284773356 26640 80.91.229.12 (18 Sep 2010 01:29:16 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 18 Sep 2010 01:29:16 +0000 (UTC) Cc: Thomas Streicher To: categories@mta.ca Original-X-From: majordomo@mlist.mta.ca Sat Sep 18 03:29:14 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 1OwmEl-0000aV-3o for gsmc-categories@m.gmane.org; Sat, 18 Sep 2010 03:29:07 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:51097) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1OwmDf-0007NV-Cz; Fri, 17 Sep 2010 22:27:59 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1OwmDc-00010b-Vo for categories-list@mlist.mta.ca; Fri, 17 Sep 2010 22:27:57 -0300 Content-Disposition: inline In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6186 Archived-At: Thomas Streicher wrote at last: >BTW under a regime which identifies equality with being isomorphic (or weakly >equivalent) it looks tempting to use functors from B^\op to Cat. These should >capture the pseudo-functors since equality and isomorphism are identified. But >writing down functoriality in type theory using \Sigma for existence amounts >to choosing a lot of not at all canonical "canonical isomorphisms". Actually, >one would get something even more general than pseudo-functors because one >wouldn't write down the coherence conditions (actually one couldn't even >since there is no honest for good equality!). Yes, you can write the coherence conditions down (although I agree that it would be easy to forget them). What you need is that a functor (pseudofunctor) P: B^\op -> Cat is not just the following data: * for each object x of B, a category P_x, * for each object x, object y, and morphism f: x -> y, a functor P_f: P_x -> P_y, * functoriality structure (and maybe coherence conditions); but in fact the following data: * for each object x of B, a category P_x, * for each object x, object y, and morphism f: x -> y, a functor P_f: P_y -> P_x, * for each object x, object y, and equal morphisms f = g: x -> y, a natural isomorphism P_{f=g}: P_f => P_g, * functoriality structure and coherence conditions. For example, given f: w -> x, g: x -> y, and h: y -> z in B, we want to compare (P_f . P_g) . P_h with P_f . (P_g . P_h). In a "kosher" treatment of category theory, these aren't equal (that would be meaningless), but there is an associator between them. As a coherence condition, we want to demand that this associator is equal (and this does have meaning) to a natural isomorphism built out of the functoriality structure isomorphisms and their inverses. As we do this, we need to compare P_{(f;g);h} and P_{f;(g;h)}. Again, it's not meaningful (much less true) that these are equal, but P_{(f;g);h = f;(g;h)} is a natural isomorphism between them. So we can write down this coherence condition after all. --Toby [For admin and other information see: http://www.mta.ca/~cat-dist/ ]