From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8282 Path: news.gmane.org!not-for-mail From: Marco Grandis Newsgroups: gmane.science.mathematics.categories Subject: Re: cleavages and choice Date: Mon, 4 Aug 2014 16:47:39 +0200 Message-ID: References: Reply-To: Marco Grandis NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 (Apple Message framework v1085) Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1407249179 31628 80.91.229.3 (5 Aug 2014 14:32:59 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Tue, 5 Aug 2014 14:32:59 +0000 (UTC) To: Toby Bartels , categories@mta.ca Original-X-From: majordomo@mlist.mta.ca Tue Aug 05 16:32:53 2014 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.1.186]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1XEfn8-0007Lr-Mq for gsmc-categories@m.gmane.org; Tue, 05 Aug 2014 16:32:42 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:50323) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1XEfm7-00036y-Pp; Tue, 05 Aug 2014 11:31:39 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1XEfm6-0006AV-Rw for categories-list@mlist.mta.ca; Tue, 05 Aug 2014 11:31:38 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8282 Archived-At: Dear Toby, The theory of monoidal categories and bicategories is already written. I may be interested in working with them, even though it is unpleasant = to recur to choice for most (non-strict) structures of these kinds. I have no interest in rewriting these theories in a different shape, = using anafunctors or similar tools, or even study such variations if = someone has given them. Of course other people may prefer the way you are saying. I just wanted to point out that there are many occasions where we are = led to use choice - at least if we want to stay within 'classical' = category theory, as expounded - say - in Mac Lane's text. Regards, Marco =20 On 03/ago/2014, at 18.30, Toby Bartels wrote: > Marco Grandis wrote in part: >=20 >> Yet you cannot define a bicategory of spans >> without assuming that such a choice has been made; >> in the same way as you cannot define the (good) monoidal structure of = Ab >> without recurring to a choice of tensor products. >=20 > A monoidal structure on a category C > is a functor C x C -> C, an object of C (aka a functor 1 -> C), > and various natural transformations satisfying some equations. > If by "functor" we mean an anafunctor, then no choice is needed. >=20 > Presumably you are thinking along these lines when you write >=20 >> Unless you want to redefine bicategories >> replacing the composition of arrows with an existence property. >=20 > My point is that anafunctors tell you automatically what to do. >=20 > Better yet, working in HoTT tells you automatically what to do. > All of this only looks complicated from a set-based perspective. >=20 >=20 > --Toby >=20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]