From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8278 Path: news.gmane.org!not-for-mail From: Toby Bartels Newsgroups: gmane.science.mathematics.categories Subject: Re: cleavages and choice Date: Sun, 3 Aug 2014 09:30:13 -0700 Message-ID: References: Reply-To: Toby Bartels NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii X-Trace: ger.gmane.org 1407156588 32351 80.91.229.3 (4 Aug 2014 12:49:48 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Mon, 4 Aug 2014 12:49:48 +0000 (UTC) To: categories@mta.ca Original-X-From: majordomo@mlist.mta.ca Mon Aug 04 14:49:42 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 1XEHht-0006PU-8z for gsmc-categories@m.gmane.org; Mon, 04 Aug 2014 14:49:41 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:50075) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1XEHhN-0007bl-O8; Mon, 04 Aug 2014 09:49:09 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1XEHhM-0002qJ-IC for categories-list@mlist.mta.ca; Mon, 04 Aug 2014 09:49:08 -0300 Content-Disposition: inline In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8278 Archived-At: Marco Grandis wrote in part: >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. 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. Presumably you are thinking along these lines when you write >Unless you want to redefine bicategories >replacing the composition of arrows with an existence property. My point is that anafunctors tell you automatically what to do. Better yet, working in HoTT tells you automatically what to do. All of this only looks complicated from a set-based perspective. --Toby [For admin and other information see: http://www.mta.ca/~cat-dist/ ]