From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6988 Path: news.gmane.org!not-for-mail From: Steve Lack Newsgroups: gmane.science.mathematics.categories Subject: Re: The tricategory of bicategories Date: Mon, 24 Oct 2011 09:59:04 +1100 Message-ID: References: Reply-To: Steve Lack NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 (Apple Message framework v1084) Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1319459388 7182 80.91.229.12 (24 Oct 2011 12:29:48 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Mon, 24 Oct 2011 12:29:48 +0000 (UTC) Cc: Categories list To: Jamie Vicary Original-X-From: majordomo@mlist.mta.ca Mon Oct 24 14:29:43 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.4]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1RIJex-0006N0-8T for gsmc-categories@m.gmane.org; Mon, 24 Oct 2011 14:29:43 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:60541) by smtpx.mta.ca with esmtp (Exim 4.76) (envelope-from ) id 1RIJdJ-0001WW-Rd; Mon, 24 Oct 2011 09:28:01 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1RIJdI-00027x-1i for categories-list@mlist.mta.ca; Mon, 24 Oct 2011 09:28:00 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6988 Archived-At: Dear Jamie, I agree that there is no canonical choice of horizontal composition of pseudonatural transformations, but that the various possible choices are, in a suitable sense, equivalent.=20 Whether or not you should be bothered by this I can't really say. But=20 perhaps it's worth pointing out that there are various different = possible descriptions of the structure of weak 3-category, and not all of them = include a chosen horizontal composition of pseudonatural transformations. Some, particularly, the simplicial approaches, include *no* choices of = compositions. Others include some choices of composition, but not this particular one.=20= For example, the notion of Gray-category does include chosen composition of 1-cells, and vertical composition of 2-cells, but does not include a = chosen horizontal composition of 2-cells.=20 Best wishes, Steve Lack. =20 On 21/10/2011, at 9:17 PM, Jamie Vicary wrote: > Dear categorists, >=20 > Suppose you have categories A, B, C, and functors S,S': A-->B, T,T': > B-->C, and natural transformations alpha: S=3D=3D>S', beta: T=3D=3D>T'. > Suppose we want to see these as part of a 2-category of categories; > then we had better know the horizontal composite of alpha and beta. > There are two possible ways to evaluate this composite: as the natural > transformation having components beta_{S'X}.T(alpha_X), and as the > natural transformation having components T'(alpha_X).beta_{S(X)}. But > these are equal, since beta is a natural transformation. So we have no > difficulty uniquely defining our horizontal composite, and obtaining a > canonical 2-category of categories. >=20 > But now suppose that A, B, C are bicategories, S,S',T,T' are > pseudofunctors, and alpha and beta are pseudonatural transformations. > Then the two possible definitions for the horizontal composite of > alpha and beta will not necessarily be equal, although of course they > will be related by an invertible modification. But then we have a > problem forming the tricategory of bicategories, pseudofunctors, > pseudonatural transformations and modifications: there is no longer a > canonical choice available for horizontal composition of pseudonatural > transformations. >=20 > Presumably this choice can be made, and a tricategory is the result, > and different choices yield equivalent tricategories. But it bothers > me that there seems to be no canonical tricategory of bicategories. > Should it? Or is my reasoning flawed? >=20 > Jamie. >=20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]