From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5462 Path: news.gmane.org!not-for-mail From: John Baez Newsgroups: gmane.science.mathematics.categories Subject: re: the definition of "evil" Date: Sun, 3 Jan 2010 09:53:39 -0800 Message-ID: References: Reply-To: John Baez NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8 X-Trace: ger.gmane.org 1262574591 23609 80.91.229.12 (4 Jan 2010 03:09:51 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Mon, 4 Jan 2010 03:09:51 +0000 (UTC) To: categories Original-X-From: categories@mta.ca Mon Jan 04 04:09:44 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.50) id 1NRdKB-0006tU-BY for gsmc-categories@m.gmane.org; Mon, 04 Jan 2010 04:09:43 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1NRcsx-0002QJ-JW for categories-list@mta.ca; Sun, 03 Jan 2010 22:41:35 -0400 In-Reply-To: Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5462 Archived-At: Dear Categorists - I'm glad Peter is trying to formulate a definition of structures that can be transported along equivalences, and I like the spirit of his definition, namely in terms of a "lifting property" where one has a 2-functor U: XCat -> Cat and one is trying to lift equivalences from Cat to XCat. But it makes me nervous when he says "isomorphic [not equivalent!]". Just as evil in category theory typically arises from definitions that impose equations between objects instead of specifying isomorphisms, evil in 2-category theory typically arises when we specify isomorphisms between objects instead of specifying equivalences. It would be sad, or at least intriguing, if the definition of "evil" was itself evil. Best, jb DEFINITION. Let X be some structure on categories. By this, I mean > that there is a given 2-category called X-Cat, whose objects are > called X-categories, whose morphisms are called X-functors, and whose > 2-cells are called X-transformations, and for which there is a given > 2-functor U to Cat, called the forgetful functor. > > We say that X is "transported along equivalences of categories" if the > following holds. Given an X-category D', with underlying category D = > U(D'), and a category C, and an equivalence (F,G,e,h) of categories D > and C, where F: D -> C, G: C -> D, and e: FG -> id_C, h: GF -> id_D, > it is then possible to find: > > (1) an X-category C' whose underlying category U(C') is isomorphic > [not equivalent!] to C. Let c : U(C') -> C be the isomorphism > (i.e., an invertible functor in Cat) with inverse c': C -> U(C'); > > (2) an X-equivalence of X-categories (F',G',e',h'), where > F': D' -> C', G': C' -> D', e': F'G' -> id_C', and h': G'F' -> id_D' > [the concept of equivalence makes sense in any 2-category]; > > such that > > (3) (U(F'), U(G'), U(e'), U(h')) = (cF, Gc', cec', h). > > Here, cF and Gc' denotes composition of functors, and cec' denotes > whiskering. > > The structure X is called "evil" iff it is not transported along > equivalences of categories. > > This finishes the definition. > > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]