From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5467 Path: news.gmane.org!not-for-mail From: Claudio Hermida Newsgroups: gmane.science.mathematics.categories Subject: re: the definition of "evil" Date: Sun, 3 Jan 2010 12:10:29 -0500 Message-ID: References: Reply-To: Claudio Hermida NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1262574735 23898 80.91.229.12 (4 Jan 2010 03:12:15 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Mon, 4 Jan 2010 03:12:15 +0000 (UTC) To: categories@mta.ca Original-X-From: categories@mta.ca Mon Jan 04 04:12:08 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 1NRdMW-0007Sy-Gy for gsmc-categories@m.gmane.org; Mon, 04 Jan 2010 04:12:08 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1NRcs4-0002Nw-Uw for categories-list@mta.ca; Sun, 03 Jan 2010 22:40:41 -0400 In-Reply-To: Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5467 Archived-At: Peter Selinger wrote: 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 =3D 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')) =3D (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. More informally, "transported along equivalences" therefore means that if D and C are equivalent, and D has an X-structure, then there is a way to equip C with an X-structure and to lift the original equivalence to an X-equivalence. There was a need for the isomorphism c in the definition, because the forgetful functor U : X-Cat -> Cat may not be strictly speaking surjective onto 0-cells in some real-life examples (and in any case, this forgetful functor may sometimes only be well-defined up to isomorphism). It is important that c is an isomorphism, rather than an equivalence, because else the definition becomes vacuous (and we are precisely interested in notions that are not well-defined up to equivalence). Also note that I didn't require the data (C',F',G',e',h') to be unique, not even up to equivalence in X-Cat. Although in practice, it will often be unique in this sense. So my definition allows for a given structure to be transported "in essentially more than one way" along a given equivalence. I am open to strengthening the definition to forbid this. Dear all, A quick pointer to a variation on this matter, which has already been aroun= d (I presented it at CT07). The nature of 2-categorical structures that transport along equivalences is nicely captured by the notion of Equ-iso-fibration: DEFINITION A 2-functor F:E =E2=86=92 B is an Equ-iso-fibration if: - it admits cartesian liftings of (adjoint) equivalences and - every hom F(X,Y):E(X,Y) =E2=86=92 B(FX,FY) admits cartesian lifitings of isomorphism 2-cells, and such liftings are preserved by precomposition (pointwise). Notice that a cartesian lifting of an adjoint equivalence is another such. The notions of cartesian liftings used here are those for 2-fibrations (as in the references below); they involve the expected 2-dimensional property for 1-cells. The liftings are strict. The most interesting example of such a structure is the forgetful functor o= f a 2-cateogory of pseudo-algebras V: Ps-T-Alg =E2=86=92 K, for a 2-monad T on a 2-category K (or a pseudo-mon= ad on a bicategory). They include all the known interesting examples of pseudo-structures, the classical such being monoidal categories. The morphisms in Ps-T-Alg are strong ones, preserving structure up to coherent isomorphism. The observation that pseudo-algebras do transport along equivalences is credited to Max Kelly in Power=C2=B4s article below. I used this notion to capture the following universality of coherence: coherence for such pseudo-algebras (meaning that we can strictify every pseudo-algebra into a equivalent strict T-algebra) is equivalent to the statement THM: The inclusion 2-functor J: T-alg =E2=86=92 Ps-T-alg makes V: Ps-T-alg = =E2=86=92 K the FREE Equ-iso-fibration over U:T-alg =E2=86=92 K In other words, the passage from strict algebras (with their strict morphisms) to pseudo-algebras (and strong morphisms) is the universal way o= f achieving transportability along equivalences. Claudio References - C.Hermida, Some properties of Fib as a fibred 2-category, JPAA, *134 (1), 83-109, 1999* - C.Hermida, Descent on 2-fibrations and 2-regular 2-categories, Applied Ca= t Str, *12(5-6), 427--459, 2004* - A.J.Power, A general coherence result, [For admin and other information see: http://www.mta.ca/~cat-dist/ ]