From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5612 Path: news.gmane.org!not-for-mail From: Richard Garner Newsgroups: gmane.science.mathematics.categories Subject: Re: Composing modifications Date: Fri, 5 Mar 2010 15:59:41 +0000 (GMT) Message-ID: References: Reply-To: Richard Garner NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; CHARSET=ISO-8859-1; FORMAT=flowed Content-Transfer-Encoding: QUOTED-PRINTABLE X-Trace: dough.gmane.org 1267831021 14238 80.91.229.12 (5 Mar 2010 23:17:01 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Fri, 5 Mar 2010 23:17:01 +0000 (UTC) To: Ronnie Brown Original-X-From: categories@mta.ca Sat Mar 06 00:16:57 2010 connect(): Connection refused 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 4.69) (envelope-from ) id 1NnglL-0004kA-2A for gsmc-categories@m.gmane.org; Sat, 06 Mar 2010 00:16:55 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1NngHU-0006fd-0t for categories-list@mta.ca; Fri, 05 Mar 2010 18:46:04 -0400 In-Reply-To: <4B8F6048.7000903@btinternet.com> Content-ID: Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5612 Archived-At: Dear Ronnie, There are a number of "cubical" notions of bicategory or=20 higher in the literature with which I am sure you are=20 familiar; the earliest being Dominic Verity's double=20 bicategories (which are really a particular kind of triple=20 category), and later the weak (or pseudo) double categories=20 studied in a series of papers by Bob Par=E9 and Marco Grandis.=20 However, none of these quite seem to fit the spirit of what=20 you are asking for, and certainly do not describe the=20 singular cubical complex of a space. One approach which is indicated in the literature is based on=20 Michael Batanin's theory of higher operads. In particular,=20 Tom Leinster, in the refined presentation of this theory=20 given in his book, discusses the possibility of using it to=20 give a notion of weak cubical omega-category, though he does=20 not go into any details. The basic idea is straightforward.=20 On the category of cubical sets (without diagonals or=20 symmetries) one has the monad T for strict cubical=20 omega-categories; and one aims to define the monad for weak=20 cubical omega-categories as a suitable deformation T' of this=20 monad. In the language of homotopy theory, one would like T'=20 to be a cofibrant replacement for T; this then ensures both=20 that T' looks sufficiently like T, and also that its algebras=20 are homotopy-invariant in a suitable sense. To make this more formal, one defines a cubical operad to be=20 a monad S on the category of cubical sets together with a=20 cartesian monad morphism S =3D> T. The existence of such a=20 monad morphism---which I think will be unique if it=20 exists---places some strong restrictions on the shapes of the=20 operations out of which the monad S is built: a typical such=20 operation will take an n_1 x n_2 x ... x n_k grid of=20 k-dimensional hypercubes, and stick them together in some way=20 to yield a single k-dimensional hypercube. This is precisely=20 what one wants for a notion of (weak) cubical category. To ensure that the monad S looks sufficiently like the monad=20 T, one needs the notion of a contraction (=3D acyclic=20 fibration). In fact, we only need to know how to say that the=20 monad morphism S =3D> T has a contraction, and this is quite=20 simple. Suppose we are given a n_1 x ... x n_k grid of=20 k-dimensional hypercubes as before. Suppose we are also given=20 operations s_1, ..., s_2k in S which indicate how to compose=20 up the (k-1)-dimensional grids of hypercubes obtained as the=20 faces of the original grid. Then we require that there should=20 be given an operation of S which acts on n_1 x ... x n_k=20 grids, and whose actions on each of the (k-1)-dimensional=20 faces are given by s_1, ..., s_2k. There now arises a category whose objects are cubical operads=20 S equipped with a contraction on S =3D> T, and whose morphisms=20 are monad maps over T which commute to the contractions in=20 the obvious sense. This category is locally presentable, and=20 hence has an initial object; which we declare to be the monad=20 T' for weak cubical omega categories. One may in fact show=20 that T' is a cofibrant replacement for T in a suitable model=20 structure on cubical operads. One may of course specialise this construction to low=20 dimensions, and I have just sat down to work it out in the=20 case where n=3D2. Rather surprisingly, it seems that there=20 might not be a finite axiomatisation of the resultant=20 structure; at least, one does not immediately leap off the=20 page which is the usual situation. The best I have been able=20 to do is the following (which is not exactly the result of=20 applying the above construction but is a perfectly good=20 surrogate). Definition. A cubical bicategory is given by sets of objects,=20 of vertical arrows, of horizontal arrows and of squares,=20 satisfying the obvious source and target criteria, together=20 with operations of identity and binary composition for=20 vertical and horizontal arrows, satisfying no laws at all;=20 and finally, for every n x m grid of squares (where possibly=20 n or m are zero), and every way of composing up the=20 horizontal and vertical boundaries using the nullary and=20 binary compositions, a composite square with those=20 boundaries. The coherence axioms which this structure must=20 satisfy say that any two ways of composing up a diagram of=20 squares must give the same answer. I would be very interested to know if anyone can extract from=20 this definition a finite collection of composition operations=20 on squares, and a finite collection of equations between=20 them, which together generate all the others. The key=20 obstable seems to be problem that identity 1-cells are not=20 strict in either direction. Richard [For admin and other information see: http://www.mta.ca/~cat-dist/ ]