Date: Sun, 18 Aug 1996 11:32:15 -0300 (ADT) Subject: Weak n-categories Date: Thu, 15 Aug 1996 04:39:47 -0700 (PDT) From: john baez It occurred to me that some people on this list might be interested in seeing the definition of weak n-categories proposed by James Dolan and myself. We are very slowly writing a paper on this, which will appear as part of the series "Higher-dimensional algebra" in Adv. Math.. (The first paper in this series is on braided monoidal 2-categories, and the second will be on 2-Hilbert spaces.) However, a sketch of the definition has been available on the web for some time; it's at http://math.ucr.edu/home/baez/ncat.def.html Also, a bunch of expository material on mathematical physics, category theory and so on can be found at http://math.ucr.edu/home/baez/README.html Sincerely, John Baez Date: Wed, 5 Feb 1997 20:33:26 -0400 (AST) Subject: weak n-categories Date: Wed, 5 Feb 1997 15:10:25 -0800 (PST) From: john baez Here is the abstract of a paper that is now available at my website. If printing it out is a problem (see below) I can mail copies to people. - John Baez ---------------------------------------------------------------------- Higher-Dimensional Algebra III: n-Categories and the Algebra of Opetopes John C. Baez and James Dolan We give a definition of weak n-categories based on the theory of operads. We work with operads having an arbitrary set S of types, or `S-operads', and given such an operad O, we denote its set of operations by elt(O). Then for any S-operad O there is an elt(O)-operad O+ whose algebras are S-operads over O. Letting I be the initial operad with a one-element set of types, and defining I(0) = I, I(i+1) = I(i)+, we call the operations of I(n-1) the `n-dimensional opetopes'. Opetopes form a category, and presheaves on this category are called `opetopic sets'. A weak n-category is defined as an opetopic set with certain properties, in a manner reminiscent of Street's simplicial approach to weak omega-categories. In a similar manner, starting from an arbitrary operad O instead of I, we define `n-coherent O-algebras', which are n times categorified analogs of algebras of O. Examples include `monoidal n-categories', `stable n-categories', `virtual n-functors' and `representable n-prestacks'. We also describe how n-coherent O-algebra objects may be defined in any (n+1)-coherent O-algebra. ----------------------------------------------------------------------- The paper is available in Postscript form on the web at http://math.ucr.edu/home/baez/op.ps The paper is 59 pages long, so this file is rather large. A compressed version is available at http://math.ucr.edu/home/baez/op.ps.Z You can download this and then (on most UNIX systems) type uncompress op.ps.Z to get the Postscript file. If you like ftp, you can also get these by anonymous ftp to math.ucr.edu They are in the directory pub/baez as the files op.ps and op.ps.Z Date: Mon, 13 Oct 1997 09:17:41 -0300 (ADT) Subject: Operads, multicategories Date: Mon, 13 Oct 1997 10:36:43 +0100 (BST) From: Tom Leinster An advertisement for an article, available by electric transmission from http://www.dpmms.cam.ac.uk/~leinster. ABSTRACT Notions of `operad' and `multicategory' abound. This work provides a single framework in which many of these various notions can be expressed. Explicitly: given a monad * on a category S, we define the term (S,*)-multicategory, subject to certain conditions on S and *. Different choices of S and * give some of the existing notions. We then describe the algebras for an (S,*)-multicategory, and finish with a selection of possible further developments. Our approach enable concise descriptions of Baez and Dolan's opetopes and Batanin's operads; both of these are included. Tom Leinster Date: Thu, 16 Oct 1997 16:53:03 -0300 (ADT) Subject: Weak higher dimensional categories Date: Thu, 16 Oct 1997 10:13:07 +0100 From: ajp@dcs.ed.ac.uk Those people interested in Tom Leinster's paper on multicategories and weak higher dimensional categories might also be interested in recent, closely related work by Claudio Hermida at McGill. I do not think there is a paper available yet, but he has given talks at Vancouver and at the recent meeting of the Canadian Math Society in Montreal, so there are probably slides available.