From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2390 Path: news.gmane.org!not-for-mail From: "John Baez" Newsgroups: gmane.science.mathematics.categories Subject: Higher-Dimensional Algebra V: 2-Groups Date: Tue, 15 Jul 2003 05:27:13 -0700 (PDT) Message-ID: <200307151227.h6FCRDn26350@math-cl-n01.ucr.edu> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241018627 3915 80.91.229.2 (29 Apr 2009 15:23:47 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:23:47 +0000 (UTC) To: categories@mta.ca (categories) Original-X-From: rrosebru@mta.ca Tue Jul 15 11:33:45 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 15 Jul 2003 11:33:45 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 19cQnn-0001l4-00 for categories-list@mta.ca; Tue, 15 Jul 2003 11:29:11 -0300 X-Mailer: ELM [version 2.5 PL6] Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 18 Original-Lines: 43 Xref: news.gmane.org gmane.science.mathematics.categories:2390 Archived-At: Here's a new paper that studies categorified groups and Lie groups: Higher-Dimensional Algebra V: 2-Groups John C. Baez and Aaron D. Lauda Abstract: A 2-group is a "categorified" version of a group, in which the underlying set G has been replaced by a category and the multiplication map m: G x G -> G has been replaced by a functor. Various versions of this notion have already been explored; our goal here is to provide a detailed introduction to two, which we call "weak" and "coherent" 2-groups. A weak 2-group is a weak monoidal category in which every morphism has an inverse and every object x has a "weak inverse": an object y such that x tensor y and y tensor x are isomorphic to 1. A coherent 2-group is a weak 2-group in which every object x is equipped with a specified weak inverse x* and isomorphisms i_x: 1 -> x tensor x*, e_x: x* tensor x -> 1 forming an adjunction. We define 2-categories of weak and coherent 2-groups, construct an "improvement" 2-functor which turns weak 2-groups into coherent ones, and prove this 2-functor is a 2-equivalence of 2-categories. We also internalize the concept of coherent 2-group, which gives a way to define topological 2-groups, Lie 2-groups, affine 2-group schemes, and the like. We conclude with a tour of examples. Diagrammatic methods are emphasized throughout - especially string diagrams. This paper will soon appear on the mathematics arXiv, but their computer seems unable to draw some of the pictures correctly, so I urge you to try this PDF version instead: http://math.ucr.edu/home/baez/hda5.pdf The next paper in this series will study categorified Lie algebras.