From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4233 Path: news.gmane.org!not-for-mail From: "Ronnie" Newsgroups: gmane.science.mathematics.categories Subject: preprints available Date: Fri, 29 Feb 2008 12:12:43 -0000 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain;charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1241019811 12334 80.91.229.2 (29 Apr 2009 15:43:31 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:43:31 +0000 (UTC) To: Original-X-From: rrosebru@mta.ca Fri Feb 29 09:40:35 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 29 Feb 2008 09:40:35 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JV5MN-0006a9-UV for categories-list@mta.ca; Fri, 29 Feb 2008 09:33:11 -0400 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 55 Original-Lines: 64 Xref: news.gmane.org gmane.science.mathematics.categories:4233 Archived-At: The following are available from my preprint page, by me unless stated = otherwise:=20 http://www.bangor.ac.uk/~mas010/brownpr.html 1) 08.04 Exact sequences of fibrations of crossed complexes, homotopy = classification of maps, and nonabelian extensions of groups=20 ABSTRACT: The classifying space of a crossed complex generalises the = construction of Eilenberg-Mac Lane spaces. We show how the theory of = fibrations of crossed complexes allows the analysis of homotopy classes = of maps from a free crossed complex to such a classifying space. This = gives results on the homotopy classification of maps from a CW-complex = to the classifying space of a crossed module and also, more generally, = of a crossed complex whose homotopy groups vanish in dimensions between = 1 and n. The results are analogous to those for the obstruction to an = abstract kernel in group extension theory.=20 2) 06.04 R. Brown, I. Morris, J. Shrimpton and C.D. Wensley=20 Graphs of morphisms of graphs=20 ABSTRACT: This is an account for the combinatorially minded reader of = various categories of directed and undirected graphs, and their = analogies with the category of sets. As an application, the = endomorphisms of a graph are in this context not only composable, giving = a monoid structure, but also have a notion of adjacency, so that the set = of endomorphisms is both a monoid and a graph. We extend Shrimpton's = (unpublished) investigations on the morphism digraphs of reflexive = digraphs to the undirected case by using an equivalence between a = category of reflexive, undirected graphs and the category of reflexive, = directed graphs with reversal. In so doing, we emphasise a picture of = the elements of an undirected graph, as involving two types of edges = with a single vertex, namely `bands' and `loops'. Such edges are = distinguished by the behaviour of morphisms with respect to these = elements. 3) Possible connections between whiskered categories and groupoids, many = object Lie algebras, automorphism structures and local-to-global = questions=20 ABSTRACT: We define the notion of whiskered categories and groupoids and = discuss potential applications and extensions, for example to a many = object Lie theory, and to resolutions of monoids. This paper is more an = outline of a possible programme or programmes than giving conclusive = results.=20 4) A new higher homotopy groupoid: the fundamental globular = $\omega$-groupoid of a filtered space=20 MSC Classification:18D10, 18G30, 18G50, 20L05, 55N10, 55N25. KEY WORDS: filtered space, higher homotopy van Kampen theorem, cubical = singular complex, free globular groupoid ABSTRACT: We show that the graded set of filter homotopy classes rel = vertices of maps from the $n$-globe to a filtered space may be given the = structure of globular $\omega$--groupoid. The proofs use an analogous = fundamental cubical $\omega$--groupoid due to the author and Philip = Higgins. This method also relates the construction to the fundamental = crossed complex of a filtered space, and this relation allows the proof = that the crossed complex associated to the free globular = $\omega$-groupoid on one element of dimension $n$ is the fundamental = crossed complex of the $n$-globe.=20 Ronnie=20