preprints available

["Ronnie" <ronnie.profbrown@btinternet.com>]

The following are available from my preprint page, by me unless stated otherwise: 

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 
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. 

2) 06.04 R. Brown, I. Morris, J. Shrimpton and C.D. Wensley 

Graphs of morphisms of graphs 

 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 We extend Shrimpton's (unpublished) investigations on the morphism digraphs of reflexive digraphs to the undirected case by using an equivalence between a category of reflexive, undirected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 

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. 

4) A new higher homotopy groupoid: the fundamental  globular $\omega$-groupoid  of a filtered space 
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. 

Ronnie