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School of Computer Science, Bangor University

Mathematics Preprints 2006

Category Theory & Homotopy Theory


06.04 BROWN, R., MORRIS, I., SHRIMPTON, J. & WENSLEY, C.D.

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 We extend Shrimpton's (unpublished) investigations on the morphism digraphs of reflexive digraphs to the undirected case by using an equivalence between a category of reflexive, undirected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

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http://www.combinatorics.org/Volume_15/Abstracts/v15i1a1.html

Published in:

Elec. J. Combinatorics, article A1 of volume 15(1), April 3, 2008.


06.09 GRATUS, J. & PORTER, T.

A spatial view of information

Abstract:

Spatial representation has two contrasting but interacting aspects: (i) representation of spaces, and (ii) representation by spaces.

In this paper we will examine two aspects that are common to both interpretations of the theme of spatial representation, namely nerve-type constructions and refinement. We consider the induced structures, in which some of the attributes of the informational context are sampled.

This research forms part of a project: Fractafolds, their geometry and topology, partially supported by a grant from the Leverhulme Trust.
This help is gratefully acknowledged.

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Published in:

Theoretical Computer Science 365 (2006) 206-215.


06.18 : BROWN, R. & SIVERA, R.

Abstract:

Crossed complexes are shown to have an algebra sufficiently rich to model the geometric inductive definition of simplices, and so to give a purely algebraic proof of the Homotopy Addition Lemma (HAL) for the boundary of a simplex. This leads to the fundamental crossed complex of a simplicial set. The main result is a normalisation theorem for this fundamental crossed complex, analogous to the usual theorem for simplicial abelian groups, but more complicated to set up and prove, because of the complications of the HAL and of the notion of homotopies for crossed complexes. We start with some historical background, and give a survey of the required basic facts on crossed complexes.

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Published in:

J. Homotopy Theory and Related Structures, (Saunders Mac Lane special issue) 2-2 (2007) 49-79.

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