Ronald Brown
Bangor University, Mathematics, Emeritus
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The main result is that two possible structures which may be imposed on an edge symmetric double category, namely a connection pair and a thin structure, are equivalent. A full proof is also given of the theorem of Spencer, that the... more
The main result is that two possible structures which may be imposed on an edge symmetric double category, namely a connection pair and a thin structure, are equivalent. A full proof is also given of the theorem of Spencer, that the category of small 2-categories is equivalent to the category of edge symmetric double categories with thin structure.
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Without Abstract
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ABSTRACT
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ABSTRACT
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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... more
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 {\it 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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Research Interests: Applied Mathematics, Mathematical Physics, Quantum Physics, Quantum Gravity, Quantum Chromodynamics, and 14 moreGeneral Relativity, Quantum Field Theory, High Energy Physics, Quantum Algebra, State Space, Conceptual Framework, High Temperature, Nuclear Reactions, Multiple Scattering, Symmetry Breaking, Spin Glass, Fourier transform, Quantum Phase Transition, and High energy
... Phys. Rev. Lett. 2009, 102, 156402, 4 pages. 4 Aguiar, MCO; Dobrosavljevic, V.; Abrahams, E.; Kotliar G. Scaling behavior of an Anderson impurity close to the Mott-Anderson transition. Phys. Rev. B 2006, 73, 115117, 7 pages. ...
In order to give an idea of the scope of applications of topological andLie groupoids, we havo appended a Bibliography on this topic - we are grateful to J. V~RSIK for help in preparing this. A sequel to this paper will deal withG-spaces... more
In order to give an idea of the scope of applications of topological andLie groupoids, we havo appended a Bibliography on this topic - we are grateful to J. V~RSIK for help in preparing this. A sequel to this paper will deal withG-spaces and topological covering mor-phisms.
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We define the notion of whiskered categories and groupoids, showing that whiskered groupoids have a commutator theory. So also do whiskered $R$-categories, thus answering questions of what might be `commutative versions' of these... more
We define the notion of whiskered categories and groupoids, showing that whiskered groupoids have a commutator theory. So also do whiskered $R$-categories, thus answering questions of what might be `commutative versions' of these theories. We relate these ideas to the theory of Leibniz algebras, but the commutator theory here does not satisfy the Leibniz identity. We also discuss potential applications and extensions, for example to resolutions of monoids.
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To give an idea of some of AG’s modes of thought. 2 To note a basic algebraic problem in homotopy theory: how do identifications in low dimensions produce high dimensional homotopy invariants. 3 To see at least one sample of AG’s methods,... more
To give an idea of some of AG’s modes of thought.
2 To note a basic algebraic problem in homotopy theory:
how do identifications in low dimensions produce high
dimensional homotopy invariants.
3 To see at least one sample of AG’s methods, cofibrations of
categories, as relevant to a basic method in homotopy theory:
homotopical excision
4 In memoriam
2 To note a basic algebraic problem in homotopy theory:
how do identifications in low dimensions produce high
dimensional homotopy invariants.
3 To see at least one sample of AG’s methods, cofibrations of
categories, as relevant to a basic method in homotopy theory:
homotopical excision
4 In memoriam
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This essay is based on a talk given by Ronald Brown in 1993 to students and staff of the Departmento de Geometr´ıa e Topolog´ıa at the University of Seville in November, 1993. The issues presented there were part of a continued debate and... more
This essay is based on a talk given by Ronald Brown in 1993 to students and staff of the Departmento
de Geometr´ıa e Topolog´ıa at the University of Seville in November, 1993. The issues presented
there were part of a continued debate and discussion at Bangor over many years, and this explains
why this is a joint paper.
The aim of the talk, and the reason for discussing these topics, was to give students an under-
standing and a sense of pride in the aims and achievements of their subject, and so help them
explain these aims and achievements to their friends and relatives.
de Geometr´ıa e Topolog´ıa at the University of Seville in November, 1993. The issues presented
there were part of a continued debate and discussion at Bangor over many years, and this explains
why this is a joint paper.
The aim of the talk, and the reason for discussing these topics, was to give students an under-
standing and a sense of pride in the aims and achievements of their subject, and so help them
explain these aims and achievements to their friends and relatives.
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Extracted from "Low Dimensional Topology" (Ed R. Brown, T. Thickstun) Proceedings of a conference on Low Dimensional Topology, Bangor 1982 LMS lecture Notes 42, 1982 The aim of the paper is to giver some main lines and intuitions of... more
Extracted from "Low Dimensional Topology" (Ed R. Brown, T. Thickstun)
Proceedings of a conference on Low Dimensional Topology, Bangor 1982
LMS lecture Notes 42, 1982
The aim of the paper is to giver some main lines and intuitions of a series of papers (1976-1981) by R. Brown and Chris Spencer and Philip HIggins on developing Higher Homotopy Seifert-van Kampen Theorems.
Proceedings of a conference on Low Dimensional Topology, Bangor 1982
LMS lecture Notes 42, 1982
The aim of the paper is to giver some main lines and intuitions of a series of papers (1976-1981) by R. Brown and Chris Spencer and Philip HIggins on developing Higher Homotopy Seifert-van Kampen Theorems.
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The aim of this article is to explain a philosophy for applying higher dimensional Seifert-van Kampen Theorems, and how the use of groupoids and strict higher groupoids resolves some foundational anomalies in algebraic topology at the... more
The aim of this article is to explain a philosophy for applying higher dimensional Seifert-van Kampen Theorems, and how the use of groupoids and strict higher groupoids resolves some foundational anomalies in algebraic topology at the border between homology and homotopy. We explain some applications to filtered spaces, and special cases of them, while a sequel will show the relevance to n-cubes of pointed spaces.
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This presentation at CT2015, Aveiro, Portugal, outlines a philosophy of using "broad" and "narrow" homotopically defined algebraic models of topological data, i.e. of certain structured topological spaces, and shows some examples of how... more
This presentation at CT2015, Aveiro, Portugal, outlines a philosophy of using "broad" and "narrow" homotopically defined algebraic models of topological data, i.e. of certain structured topological spaces, and shows some examples of how this leads to some explicit calculations, using nonabelian colimit calculations, e.g. for mapping cones.
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talk for a zoom meeting
in honor of Grothendieck
organised by
John Alexander Cruz Morales and Colin McLarty
this talk: August 28 2020
in honor of Grothendieck
organised by
John Alexander Cruz Morales and Colin McLarty
this talk: August 28 2020
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Slides of a talk in Llandudno (2013)
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Part of the title of this article is taken from writings of Einstein, which argue that we need to exercise our ability to analyse familiar concepts, to demonstrate the conditions on which their justification and usefulness depend, and the... more
Part of the title of this article is taken from writings of Einstein, which
argue that we need to exercise our ability to analyse familiar concepts, to
demonstrate the conditions on which their justification and usefulness depend,
and the way in which these developed, little by little $\ldots$. My aim is to
do this for the initial negative reactions to the seminar by E. Cech on higher
homotopy groups to the ICM meeting in Z\" urich in 1932; then the subsequent
work of Hurewicz, the use of groupoids and so the use of many base points, and
how J.H.C. Whitehead's use of free crossed modules gave rise to a successful
search for higher dimensional versions of the fundamental group and of the
theorem of Van Kampen.
argue that we need to exercise our ability to analyse familiar concepts, to
demonstrate the conditions on which their justification and usefulness depend,
and the way in which these developed, little by little $\ldots$. My aim is to
do this for the initial negative reactions to the seminar by E. Cech on higher
homotopy groups to the ICM meeting in Z\" urich in 1932; then the subsequent
work of Hurewicz, the use of groupoids and so the use of many base points, and
how J.H.C. Whitehead's use of free crossed modules gave rise to a successful
search for higher dimensional versions of the fundamental group and of the
theorem of Van Kampen.
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Part of the title of this article is taken from writings of Einstein. which argues that we need to exercise our ability to analyse familiar concepts, and to demonstrate the conditions on which their justification and usefulness depend,... more
Part of the title of this article is taken from writings of Einstein. which argues that we need to exercise our ability to analyse familiar concepts, and to demonstrate the conditions on which their justification and usefulness depend, and the way in which these developed, little by little. . .. My aim is to do this for the first negative reactions to the seminar by E. ˇ Cech on higher homotopy groups to the ICM meeting in Zürich in 1932; then the subsequent work of Hurewicz, the influence of this on the notion of space in topology, and the search for higher dimensional versions of the fundamental group and of the theorem of Van Kampen.
We generalize the van Kampen theorem for unions of non-connected spaces, due to R. Brown and A. R. Salleh, to the context where families of subspaces of the base space B are replaced with a 'large' space E equipped with a locally... more
We generalize the van Kampen theorem for unions of non-connected spaces, due to R. Brown and A. R. Salleh, to the context where families of subspaces of the base space B are replaced with a 'large' space E equipped with a locally sectionable continuous map p : E → B.