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Ronald Brown

Publications and source records attributed to Ronald Brown.

At least 19 recordsLinked to original sources
arXiv2021

Not just an idle game" (the story of higher dimensional versions of the Poincar{é} fundamental group)

The title of this article is partially taken from writings of A. Einstein. In the 1932 ICM at Zürich, when E. \vCech gave a seminar on higher homotopy groups of a pointed space and proved they were abelian for n > 1. On these grounds, H. Hopf and P.S. Aleksandrov persuaded Ĉech to withdraw his paper, so that only a small paragraph appeared in the Proceedings. This article reviews the eventual construction by the author and P.J. Higgins, of reasonably nonabelian higher dimensional versions of the fundamental group, using groupoids and many base points, and stimulated by long term work of J.H.C. Whitehead on crossed modules.

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arXiv2021

"Not just an idle game":(examining some historical conceptual arguments in homotopy theory)

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.

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arXiv2017

Modelling and Computing Homotopy Types: I

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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arXiv2011

Covering morphisms of crossed complexes and of cubical omega-groupoids with connection are closed under tensor product

The aim is the theorems of the title and the corollary that the tensor product of two free crossed resolutions of groups or groupoids is also a free crossed resolution of the product group or groupoid. The route to this corollary is through the equivalence of the category of crossed complexes with that of cubical omega-groupoids with connections where the initial definition of the tensor product lies. It is also in the latter category that we are able to apply techniques of dense subcategories to identify the tensor product of covering morphisms as a covering morphism.

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arXiv2010

Possible connections between whiskered categories and groupoids, many object Leibniz algebras, automorphism structures and local-to-global questions

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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arXiv2010

Crossed modules and the homotopy 2-type of a free loop space

The question was asked by Niranjan Ramachandran: how to describe the fundamental groupoid of LX, the free loop space of a space X? We give an answer by assuming X to be the classifying space of a crossed module over a group, and then describe completely a crossed module over a groupoid determining the homotopy 2-type of LX. The method requires detailed information on the monoidal closed structure on the category of crossed complexes.

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arXiv2009

Moore hyperrectangles on a space form a strict cubical omega-category

A question of Jack Morava is answered by generalising the notion of Moore paths to that of Moore hyperrectangles, so obtaining a strict cubical omega-category. This also has the structure of connections in the sense of Brown and Higgins, but cancellation of connections does not hold.

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arXiv2009

Algebraic Topology Foundations of Supersymmetry and Symmetry Breaking in Quantum Field Theory and Quantum Gravity: A Review

A novel algebraic topology approach to supersymmetry (SUSY) and symmetry breaking in quantum field and quantum gravity theories is presented with a view to developing a wide range of physical applications. These include: controlled nuclear fusion and other nuclear reaction studies in quantum chromodynamics, nonlinear physics at high energy densities, dynamic Jahn-Teller effects, superfluidity, high temperature superconductors, multiple scattering by molecular systems, molecular or atomic paracrystal structures, nanomaterials, ferromagnetism in glassy materials, spin glasses, quantum phase transitions and supergravity. This approach requires a unified conceptual framework that utilizes extended symmetries and quantum groupoid, algebroid and functorial representations of non-Abelian higher dimensional structures pertinent to quantized spacetime topology and state space geometry of quantum operator algebras. Fourier transforms, generalized Fourier-Stieltjes transforms, and duality relations link, respectively, the quantum groups and quantum groupoids with their dual algebraic structures; quantum double constructions are also discussed in this context in relation to quasi-triangular, quasi-Hopf algebras, bialgebroids, Grassmann-Hopf algebras and higher dimensional algebra. On the one hand, this quantum algebraic approach is known to provide solutions to the quantum Yang-Baxter equation. On the other hand, our novel approach to extended quantum symmetries and their associated representations is shown to be relevant to locally covariant general relativity theories that are consistent with either nonlocal quantum field theories or local bosonic (spin) models with the extended quantum symmetry of entangled, 'string-net condensed' (ground) states.

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arXiv2009

`Double modules', double categories and groupoids, and a new homotopical double groupoid

We give a rather general construction of double categories and so, under further conditions, double groupoids, from a structure we call a `double module'. We also give a homotopical construction of a double groupoid from a triad consisting of a space, two subspaces, and a set of base points, under a condition which also implies that this double groupoid contains two second relative homotopy groups.

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arXiv2008

Algebraic colimit calculations in homotopy theory using fibred and cofibred categories

Higher Homotopy van Kampen Theorems allow the computation as colimits of certain homotopical invariants of glued spaces. One corollary is to describe homotopical excision in critical dimensions in terms of induced modules and crossed modules over groupoids. This paper shows how fibred and cofibred categories give an overall context for discussing and computing such constructions, allowing one result to cover many cases. A useful general result is that the inclusion of a fibre of a fibred category preserves connected colimits. The main homotopical application are to pairs of spaces with several base points, but we also describe briefly the situation for triads.

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arXiv2008

Exact sequences of fibrations of crossed complexes, homotopy classification of maps, and nonabelian extensions of groups

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.

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arXiv2007

A new higher homotopy groupoid: the fundamental globular omega-groupoid of a filtered space

We use the n-globe with its skeletal filtration to define the fundamental globular omega--groupoid of a filtered space; 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.

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arXiv2007

Normalisation for the fundamental crossed complex of a simplicial set

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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