Free Crossed Resolutions of Groups and Presentations of Modules of Identities among Relations

Brown, Ronald; Salleh, Abdul Razak

doi:10.1112/s1461157000000061

Cited by 10 publications

(12 citation statements)

References 22 publications

(49 reference statements)

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“…In the case of the Hurewicz theorem, this was generalized to arbitrary topological spaces (Spanier 1966), and establishes that certain homology groups are isomorphic to 'corresponding' homotopy groups of an arbitrary topological space. Brown et al (1999 went further and generalized the van Kampen theorem, at first to fundamental groupoids on a set of base points (Brown 1967), and then, to higher dimensional algebras involving, for example, homotopy double groupoids and 2-categories . The more sensitive algebraic invariant of topological spaces seems to be, however, captured only by cohomology theory through an algebraic ring structure that is not accessible either in homology theory, or in the existing homotopy theory.…”

Section: Local-to-global (Lg) Construction Principles Consistent Withmentioning

confidence: 98%

“…The examples of topological data for which these schemes are known to work are: In fact crossed complexes are equivalent to a bewildering array of other structures, which are important for applications (Brown 1999). Cat n -groups are also equivalent to crossed n-cubes of groups.…”

Section: Wider Considerationsmentioning

confidence: 98%

“…Free crossed resolutions enable calculations with small CW-models of K(G,1)s and their maps (Brown and Razak Salleh 1999).…”

Section: Wider Considerationsmentioning

confidence: 99%

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A Conceptual Construction of Complexity Levels Theory in Spacetime Categorical Ontology: Non-Abelian Algebraic Topology, Many-Valued Logics and Dynamic Systems

2007

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A novel conceptual framework is introduced for the Complexity Levels Theory in a Categorical Ontology of Space and Time. This conceptual and formal construction is intended for ontological studies of Emergent Biosystems, Supercomplex Dynamics, Evolution and Human Consciousness. A claim is defended concerning the universal representation of an item's essence in categorical terms. As an essential example, relational structures of living organisms are well represented by applying the important categorical concept of natural transformations to biomolecular reactions and relational structures that emerge from the latter in living systems. Thus, several relational theories of living systems can be represented by natural transformations of organismic, relational structures. The ascent of man and other living organisms through adaptation, is viewed in novel categorical terms, such as variable biogroupoid representations of evolving species. Such precise but flexible evolutionary concepts will allow the further development of the unifying theme of local-to-global approaches to highly complex systems in order to represent novel patterns of relations that emerge in super-and ultra-complex systems in terms of compositions of local procedures. Solutions to such local-to-global problems in highly complex systems with 'broken symmetry' might be possible to be reached with the help of higher homotopy theorems in algebraic topology such as the generalized van Kampen theorems (HHvKT). Categories of many-valued, Łukas-iewicz-Moisil (LM) logic algebras provide useful concepts for representing the intrinsic dynamic 'asymmetry' of genetic networks in organismic development and evolution, as well as to derive novel results for (non-commutative) Quantum Logics. Furthermore, as recently pointed out by Baianu and Poli (Theory and applications of ontology, vol 1. Springer, Berlin, in press), LM-logic algebras may also provide the appropriate framework for future developments of the ontological theory of levels with its complex/entangled/intertwined ramifications in psychology, sociology and ecology. As shown in the preceding two papers in this issue, a paradigm shift towards non-commutative, or non-Abelian, theories of highly complex dynamics-which is presently unfolding in physics, mathematics, life and cognitive sciences-may be implemented through realizations of higher dimensional algebras in neurosciences and psychology, as well as in human genomics, bioinformatics and interactomics.Keywords Categorical ontology and the theory of levels Á Formal foundation and relational structure of categorical ontology and emergent complexity theories Á Ontological essence and Universal properties of items Á Mathematical categories, Groupoids, Locally Lie groupoids, Groupoid Atlas, Stacks, Fibred categories Á Relational biology principles Á Higher homotopy-General Van Kampen Theorems (HHvKT) and Non-Abelian algebraic topology (NAAT) Á Non-commutativity of diagrams and non-Abelian theories-Non-Abelian categorical ontology Á Non-commutative topological...

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Section: Local-to-global (Lg) Construction Principles Consistent Withmentioning

confidence: 98%

Section: Wider Considerationsmentioning

confidence: 98%

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A Conceptual Construction of Complexity Levels Theory in Spacetime Categorical Ontology: Non-Abelian Algebraic Topology, Many-Valued Logics and Dynamic Systems

2007

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“…Free crossed resolutions enable calculations with small CW-models of K(G,1)s and their maps (Brown and Razak Salleh 1999). • Also, they have an interesting relation with the Moore complex of simplicial groups and of simplicial groupoids.…”

Section: Crossed Complexesmentioning

confidence: 99%

Categorical Ontology of Complex Spacetime Structures: The Emergence of Life and Human Consciousness

2007

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A categorical ontology of space and time is presented for emergent biosystems, super-complex dynamics, evolution and human consciousness. Relational structures of organisms and the human mind are naturally represented in nonabelian categories and higher dimensional algebra. The ascent of man and other organisms through adaptation, evolution and social co-evolution is viewed in categorical terms as variable biogroupoid representations of evolving species. The unifying theme of local-to-global approaches to organismic development, evolution and human consciousness leads to novel patterns of relations that emerge in superand ultra-complex systems in terms of colimits of biogroupoids, and more generally, as compositions of local procedures to be defined in terms of locally Lie groupoids. Solutions to such local-to-global problems in highly complex systems with 'broken symmetry' may be found with the help of generalized van Kampen theorems in algebraic topology such as the Higher Homotopy van Kampen theorem (HHvKT). Primordial organism structures are predicted from the simplest metabolic-repair systems extended to self-replication through autocatalytic reactions. The intrinsic dynamic 'asymmetry' of genetic networks in organismic development and evolution is investigated in terms of categories of many-valued, Łukasiewicz-Moisil logic algebras and then compared with those obtained for (non-commutative) quantum logics. The claim is defended in this essay that human consciousness is unique and should be viewed as an ultra-complex, global process of processes. The emergence of consciousness and its existence seem dependent upon an extremely complex structural and functional unit with an asymmetric network topology and connectivitiesthe human brain-that developed through societal co-evolution, elaborate language/ symbolic communication and 'virtual', higher dimensional, non-commutative processes involving separate space and time perceptions. Philosophical theories of the mind are approached from the theory of levels and ultra-complexity viewpoints which throw new light on previous representational hypotheses and proposed semantic models in cognitive science. Anticipatory systems and complex causality at the top levels of reality are also discussed in the context of the ontological theory of levels with its complex/entangled/intertwined ramifications in psychology, sociology and ecology. The presence of strange attractors in modern society dynamics gives rise to very serious concerns for the future of mankind and the continued persistence of a multi-stable biosphere. A paradigm shift towards non-commutative, or nonAbelian, theories of highly complex dynamics is suggested to unfold now in physics, mathematics, life and cognitive sciences, thus leading to the realizations of higher dimensional algebras in neurosciences and psychology, as well as in human genomics, bioinformatics and interactomics.Keywords Space, time, chronotopoids and spacetime, ST Á ST in automata vs. quantum automata and organisms Á Categorical ontology an...

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“…(ii) Several authors have developed sophisticated and powerful methods for computing identities among relators (and for tackling the related problem of computing free ZGresolutions). See for example the survey papers [4] and [7], and more recent papers such as [1], [10], [22] and [25]. Our second motivation was a desire to compare these methods with the relatively naive computer techniques described below.…”

Section: Three-dimensional Presentationsmentioning

confidence: 99%

Three-Dimensional Presentations for the Groups of Order at Most 30

Ellis¹,

Kholodna

1999

LMS J. Comput. Math.

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For each group G of order up to 30 we compute a small 3-dimensional CW-space X with π1X≌ G and π2X = 0, and we quantify the ‘efficiency’ of X. Furthermore, we give a theoretical result for treating the case when G is a semi-direct product of two groups for which 3-presentations are known. We also describe the ZG-module structure on the second homotopy group π2X2 of the 2-skeleton of X. This module structure can in principle be used to determine the co-homology groups H2(G, A) and H3(G, A) with coefficients in a ZG-module A. Our computations, which involve the Todd–Coxeter procedure for coset enumeration and the LLL algorithm for finding bases of integer lattices, are rather naive in that the LLL algorithm is applied to matrices of dimension a multiple of |G|. Thus, in their present form, our techniques can be used only on small groups (say of order up to several hundred). They can in principle be used to construct (crossed) ZG-resolutions of Z, but again, only for small G. The paper is accompanied by two attachment files. The first of these is a summary of our computations in HTML format. The second contains various GAP programs used in the computations.

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Free Crossed Resolutions of Groups and Presentations of Modules of Identities among Relations

Cited by 10 publications

References 22 publications

A Conceptual Construction of Complexity Levels Theory in Spacetime Categorical Ontology: Non-Abelian Algebraic Topology, Many-Valued Logics and Dynamic Systems

A Conceptual Construction of Complexity Levels Theory in Spacetime Categorical Ontology: Non-Abelian Algebraic Topology, Many-Valued Logics and Dynamic Systems

Categorical Ontology of Complex Spacetime Structures: The Emergence of Life and Human Consciousness

Three-Dimensional Presentations for the Groups of Order at Most 30

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