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non-Abelian theory
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(Topic)
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ETAC and ETAS axiom interpretations that do not satisfy-in addition to the ETAC or ETAS axioms- the $Ab1$ to $Ab6$ axioms for an abelian category are all examples on non-Abelian categories; a more detailed list is also presented next.
Remark 0.1 In a general sense, any Abelian category (or abelian category) can be regarded as a `good' model for the category of Abelian, or commutative, groups. Furthermore, in an Abelian category $Ab$ every class, or set, of morphisms $Hom_{Ab}(-,-)$ forms an Abelian (or commutative) group. There are several strict definitions of Abelian categories involving 3, 4 or up to 6 axioms defining the Abelian character of a category. To illustrate non-Abelian theories it is useful to consider non-Abelian structures so that specific properties determined by the non-Abelian set of axioms become `transparent' in terms of the properties of objects for example for concrete
categories that have objects; such examples are presented separately as non-Abelian structures.
The following is only a short list of non-Abelian theories:
- Non-Abelian algebraic topology, including also non-Abelian homological algebra; non-Abelian algebraic topology overview and R. Brown 2008 preprint, ([1,2]).
(See also the recent book exposition with the title ``Nonabelian Algebraic Topology'' vol. 1 by Brown and Sivera,(respectively, vol. 2 with Higgins, in preparation).
- Non-Abelian quantum algebraic topology;
- Non-Abelian gauge field theory (in Quantum Physics);
- Noncommutative geometry;
- The axiomatic theory of supercategories (ETAS);
- Higher dimensional algebra (HDA)
- $LM_n$ Logic algebras;
- Non-Abelian categorical ontology ([3]).
The following alternative definition by Barry Mitchell of an Abelian category should also be mentioned as ``an exact additive category with finite products.''.
He also published in his textbook the following theorem: (Theorem 20.1, on p.33 of Barry Mitchell in ``Theory of Catgeories'', 1965, Academic Press: New York and London):
Theorem 0.1 ``The following statements are equivalent:
- (a) $Ab$ is an abelian category;
- (b) $Ab$ has kernels, cokernels, finite products, finite coproducts, and is both normal and conormal;
- (c) $Ab$ has pushouts and pullbacks and is both normal and conormal.''
- 1
- R. Brown et al. 2008. ``Non-Abelian Algebraic Topology''. vols. 1 and 2. (Preprint).
- 2
- R. Brown. 2008. Higher Dimensional Algebra Preprint as pdf and ps docs. at $arXiv:math/0212274v6 [math.AT]$
- 3
- I. C. Baianu, R. Brown and J. F. Glazebrook. 2007, A Non-Abelian Categorical Ontology and Higher Dimensional Algebra of Spacetimes and Quantum Gravity., Axiomathes, 17: 353-408.
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"non-Abelian theory" is owned by bci1. [ full author list (2) ]
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See Also: non-Abelian structures, abelian category, supplemental axioms for an Abelian category, higher dimensional generalized Van Kampen theorems (HD-VKT), axiomatic theory of supercategories and metacategories, algebraic category of LMn logic algebras, categorical quantum logics as quantum LM-algebraic logic, non-commuting graph, topic on the algebraic foundations of quantum algebraic topology, bibliography for axiomatics and mathematics foundations in categories
| Other names: |
non-Abelian, nonabelian |
| Also defines: |
non-Abelian character, non-Abelian theories, nonabelian examples |
| Keywords: |
quantum non--Abelian algebraic topology (QNAAT) |
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Cross-references: theorem, categorical ontology, logic algebras, HDA, higher dimensional algebra, supercategories, axiomatic, noncommutative geometry, field, gauge, non-Abelian Quantum Algebraic Topology, topology, algebraic, concrete categories, objects, terms, properties, useful, character, definitions, strict, morphisms, class, groups, commutative, categories, ETAS, interpretations, ETAS axiom, ETAC, abelian category, theory, abelian, axioms, satisfy
There are 25 references to this entry.
This is version 54 of non-Abelian theory, born on 2008-07-14, modified 2009-02-02.
Object id is 10789, canonical name is NonAbelianTheories.
Accessed 2396 times total.
Classification:
| AMS MSC: | 03G20 (Mathematical logic and foundations :: Algebraic logic :: Lukasiewicz and Post algebras) | | | 03G30 (Mathematical logic and foundations :: Algebraic logic :: Categorical logic, topoi) | | | 03G12 (Mathematical logic and foundations :: Algebraic logic :: Quantum logic) | | | 18A15 (Category theory; homological algebra :: General theory of categories and functors :: Foundations, relations to logic and deductive systems) | | | 18-00 (Category theory; homological algebra :: General reference works ) |
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