PlanetPhysics/Topic on Foundations of Quantum Algebraic Topology

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topic on the Algebraic Foundations of Quantum Algebraic Topology

This is a contributed topic on the [[../CoIntersections/|algebraic]] foundations of [[../TriangulationMethodsForQuantizedSpacetimes2/|Quantum Algebraic Topology]] ([[../QuantumOperatorAlgebra5/|QAT]])

(A.) Quantum Algebraic Topology (QAT) is defined as the mathematical and physical study of [[../GeneralTheory/|general theories]] of quantum [[../TrivialGroupoid/|algebraic structures]] from the standpoint of [[../CubicalHigherHomotopyGroupoid/|algebraic topology]], [[../TrivialGroupoid/|category theory]] and their [[../AbelianCategory3/|non-Abelian]] extensions in [[../2Groupoid2/|higher dimensional algebra]] and [[../SuperCategory6/|supercategories]] in [[../Bijective/|relation]] to, or petinent to, [[../SpaceTimeQuantizationInQuantumGravityTheories/|quantum theories]], Quantum Field Theories, [[../SR/|general relativity]] and its Quantum extensions, [[../LQG2/|quantum gravity]].

(B). Several suggested new QAT topics are:

  1. [[../PoissonRing/|Poisson algebras]], [[../QuantizationMethods/|quantization methods]] and [[../HamiltonianAlgebroid3/|Hamiltonian algebroids]]
  1. [[../QAT2/|K-S theorem]] and its Quantum algebraic consequences in QAT
  1. Logic Lattice algebras or Many-Valued (MV) Logic algebras
  1. Quantum MV-Logic algebras and Failed to parse (unknown function "\L"): {\displaystyle \L{}-M_n} -noncommutative algebras
  1. [[../Groupoid/|quantum operator algebras]] ( such as : involution, *-algebras, or *-algebras, [[../LocallyCompactQuantumGroup/|von Neumann algebras]], JB- and JL- algebras, C* - or C*- algebras, etc.
  1. Quantum von Neumann algebra and subfactors
  1. Kac-Moody and K-algebras
  1. [[../QuantumOperatorAlgebra5/|Hopf algebras]], Quantum [[../TrivialGroupoid/|groups]] and [[../QuantumGroup4/|quantum group]] algebras
  1. Quantum [[../QuantumOperatorAlgebra5/|groupoids]] and weak Hopf C*-algebras
  1. [[../GroupoidCConvolutionAlgebra/|groupoid C*-convolution algebras]] and *-Convolution [[../Algebroids/|algebroids]]
  2. [[../NonAbelianQuantumAlgebraicTopology3/|quantum spacetimes]] and Quantum Fundamental Groupoids
  1. Quantum Double Algebras
  1. Quantum Gravity, [[../Supersymmetry/|supersymmetries]], [[../AntiCommutationRelations/|supergravity]], [[../NewtonianMechanics/|superalgebras]] and graded `[[../BilinearMap/|Lie' algebras]]
  2. Quantum [[../CategoryOfLogicAlgebras/|categorical algebra]] and Higher Dimensional, Failed to parse (unknown function "\L"): {\displaystyle \L{}-M_n} - Toposes
  1. Quantum [[../RCategory/|R-categories]], [[../RDiagram/|R-supercategories]] and Symmetry Breaking
  1. [[../ExtendedQuantumSymmetries/|extended quantum symmetries]] in Higher Dimensional Algebras ([[../2Groupoid2/|HDA]]), such as: \\

algebroids, [[../GeneralizedSuperalgebras/|double algebroids]], categorical algebroids, [[../WeakHomotopy/|double groupoids]], \\ [[../AssociatedGroupoidAlgebraRepresentations/|convolution]] algebroids, groupoid C* -convolution algebroids

  1. Universal algebras in R-Supercategories
  1. Supercategorical algebras (SA) as concrete interpretations of the Theory of Elementary Abstract Supercategories ([[../ETACAxioms/|ETAS]]).
  1. Quantum [[../ModuleAlgebraic/|non-Abelian algebraic topology]] (QNAAT)
  1. [[../NoncommutativeGeometry4/|noncommutative geometry]], [[../NAQAT2/|quantum geometry]], and Non-Abelian Quantum Algebraic Geometry
  1. Other -- Miscellaneous \textbf{[please add here your additions, changes, editing,

remarks, proofs, conjectures, and so on...]}

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  1. ↑ Alfsen, E.M. and F. W. Schultz: Geometry of State Spaces of Operator Algebras , Birk\"auser, Boston--Basel--Berlin (2003).
  2. ↑ 2.0 2.1 Atyiah, M.F. 1956. On the Krull-Schmidt theorem with applications to sheaves. Bull. Soc. Math. France , 84 : 307--317. Cite error: Invalid <ref> tag; name "AMF56" defined multiple times with different content
  3. ↑ Awodey, S. \& Butz, C., 2000, Topological Completeness for Higher Order Logic., Journal of Symbolic Logic, 65, 3, 1168--1182.
  4. ↑ Awodey, S., 1996, "Structure in Mathematics and Logic: A Categorical Perspective", Philosophia Mathematica, 3, 209--237.
  5. ↑ Awodey, S., 2004, "An Answer to Hellman's Question: Does Category Theory Provide a Framework for Mathematical Structuralism", Philosophia Mathematica, 12, 54--64.
  6. ↑ Awodey, S., 2006, Category Theory, Oxford: Clarendon Press.
  7. ↑ Baez, J. \& Dolan, J., 1998a, "Higher-Dimensional Algebra III. n-Categories and the Algebra of Opetopes", Advances in Mathematics, 135, 145--206.
  8. ↑ Baez, J. \& Dolan, J., 2001, "From Finite Sets to Feynman Diagrams", Mathematics Unlimited -- 2001 and Beyond, Berlin: Springer, 29--50.
  9. ↑ Baez, J., 1997, "An Introduction to n-Categories", Category Theory and Computer Science, Lecture Notes in Computer Science, 1290, Berlin: Springer-Verlag, 1--33.
  10. ↑ Baianu, I.C.: 1970, Organismic Supercategories: II. On Multistable Systems. Bulletin of Mathematical Biophysics , 32 : 539-561.
  11. ↑ Baianu, I.C.: 1971b, Categories, Functors and Quantum Algebraic Computations, in P. Suppes (ed.), Proceed. Fourth Intl. Congress Logic-Mathematics-Philosophy of Science , September 1--4, 1971, Bucharest.
  12. ↑ Baianu, I.C. and D. Scripcariu: 1973, On Adjoint Dynamical Systems. Bulletin of Mathematical Biophysics , 35 (4), 475--486.
  13. ↑ Baianu, I.C.: 1973, Some Algebraic Properties of (M,R) -- Systems. Bulletin of Mathematical Biophysics 35 , 213-217.
  14. ↑ Baianu, I.C.: 1977, A Logical Model of Genetic Activities in \L ukasiewicz Algebras: The Non-linear Theory. Bulletin of Mathematical Biology , 39 : 249-258.
  15. ↑ Baianu, I. C., Glazebrook, J. F. and G. Georgescu: 2004, Categories of Quantum Automata and N-Valued \L ukasiewicz Algebras in Relation to Dynamic Bionetworks, (M,R) --Systems and Their Higher Dimensional Algebra, Abstract and Preprint of Report : Failed to parse (syntax error): {\displaystyle \\http://www.ag.uiuc.edu/fs401/QAuto.pdf } and http://www.medicalupapers.com/quantum+automata+math+categories+baianu/
  16. ↑ Baianu, I.C., R. Brown and J.F. Glazebrook. : 2007a, Categorical Ontology of Complex Spacetime Structures: The Emergence of Life and Human Consciousness, Axiomathes, 17: 35-168.
  17. ↑ Baianu, I.C., R. Brown and J. F. Glazebrook: 2007b, A Non-Abelian, Categorical Ontology of Spacetimes and Quantum Gravity, Axiomathes, 17: 169-225.
  18. ↑ Baianu, I. C. et al. 2008. Quantum Non-Abelian Algebraic Topology (QNAAT): PM Exposition lec id=68.
  19. ↑ Barr, M. and Wells, C., 1985, Toposes, Triples and Theories, New York: Springer-Verlag.
  20. ↑ Barr, M. and Wells, C., 1999, Category Theory for Computing Science, Montreal: CRM.
  21. ↑ Bell, J. L., 1981, "Category Theory and the Foundations of Mathematics", British Journal for the Philosophy of Science, 32, 349--358.
  22. ↑ Bell, J. L., 1982, "Categories, Toposes and Sets", Synthese, 51, 3, 293--337.
  23. ↑ Bell, J. L., 1986, "From Absolute to Local Mathematics", Synthese, 69, 3, 409--426.
  24. ↑ Bell, J. L., 1988, Toposes and Local Set Theories: An Introduction, Oxford: Oxford University Press.
  25. ↑ Birkoff, G. \& Mac Lane, S., 1999, Algebra, 3rd ed., Providence: AMS.
  26. ↑ Borceux, F.: 1994, Handbook of Categorical Algebra , vols: 1--3, in Encyclopedia of Mathematics and its Applications 50 to 52 , Cambridge University Press.
  27. ↑ Bourbaki, N. 1961 and 1964: Alg\`{e bre commutative.}, in \`{E}l\'{e}ments de Math\'{e}matique., Chs. 1--6., Hermann: Paris.
  28. ↑ Brown, R. and G. Janelidze: 2004, Galois theory and a new homotopy double groupoid of a map of spaces, \emph{Applied Categorical Structures} 12 : 63-80.
  29. ↑ Brown, R., Higgins, P. J. and R. Sivera,: 2007a, \emph{Non-Abelian Algebraic Topology}, in preparation.\\ http://www.bangor.ac.uk/~mas010/nonab-a-t.html ; \\ http://www.bangor.ac.uk/~mas010/nonab-t/partI010604.pdf
  30. ↑ Brown, R., Glazebrook, J. F. and I.C. Baianu.: 2007b, A Conceptual, Categorical and Higher Dimensional Algebra Framework of Universal Ontology and the Theory of Levels for Highly Complex Structures and Dynamics., Axiomathes (17): 321--379.
  31. ↑ Brown, R., Hardie, K., Kamps, H. and T. Porter: 2002, The homotopy double groupoid of a Hausdorff space., Theory and Applications of Categories 10 , 71-93.
  32. ↑ Brown, R., and Hardy, J.P.L.:1976, Topological groupoids I: universal constructions, Math. Nachr. , 71: 273-286.
  33. ↑ Brown, R. and Spencer, C.B.: 1976, Double groupoids and crossed modules, Cah. Top. G\'{e om. Diff.} 17 , 343-362.
  34. ↑ Brown R and Razak Salleh A (1999) Free crossed resolutions of groups and presentations of modules of identities among relations. LMS J. Comput. Math. , 2 : 25--61.
  35. ↑ 35.0 35.1 Buchsbaum, D. A.: 1955, Exact categories and duality., Trans. Amer. Math. Soc. 80 : 1-34. Cite error: Invalid <ref> tag; name "BDA55" defined multiple times with different content
  36. ↑ Bunge, M. and S. Lack: 2003, Van Kampen theorems for toposes, Adv. in Math. 179 , 291-317.
  37. ↑ Bunge, M., 1984, "Toposes in Logic and Logic in Toposes", Topoi, 3, no. 1, 13-22.
  38. ↑ Bunge M, Lack S (2003) Van Kampen theorems for toposes. Adv Math , \textbf {179}: 291-317.
  39. ↑ Cartan, H. and Eilenberg, S. 1956. Homological Algebra , Princeton Univ. Press: Pinceton.
  40. ↑ Cohen, P.M. 1965. Universal Algebra , Harper and Row: New York, London and Tokyo.
  41. ↑ Connes A 1994. Noncommutative geometry . Academic Press: New York.
  42. ↑ Croisot, R. and Lesieur, L. 1963. Alg\`ebre noeth\'erienne non-commutative. , Gauthier-Villard: Paris.
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