Talk:PlanetPhysics/Topic on Foundations of Quantum Algebraic Topology
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%% Title: Quantum Algebraic Topology Foundations
%%I. C. Baianu, J. F. Glazebrook and R. Brown
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\begin{document}
\section{topic on the Algebraic Foundations of Quantum Algebraic Topology}
This is a contributed topic on the \htmladdnormallink{algebraic}{http://planetphysics.us/encyclopedia/CoIntersections.html} foundations of \htmladdnormallink{Quantum Algebraic Topology}{http://planetphysics.us/encyclopedia/TriangulationMethodsForQuantizedSpacetimes2.html} (\htmladdnormallink{QAT}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra5.html})
\textbf{(A.)} \emph{Quantum Algebraic Topology (QAT)} is defined as the mathematical and physical study of \htmladdnormallink{general theories}{http://planetphysics.us/encyclopedia/GeneralTheory.html} of quantum \htmladdnormallink{algebraic structures}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} from the standpoint of \htmladdnormallink{algebraic topology}{http://planetphysics.us/encyclopedia/CubicalHigherHomotopyGroupoid.html}, \htmladdnormallink{category theory}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} and their \htmladdnormallink{non-Abelian}{http://planetphysics.us/encyclopedia/AbelianCategory3.html} extensions in \htmladdnormallink{higher dimensional algebra}{http://planetphysics.us/encyclopedia/2Groupoid2.html} and \htmladdnormallink{supercategories}{http://planetphysics.us/encyclopedia/SuperCategory6.html} in \htmladdnormallink{relation}{http://planetphysics.us/encyclopedia/Bijective.html} to, or petinent to, \htmladdnormallink{quantum theories}{http://planetphysics.us/encyclopedia/SpaceTimeQuantizationInQuantumGravityTheories.html}, Quantum Field Theories,
\htmladdnormallink{general relativity}{http://planetphysics.us/encyclopedia/SR.html} and its Quantum extensions, \htmladdnormallink{quantum gravity}{http://planetphysics.us/encyclopedia/LQG2.html}.
\textbf{(B). Several suggested new QAT topics are:}
\begin{enumerate}
\item \htmladdnormallink{Poisson algebras}{http://planetphysics.us/encyclopedia/PoissonRing.html}, \htmladdnormallink{quantization methods}{http://planetphysics.us/encyclopedia/QuantizationMethods.html} and \htmladdnormallink{Hamiltonian algebroids}{http://planetphysics.us/encyclopedia/HamiltonianAlgebroid3.html}
\item \htmladdnormallink{K-S theorem}{http://planetphysics.us/encyclopedia/QAT2.html} and its Quantum algebraic consequences in QAT
\item Logic Lattice algebras or Many-Valued (MV) Logic algebras
\item Quantum MV-Logic algebras and $\L{}-M_n$-noncommutative algebras
\item \htmladdnormallink{quantum operator algebras}{http://planetphysics.us/encyclopedia/Groupoid.html} ( such as : involution, *-algebras, or $*$-algebras, \htmladdnormallink{von Neumann algebras}{http://planetphysics.us/encyclopedia/LocallyCompactQuantumGroup.html}, JB- and JL- algebras, $C^*$ - or C*- algebras, etc.
\item Quantum von Neumann algebra and subfactors
\item Kac-Moody and K-algebras
\item \htmladdnormallink{Hopf algebras}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra5.html}, Quantum \htmladdnormallink{groups}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} and \htmladdnormallink{quantum group}{http://planetphysics.us/encyclopedia/QuantumGroup4.html} algebras
\item Quantum \htmladdnormallink{groupoids}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra5.html} and weak Hopf $C^*$-algebras
\item \htmladdnormallink{groupoid C*-convolution algebras}{http://planetphysics.us/encyclopedia/GroupoidCConvolutionAlgebra.html} and *-Convolution \htmladdnormallink{algebroids}{http://planetphysics.us/encyclopedia/Algebroids.html}
\item \htmladdnormallink{quantum spacetimes}{http://planetphysics.us/encyclopedia/NonAbelianQuantumAlgebraicTopology3.html} and Quantum Fundamental Groupoids
\item Quantum Double Algebras
\item Quantum Gravity, \htmladdnormallink{supersymmetries}{http://planetphysics.us/encyclopedia/Supersymmetry.html}, \htmladdnormallink{supergravity}{http://planetphysics.us/encyclopedia/AntiCommutationRelations.html}, \htmladdnormallink{superalgebras}{http://planetphysics.us/encyclopedia/NewtonianMechanics.html} and graded `\htmladdnormallink{Lie' algebras}{http://planetphysics.us/encyclopedia/BilinearMap.html}
\item Quantum \htmladdnormallink{categorical algebra}{http://planetphysics.us/encyclopedia/CategoryOfLogicAlgebras.html} and Higher Dimensional, $\L{}-M_n$- Toposes
\item Quantum \htmladdnormallink{R-categories}{http://planetphysics.us/encyclopedia/RCategory.html}, \htmladdnormallink{R-supercategories}{http://planetphysics.us/encyclopedia/RDiagram.html} and Symmetry Breaking
\item \htmladdnormallink{extended quantum symmetries}{http://planetphysics.us/encyclopedia/ExtendedQuantumSymmetries.html} in Higher Dimensional Algebras (\htmladdnormallink{HDA}{http://planetphysics.us/encyclopedia/2Groupoid2.html}), such as: \\
algebroids, \htmladdnormallink{double algebroids}{http://planetphysics.us/encyclopedia/GeneralizedSuperalgebras.html}, categorical algebroids, \htmladdnormallink{double groupoids}{http://planetphysics.us/encyclopedia/WeakHomotopy.html}, \\
\htmladdnormallink{convolution}{http://planetphysics.us/encyclopedia/AssociatedGroupoidAlgebraRepresentations.html} algebroids, groupoid $C^*$ -convolution algebroids
\item Universal algebras in R-Supercategories
\item Supercategorical algebras (SA) as concrete interpretations of the Theory of Elementary Abstract Supercategories (\htmladdnormallink{ETAS}{http://planetphysics.us/encyclopedia/ETACAxioms.html}).
\item Quantum \htmladdnormallink{non-Abelian algebraic topology}{http://planetphysics.us/encyclopedia/ModuleAlgebraic.html} (QNAAT)
\item \htmladdnormallink{noncommutative geometry}{http://planetphysics.us/encyclopedia/NoncommutativeGeometry4.html}, \htmladdnormallink{quantum geometry}{http://planetphysics.us/encyclopedia/NAQAT2.html}, and Non-Abelian Quantum Algebraic Geometry
\item Other -- Miscellaneous \textbf{[please add here your additions, changes, editing,
remarks, proofs, conjectures, and so on...]}
\end{enumerate}
\begin{thebibliography} {9}
\bibitem{AS}
Alfsen, E.M. and F. W. Schultz: \emph{Geometry of State Spaces of Operator Algebras}, Birk\"auser, Boston--Basel--Berlin (2003).
\bibitem{AMF56}
Atyiah, M.F. 1956. On the Krull-Schmidt theorem with applications to sheaves.
\emph{Bull. Soc. Math. France}, \textbf{84}: 307--317.
\bibitem{AMF56}
Auslander, M. 1965. Coherent Functors. \emph{Proc. Conf. Cat. Algebra, La Jolla},
189--231.
\bibitem{AS-BC2k}
Awodey, S. \& Butz, C., 2000, Topological Completeness for Higher Order Logic., Journal of Symbolic Logic, 65, 3, 1168--1182.
\bibitem{AS96}
Awodey, S., 1996, "Structure in Mathematics and Logic: A Categorical Perspective", Philosophia Mathematica, 3, 209--237.
\bibitem{AS2k4}
Awodey, S., 2004, "An Answer to Hellman's Question: Does Category Theory Provide a Framework for Mathematical Structuralism", Philosophia Mathematica, 12, 54--64.
\bibitem{AS2k6}
Awodey, S., 2006, Category Theory, Oxford: Clarendon Press.
\bibitem{BAJ-DJ98a}
Baez, J. \& Dolan, J., 1998a, "Higher-Dimensional Algebra III. n-Categories and the Algebra of Opetopes", Advances in Mathematics, 135, 145--206.
\bibitem{BAJ-DJ2k1}
Baez, J. \& Dolan, J., 2001, "From Finite Sets to Feynman Diagrams", Mathematics Unlimited -- 2001 and Beyond, Berlin: Springer, 29--50.
\bibitem{BAJ-DJ97}
Baez, J., 1997, "An Introduction to n-Categories", Category Theory and Computer Science, Lecture Notes in Computer Science, 1290, Berlin: Springer-Verlag, 1--33.
\bibitem{ICB3}
Baianu, I.C.: 1970, Organismic Supercategories: II. On Multistable Systems. \emph{Bulletin of Mathematical Biophysics}, \textbf{32}: 539-561.
\bibitem{ICB4}
Baianu, I.C.: 1971b, Categories, Functors and Quantum Algebraic
Computations, in P. Suppes (ed.), \emph{Proceed. Fourth Intl. Congress Logic-Mathematics-Philosophy of Science}, September 1--4, 1971, Bucharest.
\bibitem{ICBs5}
Baianu, I.C. and D. Scripcariu: 1973, On Adjoint Dynamical Systems. \emph{Bulletin of Mathematical Biophysics}, \textbf{35}(4), 475--486.
\bibitem{ICB5}
Baianu, I.C.: 1973, Some Algebraic Properties of \emph{\textbf{(M,R)}} -- Systems. \emph{Bulletin of Mathematical Biophysics} \textbf{35}, 213-217.
\bibitem{ICB6}
Baianu, I.C.: 1977, A Logical Model of Genetic Activities in \L ukasiewicz Algebras: The Non-linear Theory. \emph{Bulletin of Mathematical Biology},
\textbf{39}: 249-258.
\bibitem{Bgg2}
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, \textbf{(M,R)}--Systems and Their Higher Dimensional Algebra, \emph{Abstract and Preprint of Report}: $\\http://www.ag.uiuc.edu/fs401/QAuto.pdf $ and $http://www.medicalupapers.com/quantum+automata+math+categories+baianu/$
\bibitem{Bbg3}
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.
\bibitem{Bggb4}
Baianu, I.C., R. Brown and J. F. Glazebrook: 2007b, A Non-Abelian, Categorical Ontology of Spacetimes and Quantum Gravity, Axiomathes, 17: 169-225.
\bibitem{QNAAT}
Baianu, I. C. et al. 2008. Quantum Non-Abelian Algebraic Topology (QNAAT): PM Exposition lec $id=68$.
\bibitem{Ba-We85}
Barr, M. and Wells, C., 1985, Toposes, Triples and Theories, New York: Springer-Verlag.
\bibitem{BM-CW99}
Barr, M. and Wells, C., 1999, Category Theory for Computing Science, Montreal: CRM.
\bibitem{BJL81}
Bell, J. L., 1981, "Category Theory and the Foundations of Mathematics", British Journal for the Philosophy of Science, 32, 349--358.
\bibitem{BJL82}
Bell, J. L., 1982, "Categories, Toposes and Sets", Synthese, 51, 3, 293--337.
\bibitem{BJL86}
Bell, J. L., 1986, "From Absolute to Local Mathematics", Synthese, 69, 3, 409--426.
\bibitem{BJL88}
Bell, J. L., 1988, Toposes and Local Set Theories: An Introduction, Oxford: Oxford University Press.
\bibitem{BG-MCLS99}
Birkoff, G. \& Mac Lane, S., 1999, Algebra, 3rd ed., Providence: AMS.
\bibitem{Borceux94}
Borceux, F.: 1994, \emph{Handbook of Categorical Algebra}, vols: 1--3,
in {\em Encyclopedia of Mathematics and its Applications} \textbf{50} to \textbf{52}, Cambridge University Press.
\bibitem{Bourbaki1}
Bourbaki, N. 1961 and 1964: \emph{Alg\`{e}bre commutative.},
in \`{E}l\'{e}ments de Math\'{e}matique., Chs. 1--6., Hermann: Paris.
\bibitem{BJk4}
Brown, R. and G. Janelidze: 2004, Galois theory and a new homotopy
double groupoid of a map of spaces, \emph{Applied Categorical
Structures} \textbf{12}: 63-80.
\bibitem{BHR2}
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
\bibitem{BGB2k7b}
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., \emph{Axiomathes} (17): 321--379.
\bibitem{Br-Har-Ka-Po2k2}
Brown, R., Hardie, K., Kamps, H. and T. Porter: 2002, The homotopy double groupoid of a Hausdorff space., \emph{Theory and Applications of Categories} \textbf{10}, 71-93.
\bibitem{Br-Hardy76}
Brown, R., and Hardy, J.P.L.:1976, Topological groupoids I: universal constructions, \emph{Math. Nachr.}, 71: 273-286.
\bibitem{Br-Sp76}
Brown, R. and Spencer, C.B.: 1976, Double groupoids and crossed modules, \emph{Cah. Top. G\'{e}om. Diff.} \textbf{17}, 343-362.
\bibitem{BR-SCB76}
Brown R and Razak Salleh A (1999) Free crossed resolutions of groups and presentations of modules of
identities among relations. {\em LMS J. Comput. Math.}, \textbf{2}: 25--61.
\bibitem{BDA55}
Buchsbaum, D. A.: 1955, Exact categories and duality., Trans. Amer. Math. Soc. \textbf{80}: 1-34.
\bibitem{BDA55}
Buchsbaum, D. A.: 1969, A note on homology in categories., Ann. of Math. \textbf{69}: 66-74.
\bibitem{BL2k3}
Bunge, M. and S. Lack: 2003, Van Kampen theorems for toposes, \emph{Adv. in Math.} \textbf{179}, 291-317.
\bibitem{BM84}
Bunge, M., 1984, "Toposes in Logic and Logic in Toposes", Topoi, 3, no. 1, 13-22.
\bibitem{BM-LS2k3}
Bunge M, Lack S (2003) Van Kampen theorems for toposes. {\em Adv Math}, \textbf {179}: 291-317.
\bibitem{CH-ES56}
Cartan, H. and Eilenberg, S. 1956. {\em Homological Algebra}, Princeton Univ. Press: Pinceton.
\bibitem{CPM65}
Cohen, P.M. 1965. {\em Universal Algebra}, Harper and Row: New York, London and Tokyo.
\bibitem{CA94}
Connes A 1994. \emph{Noncommutative geometry}. Academic Press: New York.
\bibitem{CR-LL63}
Croisot, R. and Lesieur, L. 1963. \emph{Alg\`ebre noeth\'erienne non-commutative.}, Gauthier-Villard: Paris.
\end{thebibliography}
\end{document}