TopicOnAlgebraicFoundationsOfQuantumAlgebraicTopology 2009-01-19 21:20:37 2009-01-26 12:54:39 Topic K-S theorem quantum algebraic topology algebraic foundations of quantum algebraic topology Kochen-Specker theorem (K-S theorem) This is a contributed topic on quantum algebraic topology (QAT) introducing mathematical concepts of QAT based on algebraic topology (AT), category theory (CT) and their non-Abelian extensions in higher dimensional algebra (HDA) and supercategories. \subsection{Introduction} \emph{Quantum algebraic topology (QAT)} is an area of physical mathematics and mathematical physics concerned with the foundation and study of general theories of quantum algebraic structures from the standpoint of algebraic topology, category theory, as well as non-Abelian extensions of AT and CT in higher dimensional algebra and supercategories. \subsubsection{The following are examples of QAT topics:} \begin{enumerate} \item Poisson algebras, Quantization methods and Hamiltonian algebroids \item K-S theorem and its quantum algebraic consequences in QAT \item Logic lattice algebras and many-valued (MV) logic algebras \item Quantum MV-logic algebras and $\L{}-M_n$-noncommutative algebras \item Quantum operator algebras ( such as : involution, *-algebras, or $*$-algebras, von Neumann algebras, , JB- and JL- algebras, $C^*$ - or C*- algebras, \item Quantum von Neumann algebra and subfactors \item Kac-Moody and K-algebras \item Quantum groups, quantum group algebras and Hopf algebras \item Quantum groupoids and weak Hopf $C^*$-algebras \item Groupoid C*-convolution algebras and *-convolution algebroids \item \PMlinkname{Quantum spacetimes}{QuantumSpaceTimes} and quantum fundamental groupoids \item Quantum double algebras \item Quantum gravity, supersymmetries, supergravity, superalgebras and graded `Lie' algebras \item Quantum categorical algebra and higher dimensional, $\L{}-M_n$- toposes \item Quantum R-categories, R-supercategories and symmetry breaking \item Extended quantum symmetries in higher dimensional algebras (HDA), such as: algebroids, double algebroids, categorical algebroids, double groupoids,convolution algebroids, and groupoid $C^*$ -convolution algebroids \item Universal algebras in R-supercategories \item Supercategorical algebras (SA) as concrete interpretations of the theory of elementary abstract supercategories (ETAS). \item \PMlinkexternal{Non-Abelian quantum algebraic topology (NAQAT)}{http://planetmath.org/?op=getobj&from=papers&id=410} \item Noncommutative geometry, quantum geometry, and non-Abelian quantum algebraic geometry \item Kochen-Specker theorem (K-S theorem) \item Other -- Miscellaneous \end{enumerate} \begin{thebibliography} {9} \bibitem{AS} Alfsen, E.M. and F. 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