# PlanetPhysics/Topic on Algebraic Foundations of Quantum Algebraic Topology

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.

### Introduction

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.

#### The following are examples of QAT topics:

* Poisson algebras, quantization methods and Hamiltonian algebroids
* K-S theorem and its quantum algebraic consequences in QAT
* Logic lattice algebras and many-valued (MV) logic algebras
* Quantum MV-logic algebras and Failed to parse (unknown function "\L"): {\displaystyle \L-M_n} -noncommutative algebras
* quantum operator algebras ( such as : involution, *-algebras, or $*$-algebras, von Neumann algebras,
, JB- and JL- algebras, $C^{*}$ - or C*- algebras,
* Quantum von Neumann algebra and subfactors
* Kac-Moody and K-algebras
* quantum groups, quantum group algebras and Hopf algebras
* quantum groupoids and weak Hopf $C^{*}$-algebras
* groupoid C*-convolution algebras and *-convolution algebroids
* Quantum spacetimes and quantum fundamental groupoids
* Quantum double algebras
* quantum gravity, supersymmetries, supergravity, superalgebras and graded `Lie' algebras
* Quantum categorical algebra and higher dimensional, Failed to parse (unknown function "\L"): {\displaystyle \L-M_n} - toposes
* Quantum R-categories, R-supercategories and symmetry breaking
* extended quantum symmetries in higher dimensional algebras (HDA), such as:
algebroids, double algebroids, categorical algebroids, double groupoids,convolution algebroids, and groupoid $C^{*}$ -convolution algebroids
* Universal algebras in R-supercategories
* Supercategorical algebras (SA) as concrete interpretations of the theory of elementary abstract supercategories (ETAS).
* Non-Abelian quantum algebraic topology (NAQAT)
* noncommutative geometry, quantum geometry, and non-Abelian quantum algebraic geometry
* Kochen-Specker theorem (K-S theorem)
* Other -- Miscellaneous
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* ↑ Alfsen, E.M. and F. W. Schultz: Geometry of State Spaces of Operator Algebras , Birk\"auser, Boston--Basel--Berlin (2003).
* ↑ Atiyah, M.F. 1956. On the Krull-Schmidt theorem with applications to sheaves. Bull. Soc. Math. France , 84 : 307--317.
* ↑ Awodey, S., 2006, Category Theory, Oxford: Clarendon Press.
* ↑ Baez, J. \& Dolan, J., 1998a, Higher-Dimensional Algebra III. n-Categories and the Algebra of Opetopes, Advances in Mathematics, 135, 145--206.
* ↑ Baez, J. \& Dolan, J., 2001, From Finite Sets to Feynman Diagrams, Mathematics Unlimited -- 2001 and Beyond, Berlin: Springer, 29--50.
* ↑ Baez, J., 1997, An Introduction to n-Categories, Category Theory and Computer Science, Lecture Notes in Computer Science, 1290, Berlin: Springer-Verlag, 1--33.
* ↑ 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.
* ↑ 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
* ↑ Baianu, I.C., R. Brown and J. F. Glazebrook: 2007b, A Non-Abelian, Categorical Ontology of Spacetimes and Quantum Gravity, Axiomathes, 17: 169-225.
* ↑ Barr, M. and Wells, C., 1985, Toposes, Triples and Theories, New York: Springer-Verlag.
* ↑ Barr, M. and Wells, C., 1999, Category Theory for Computing Science, Montreal: CRM.
* ↑ Bell, J. L., 1981, Category Theory and the Foundations of Mathematics, British Journal for the Philosophy of Science, 32, 349--358.
* ↑ Bell, J. L., 1982, Categories, Toposes and Sets, Synthese, 51, 3, 293--337.
* ↑ Bell, J. L., 1986, From Absolute to Local Mathematics, Synthese, 69, 3, 409--426.
* ↑ Birkoff, G. \& Mac Lane, S., 1999, Algebra, 3rd ed., Providence: AMS.
* ↑ Borceux, F.: 1994, Handbook of Categorical Algebra , vols: 1--3, in Encyclopedia of Mathematics and its Applications 50 to 52 , Cambridge University Press.
* ↑ Bourbaki, N. 1961 and 1964: Alg\`{e bre commutative.}, in \'{E}l\'{e}ments de Math\'{e}matique., Chs. 1--6., Hermann: Paris.
* ↑ Brown, R. and G. Janelidze: 2004, Galois theory and a new homotopy double groupoid of a map of spaces, Applied Categorical Structures 12 : 63-80.
* ↑ Brown, R., Higgins, P. J. and R. Sivera,: 2008, Non-Abelian Algebraic Topology , (vol.2 in preparation).
* ↑ 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.
* ↑ Brown, R., and Hardy, J.P.L.:1976, Topological groupoids I: universal constructions, Math. Nachr. , 71: 273-286.
* ↑ Brown, R. and Spencer, C.B.: 1976, Double groupoids and crossed modules, Cah. Top. G\'{e om. Diff.} 17 , 343-362.
* ↑ Brown R, Razak Salleh A (1999) Free crossed resolutions of groups and presentations of modules of identities among relations. LMS J. Comput. Math. , 2 : 25--61.
* 1 2 Buchsbaum, D. A.: 1955, Exact categories and duality., Trans. Amer. Math. Soc. 80 : 1-34.
* ↑ Bunge, M. and S. Lack: 2003, Van Kampen theorems for toposes, Adv. in Math. 179 , 291-317.
* ↑ Bunge, M., 1984, Toposes in Logic and Logic in Toposes, Topoi , 3, no. 1, 13-22.
* ↑ Bunge M, Lack S (2003) Van Kampen theorems for toposes. Adv Math , \textbf {179}: 291-317.
* ↑ Cartan, H. and Eilenberg, S. 1956. Homological Algebra , Princeton Univ. Press: Pinceton.
* ↑ Cohen, P.M. 1965. Universal Algebra , Harper and Row: New York, London and Tokyo.
* ↑ Connes A 1994. Noncommutative geometry . Academic Press: New York.
* ↑ Croisot, R. and Lesieur, L. 1963. Alg\`ebre noeth\'erienne non-commutative. , Gauthier-Villard: Paris.