View source for PlanetPhysics/Non Abelian Quantum Algebraic Topology 2

==Non-Abelian Quantum Algebraic Topology (NAQAT)==

This is a new contributed topic (under construction).

''Quantum Algebraic Topology''  is the area of [[../NonNewtonian2/|theoretical physics]] and [[../NonNewtonian2/|physical mathematics]] concerned with the applications of [[../CubicalHigherHomotopyGroupoid/|algebraic topology]] methods, results and constructions (including its extensions to [[../TrivialGroupoid/|category theory]], [[../GrothendieckTopos/|topos]] Theory and [[../HigherDimensionalAlgebra2/|higher dimensional algebra]]) to fundamental quantum physics problems, such as the [[../CategoricalGroupRepresentation/|representations]] of Quantum spacetimes and Quantum State Spaces in [[../LQG2/|quantum gravity]], in arbitrary [[../CosmologicalConstant2/|reference frames]].
Non--Abelian gauge [[../CosmologicalConstant2/|field]] theories can also be formalized or presented in the [[../QATs/|QAT]] framework.

Perhaps the neighbor areas with which QAT overlaps significantly
are: [[../CoIntersections/|algebraic]] [[../SpaceTimeQuantizationInQuantumGravityTheories/|quantum field theories]] (AQFT[[../MathematicalFoundationsOfQuantumTheories/|)/local quantum physics]] (LQP), Axiomatic QFT, Lattice [[../HotFusion/|QFT]] ([[../LQG2/|LQFT]]) and [[../Supersymmetry/|supersymmetry/]]. One can also claim overlap with various [[../CoIntersections/|topological]] Field Theories (TFT), or Topological Quantum Field Theories (TQFT), [[../ThinEquivalence/|homotopy]] QFT ([[../NonNewtonian2/|HQFT]]), Dilaton, and Lattice Quantum Gravity (respectively, DQG and [[../LQG2/|LQG]]) theories.

===Applications of the Van Kampen Theorem to Crossed Complexes. Representations of Quantum Space-Time in terms of Quantum Crossed Complexes over a Quantum Groupoid.===

There are several possible applications of the [[../SingularComplexOfASpace/|generalized Van Kampen theorem]] in the development of physical representations of a
quantized [[../SR/|space-time]] `geometry' For example, a possible
application of the generalized van Kampen theorem is the
construction of the initial, quantized space-time as the
''unique colimit''  of ''quantum causal sets (posets)''  in terms of
''the nerve of an open [[../CubicalHigherHomotopyGroupoid/|covering]]'' <math>N '''U''' </math> of the topological space <math>X</math> that would be isomorphic to a <math>k</math>-simplex <math>K</math> underlying <math>X</math>.
The corresponding,''[[../AbelianCategory3/|noncommutative]]'' algebra <math>\Omega</math> associated
with the finitary <math>T_0</math>-poset <math>P(S)</math> is ''the Rota algebra''
<math>\Omega</math>, and the ''quantum topology''  <math>T_0</math> is defined by the partial ordering arrows for regions that can overlap, or superpose, coherently (in the quantum sense) with each other. When the poset <math>P(S)</math> contains <math>2N</math> points we write this as <math>P_{2N}(S)</math>. The ''unique''  (up to an [[../IsomorphicObjectsUnderAnIsomorphism/|isomorphism]]) <math>P(S)</math> in the ''colimit'' , <math>\lim_\leftarrow P_N{X}</math>, recovers a space homeomorphic to <math>X</math>. Other non-Abelian results derived from the generalized van Kampen theorem were discussed by Brown, Hardie, Kamps and Porter, and also by Brown, Higgins and Sivera.

===Local--to--Global (LG) Construction Principles consistent with Quantum Axiomatics===

A novel approach to [[../SUSY2/|QST]] construction in AQFT may involve the
use of fundamental [[../Formula/|theorems]] of algebraic topology generalised from
topological spaces to spaces with structure, such as a filtration,
or as an <math>n</math>-cube of spaces. In this [[../Cod/|category]] are the generalized,
''\htmladdnormallink{higher homotopy'' {http://planetphysics.us/encyclopedia/ModuleAlgebraic.html} Seifert-van Kampen theorems (HHSvKT)} of
Algebraic Topology with novel and unique non-Abelian applications.
Such theorems have allowed some new calculations of homotopy [[../Bijective/|types]] of topological spaces. They have also allowed new proofs and
generalisations of the classical [[../ModuleAlgebraic/|relative Hurewicz theorem]] by R.
Brown and coworkers. One may find links of such results
to the expected \emph [`non-commutative']{http://planetphysics.us/encyclopedia/AbelianCategory3.html} geometrical structure of
quantized space--time.

See also the Exposition on NAQAT at: http://aux.planetphysics.org/files/lec/61/ANAQAT20e[[../LebesgueMeasure/|.pdf]]

{{CourseCat}}