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\begin{document}
\section{Algebraic topology topics}
\subsection{Introduction}
\emph{\htmladdnormallink{algebraic topology}{http://planetphysics.us/encyclopedia/ModuleAlgebraic.html}} (AT) utilizes \htmladdnormallink{algebraic}{http://planetphysics.us/encyclopedia/CoIntersections.html} approaches to solve \htmladdnormallink{topological}{http://planetphysics.us/encyclopedia/CoIntersections.html} problems,
such as the \htmladdnormallink{classification}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} of surfaces, proving \htmladdnormallink{duality}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} \htmladdnormallink{theorems}{http://planetphysics.us/encyclopedia/Formula.html} for \htmladdnormallink{manifolds}{http://planetphysics.us/encyclopedia/NoncommutativeGeometry4.html} and
approximation theorems for topological spaces. A central problem in algebraic topology
is to find algebraic invariants of topological spaces, which is usually carried out by means
of \htmladdnormallink{homotopy}{http://planetphysics.us/encyclopedia/ThinEquivalence.html}, homology and \htmladdnormallink{cohomology groups}{http://planetphysics.us/encyclopedia/CohomologyTheoryOnCWComplexes.html}. There are close connections between algebraic topology,
Algebraic Geometry (AG), and \htmladdnormallink{non-commutative geometry}{http://planetphysics.us/encyclopedia/NAQAT2.html} / NAAT. On the other hand, there are also close ties between algebraic geometry and number
theory.
\subsection{Background}
Latin quote: ``{\em Non multa sed multum}''
\subsection{Outline}
\begin{enumerate}
\item \htmladdnormallink{homotopy theory}{http://planetphysics.us/encyclopedia/CubicalHigherHomotopyGroupoid.html} and \htmladdnormallink{fundamental groups}{http://planetphysics.us/encyclopedia/HomotopyCategory.html} \item Topology and \htmladdnormallink{groupoids}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra5.html}; van Kampen theorem
\item Homology and \htmladdnormallink{cohomology theories}{http://planetphysics.us/encyclopedia/NoncommutativeGeometry4.html}
\item Duality
\item \htmladdnormallink{category theory applications}{http://planetphysics.us/encyclopedia/CategoricalOntology.html} in algebraic topology
\item \htmladdnormallink{indexes of category}{http://planetphysics.us/encyclopedia/IndexOfCategories.html}, \htmladdnormallink{functors}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} and \htmladdnormallink{natural transformations}{http://planetphysics.us/encyclopedia/VariableCategory2.html}
\item \htmladdnormallink{Grothendieck's Descent theory}{http://www.uclouvain.be/17501.html}
\item `\htmladdnormallink{Anabelian Geometry}{http://planetphysics.us/encyclopedia/IsomorphismClass.html}' \item Categorical Galois theory
\item \htmladdnormallink{higher dimensional algebra}{http://planetphysics.us/encyclopedia/InfinityGroupoid.html} (\htmladdnormallink{HDA}{http://planetphysics.us/encyclopedia/2Groupoid2.html})
\item \htmladdnormallink{Quantum Algebraic Topology}{http://planetphysics.us/encyclopedia/TriangulationMethodsForQuantizedSpacetimes2.html} (\htmladdnormallink{QAT}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra5.html})
\item Quantum Geometry
\item Non-Abelian algebraic topology (NAAT)
\end{enumerate}
\subsection{Homotopy theory and fundamental groups}
\begin{enumerate}
\item Homotopy
\item \htmladdnormallink{fundamental group}{http://planetphysics.us/encyclopedia/SingularComplexOfASpace.html} of a space
\item Fundamental theorems
\item \htmladdnormallink{Van Kampen theorem}{http://planetphysics.us/encyclopedia/VanKampenTheorems.html} \item Whitehead \htmladdnormallink{groups}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}, torsion and towers
\item Postnikov towers
\end{enumerate}
\subsection{Topology and Groupoids}
\begin{enumerate}
\item Topology definition, axioms and basic \htmladdnormallink{concepts}{http://planetphysics.us/encyclopedia/PreciseIdea.html} \item \htmladdnormallink{fundamental groupoid}{http://planetphysics.us/encyclopedia/CubicalHigherHomotopyGroupoid.html} \item \htmladdnormallink{topological groupoid}{http://planetphysics.us/encyclopedia/GroupoidHomomorphism2.html} \item van Kampen theorem for groupoids
\item Groupoid \htmladdnormallink{pushout}{http://planetphysics.us/encyclopedia/Pushout.html} theorem
\item \htmladdnormallink{double groupoids}{http://planetphysics.us/encyclopedia/WeakHomotopy.html} and \htmladdnormallink{crossed modules}{http://planetphysics.us/encyclopedia/CubicalHigherHomotopyGroupoid.html}
\item new4
\end{enumerate}
\subsection{Homology theory}
\begin{enumerate}
\item \htmladdnormallink{homology group}{http://planetphysics.us/encyclopedia/ExtendedHurewiczFundamentalTheorem.html} \item Homology sequence
\item Homology complex
\item new4
\end{enumerate}
\subsection{Cohomology theory}
\begin{enumerate}
\item Cohomology group
\item Cohomology sequence
\item DeRham cohomology
\item new4
\end{enumerate}
\subsection{Duality in algebraic topology and category theory}
\begin{enumerate}
\item Tanaka-Krein duality
\item Grothendieck duality
\item \htmladdnormallink{categorical duality}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} \item \htmladdnormallink{tangled duality}{http://planetphysics.us/encyclopedia/DualityAndTriality.html} \item DA5
\item DA6
\item DA7
\end{enumerate}
\subsection{Category theory applications}
\begin{enumerate}
\item \htmladdnormallink{abelian categories}{http://planetphysics.us/encyclopedia/AbelianCategory2.html}
\item Topological \htmladdnormallink{category}{http://planetphysics.us/encyclopedia/Cod.html} \item \htmladdnormallink{fundamental groupoid functor}{http://planetphysics.us/encyclopedia/QuantumFundamentalGroupoid.html} \item Categorical Galois theory
\item Non-Abelian algebraic topology
\item Group category
\item \htmladdnormallink{groupoid category}{http://planetphysics.us/encyclopedia/GroupoidCategory3.html} \item $\mathcal{T}op$ category
\item \htmladdnormallink{topos}{http://planetphysics.us/encyclopedia/GrothendieckTopos.html} and topoi axioms
\item \htmladdnormallink{generalized toposes}{http://planetphysics.us/encyclopedia/ManyValuedLogicSubobjectClassifiers.html} \item Categorical logic and algebraic topology
\item \htmladdnormallink{meta-theorems}{http://planetphysics.us/encyclopedia/MetaTheorems.html} \item Duality between spaces and algebras
\end{enumerate}
\subsection{Index of categories}
The following is a listing of categories relevant to algebraic topology:
\begin{enumerate}
\item \htmladdnormallink{Algebraic categories}{http://www.uclouvain.be/17501.html}
\item Topological category
\item Category of sets, Set
\item Category of topological spaces
\item \htmladdnormallink{category of Riemannian manifolds}{http://planetphysics.us/encyclopedia/CategoryOfRiemannianManifolds.html} \item Category of CW-complexes
\item Category of Hausdorff spaces
\item \htmladdnormallink{category of Borel spaces}{http://planetphysics.us/encyclopedia/CategoryOfBorelSpaces.html} \item Category of CR-complexes
\item Category of \htmladdnormallink{graphs}{http://planetphysics.us/encyclopedia/Cod.html} \item Category of \htmladdnormallink{spin networks}{http://planetphysics.us/encyclopedia/SimplicialCWComplex.html} \item Category of groups
\item Galois category
\item Category of fundamental groups
\item Category of \htmladdnormallink{Polish groups}{http://planetphysics.us/encyclopedia/InvariantBorelSet.html}
\item Groupoid category
\item \htmladdnormallink{category of groupoids}{http://planetphysics.us/encyclopedia/GroupoidCategory.html} (or groupoid category)
\item \htmladdnormallink{category of Borel groupoids}{http://planetphysics.us/encyclopedia/CategoryOfBorelGroupoids.html} \item Category of fundamental groupoids
\item Category of functors (or \htmladdnormallink{functor category}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html})
\item Double groupoid category
\item \htmladdnormallink{double category}{http://planetphysics.us/encyclopedia/HorizontalIdentities.html} \item \htmladdnormallink{category of Hilbert spaces}{http://planetphysics.us/encyclopedia/CategoryOfHilbertSpaces.html} \item \htmladdnormallink{category of quantum automata}{http://planetphysics.us/encyclopedia/CategoryOfQuantumAutomata.html} \item \htmladdnormallink{R-category}{http://planetphysics.us/encyclopedia/RCategory.html} \item Category of \htmladdnormallink{algebroids}{http://planetphysics.us/encyclopedia/Algebroids.html} \item Category of \htmladdnormallink{double algebroids}{http://planetphysics.us/encyclopedia/GeneralizedSuperalgebras.html}
\item Category of \htmladdnormallink{dynamical systems}{http://planetphysics.us/encyclopedia/ContinuousGroupoidHomomorphism.html}
\end{enumerate}
\subsection{Index of functors}
\emph{The following is a contributed listing of functors:}
\begin{enumerate}
\item Covariant functors
\item Contravariant functors
\item \htmladdnormallink{adjoint functors}{http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html}
\item \htmladdnormallink{preadditive functors}{http://planetphysics.us/encyclopedia/PreadditiveFunctor.html}
\item Additive functor
\item \htmladdnormallink{representable functors}{http://planetphysics.us/encyclopedia/CategoryOfLogicAlgebras.html}
\item Fundamental groupoid functor
\item Forgetful functors
\item Grothendieck group functor
\item Exact functor
\item Multi-functor
\item \htmladdnormallink{section functors}{http://planetphysics.us/encyclopedia/RightAdjointFunctor.html}
\item NT2
\item NT3
\end{enumerate}
\subsection{Index of natural transformations}
\emph{The following is a contributed listing of natural transformations:}
\begin{enumerate}
\item \htmladdnormallink{natural equivalence}{http://planetphysics.us/encyclopedia/IsomorphismClass.html} \item Natural transformations in a \htmladdnormallink{2-category}{http://planetphysics.us/encyclopedia/2Category.html} \item NT3
\item NT1
\item NT2
\item NT3
\end{enumerate}
\subsection{Grothendieck proposals}
\begin{enumerate}
\item Esquisse d'un Programme
\item
\htmladdnormallink{Pursuing Stacks}{http://www.math.jussieu.fr/~leila/grothendieckcircle/stacks.ps}
\item S2
\item S3
\item S4
\end{enumerate}
\subsection{Descent theory}
\begin{enumerate}
\item D1
\item D2
\item D3
\item D4
\end{enumerate}
\subsection{Higher dimensional algebra (HDA)}
\begin{enumerate}
\item Categorical groups
\item Double groupoids
\item Double algebroids
\item Bi-algebroids
\item $R$-algebroid
\item $2$-category
\item $n$-category
\item \htmladdnormallink{super-category}{http://planetphysics.us/encyclopedia/SuperCategory6.html} \item weak \htmladdnormallink{n-categories}{http://planetphysics.us/encyclopedia/InfinityGroupoid.html} \item Bi-dimensional Geometry
\item \htmladdnormallink{Noncommutative geometry}{http://planetphysics.us/encyclopedia/NoncommutativeGeometry.html}
\item Higher-Homotopy theories
\item Higher-Homotopy Generalized van Kampen Theorem (HGvKT)
\item H1
\item H2
\item H3
\item H4
\end{enumerate}
\subsubsection{Axioms of cohomology theory}
\begin{enumerate}
\item A1
\item A2
\item A3
\item A4
\item A5
\item A6
\item A7
\end{enumerate}
\subsubsection{Axioms of homology theory}
\begin{enumerate}
\item A1
\item A2
\item A3
\item A4
\item A5
\item A6
\end{enumerate}
\subsection{Quantum algebraic topology (QAT)}
\textbf{(a). Quantum algebraic topology} is described as \emph{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 algebraic topology, \htmladdnormallink{category theory}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} and
their \htmladdnormallink{non-Abelian}{http://planetphysics.us/encyclopedia/AbelianCategory3.html} extensions in higher dimensional algebra and \htmladdnormallink{supercategories}{http://planetphysics.us/encyclopedia/SuperCategory6.html}}
\begin{enumerate}
\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/CoordinateSpace.html},
, JB- and JL- algebras, $C^*$ - or C*- algebras,
\item Quantum von Neumann algebra and subfactors; Jone's towers and subfactors
\item Kac-Moody and K-algebras
\item categorical groups
\item \htmladdnormallink{Hopf algebras}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra5.html}, quantum Groups and \htmladdnormallink{quantum group}{http://planetphysics.us/encyclopedia/QuantumGroup4.html} algebras
\item \htmladdnormallink{quantum groupoids}{http://planetphysics.us/encyclopedia/WeakHopfAlgebra.html} and weak Hopf $C^*$-algebras
\item \htmladdnormallink{groupoid C*-convolution algebras}{http://planetphysics.us/encyclopedia/GroupoidCConvolutionAlgebra.html} and *-convolution algebroids
\item \htmladdnormallink{quantum spacetimes}{http://planetphysics.us/encyclopedia/NonAbelianQuantumAlgebraicTopology3.html} and \htmladdnormallink{quantum fundamental groupoids}{http://planetphysics.us/encyclopedia/QuantumFundamentalGroupoid4.html}
\item Quantum double Algebras
\item \htmladdnormallink{quantum gravity}{http://planetphysics.us/encyclopedia/LQG2.html}, \htmladdnormallink{supersymmetries}{http://planetphysics.us/encyclopedia/Supersymmetry.html}, \htmladdnormallink{supergravity}{http://planetphysics.us/encyclopedia/AntiCommutationRelations.html}, \htmladdnormallink{superalgebras}{http://planetphysics.us/encyclopedia/MathematicalFoundationsOfQuantumTheories.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 R-categories, \htmladdnormallink{R-supercategories}{http://planetphysics.us/encyclopedia/RDiagram.html} and \htmladdnormallink{spontaneous symmetry breaking}{http://planetphysics.us/encyclopedia/LongRangeCoupling.html} \item \htmladdnormallink{Non-Abelian Quantum Algebraic Topology}{http://planetphysics.us/encyclopedia/NonAbelianQuantumAlgebraicTopology3.html} (NA-QAT): closely related to NAAT and HDA.
\end{enumerate}
\subsection{Quantum Geometry}
\begin{enumerate}
\item \htmladdnormallink{Quantum Geometry overview}{http://planetphysics.us/encyclopedia/QuantumGeometry2.html}
\item Quantum non-commutative geometry
\end{enumerate}
\subsection{Non-Abelian Algebraic Topology (NAAT)}
\begin{enumerate}
\item \htmladdnormallink{non-Abelian categories}{http://planetphysics.us/encyclopedia/AbelianCategory3.html}
\item \htmladdnormallink{non-commutative}{http://planetphysics.us/encyclopedia/AbelianCategory3.html} groupoids (including non-Abelian groups)
\item Generalized van Kampen theorems
\item \htmladdnormallink{Noncommutative Geometry (NCG)}{http://planetphysics.us/encyclopedia/NoncommutativeGeometry.html}
\item Non-commutative `spaces' of \htmladdnormallink{functions}{http://planetphysics.us/encyclopedia/Bijective.html} \item new4
\end{enumerate}
\subsection{12}
\begin{enumerate}
\item new1
\item new2
\item new3
\item new4
\end{enumerate}
\subsection{13}
\begin{enumerate}
\item new1
\item new2
\item new3
\item new4
\end{enumerate}
\subsection{14}
\subsection{References}
\htmladdnormallink{Bibliography on Category theory, AT and QAT}{http://planetmath.org/?op=getobj&from=objects&id=10746}
\subsubsection{Textbooks and Expositions:}
\begin{enumerate}
\item A \htmladdnormallink{Textbook1}{http://planetmath.org/?op=getobj&from=books&id=172}
\item A \htmladdnormallink{Textbook2}{http://planetmath.org/?op=getobj&from=books&id=156}
\item A \htmladdnormallink{Textbook3}{http://planetmath.org/?op=getobj&from=books&id=159}
\item A \htmladdnormallink{Textbook4}{http://planetmath.org/?op=getobj&from=books&id=160}
\item A \htmladdnormallink{Textbook5}{http://planetmath.org/?op=getobj&from=books&id=153}
\item A \htmladdnormallink{Textbook6}{http://planetmath.org/?op=getobj&from=lec&id=68}
\item A \htmladdnormallink{Textbook7}{http://planetmath.org/?op=getobj&from=books&id=158}
\item A \htmladdnormallink{Textbook8}{http://planetmath.org/?op=getobj&from=lec&id=75}
\item A \htmladdnormallink{Textbook9}{http://planetmath.org/?op=getobj&from=lec&id=73}
\item A \htmladdnormallink{Textbook10}{http://planetmath.org/?op=getobj&from=books&id=174}
\item A \htmladdnormallink{Textbook11}{http://planetmath.org/?op=getobj&from=books&id=169}
\item A \htmladdnormallink{Textbook12}{http://planetmath.org/?op=getobj&from=books&id=178}
\item A \htmladdnormallink{Textbook13}{http://www.math.cornell.edu/~hatcher/VBKT/VB.pdf}
\item new1
\item new2
\item new3
\item new4
\end{enumerate}
\end{document}
