Talk:PlanetPhysics/Algebraic Topology

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\section{Algebraic topology}

\subsection{Introduction} \emph{Algebraic topology} (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), \htmladdnormallink{non-commutative geometry}{http://planetphysics.us/encyclopedia/NAQAT2.html} and, of course, its most recent development-- non-Abelian Algebraic Topology (NAAT). On the other hand, there are also close ties between algebraic geometry and number theory.

\subsection{Outline} \begin{enumerate}

\item Homotopy theory 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/TrivialGroupoid.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/2Groupoid2.html} (\htmladdnormallink{HDA}{http://planetphysics.us/encyclopedia/2Groupoid2.html}) \item \htmladdnormallink{Non-Abelian Quantum Algebraic Topology}{http://planetphysics.us/encyclopedia/NonAbelianQuantumAlgebraicTopology3.html} (NAQAT) \item Quantum Geometry \item \htmladdnormallink{Non-Abelian algebraic topology (NAAT)}{http://planetphysics.org/encyclopedia/NonAbelianAlgebraicTopology6.html} \end{enumerate}

\subsection{Homotopy theory and fundamental groups} \begin{enumerate} \item Homotopy \item Fundamental group 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 Fundamental groupoid \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 crossed modules \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 Fundamental groupoid functor \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/PolishGroup.html} \item Groupoid category \item \htmladdnormallink{category of groupoids}{http://planetphysics.us/encyclopedia/GroupoidCategory4.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{Non-Abelian Algebraic Topology (NAAT)}

\begin{enumerate} \item \htmladdnormallink{An overview of Nonabelian Algebraic Topology}{http://arxiv.org/PS_cache/math/pdf/0407/0407275v2.pdf} \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 \htmladdnormallink{Non-Abelian Algebraic Topology textbook}{http://planetphysics.us/encyclopedia/NonAbelianAlgebraicTopology5.html}

\end{enumerate}

\subsubsection{References for NAAT}

\begin{enumerate}

\item [1] M. Alp and C. D. Wensley, XMod, Crossed modules and Cat1--groups: a GAP4 package,(2004) (http://www.maths.bangor.ac.uk/chda/)

\item [2] R. Brown, Elements of Modern Topology, McGraw Hill, Maidenhead, 1968. second edition as Topology: a geometric account of general topology, homotopy \htmladdnormallink{types}{http://planetphysics.us/encyclopedia/Bijective.html}, and the fundamental groupoid, Ellis Horwood, Chichester (1988) 460 pp.

\item [3] R. Brown, \htmladdnormallink{`Higher dimensional group theory'}{http://www.bangor.ac.uk/âˆ¼mas010/hdaweb2.htm}

\item [4] R. Brown\htmladdnormallink{.`crossed complexes}{http://planetphysics.us/encyclopedia/SingularComplexOfASpace.html} and homotopy groupoids as non commutative tools for higher dimensional local--to--global problems', Proceedings of the \htmladdnormallink{fields}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} Institute Workshop on Categorical Structures for Descent and Galois Theory, Hopf Algebras and Semiabelian Categories, September 23--28, 2002, Contemp. Math. (2004). (to appear), UWB Math Preprint 02.26\htmladdnormallink{.pdf}{http://planetphysics.us/encyclopedia/LebesgueMeasure.html} (30 pp.)

\item [5] R. Brown and P. J. Higgins, On the connection between the second relative \htmladdnormallink{homotopy groups}{http://planetphysics.us/encyclopedia/ExtendedHurewiczFundamentalTheorem.html} of some related spaces, Proc.London Math. Soc., (3) 36 (1978) 193--212.

\item [6] R. Brown and R. Sivera, `Nonabelian algebraic topology', (in preparation) Part I is downloadable from (http://www.bangor.ac.uk/~mas010/nonab-a-t.html)

\item [7] R. Brown and C. B. Spencer, Double groupoids and crossed modules, Cahiers Top. G'/eom.Diff., 17 (1976) 343--362.

\item [8] R. Brown and C. D.Wensley, `\htmladdnormallink{computation}{http://planetphysics.us/encyclopedia/LQG2.html} and homotopical applications of induced crossed modules', J. Symbolic Computation, 35 (2003) 59--72.

\item [9] The GAP Group, 2004, GAP --Groups, \htmladdnormallink{algorithms}{http://planetphysics.us/encyclopedia/RecursiveFunction.html}, and \htmladdnormallink{programming}{http://planetphysics.us/encyclopedia/ComputerProgram.html}, version 4.4 , Technical report, (http://www.gap-system.org)

\item [10] A. Grothendieck, `Pursuing \htmladdnormallink{stacks',}{http://planetphysics.us/encyclopedia/GrothendiecksEsquisseDunProgramme.html} 600p, 1983, distributed from Bangor. Now being edited by G. Maltsiniotis for the SMF.

\item [11] P. J. Higgins, 1971, Categories and Groupoids, Van Nostrand, New York. Reprint Series, Theory and Appl. Categories (to appear).

\item [12] V. Sharko, 1993, Functions on manifolds: algebraic and topological aspects, number 131 in Translations of Mathematical Monographs, American Mathematical Society. \end{enumerate}

\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:}

\end{enumerate}

\subsection{Algebraic Topology and Groupoids} \begin{enumerate} \item Ronald Brown: Topology and Groupoids, BookSurge LLC (2006). \item Ronald Brown R, P.J. Higgins, and R. Sivera.: \emph{``Non-Abelian algebraic topology"}. http://www. bangor.ac.uk/mas010/nonab-a-t.html; http://www.bangor.ac.uk/mas010/nonab-t/partI010604.pdf , Springer: in press (2010). \item R. Brown and J.-L. Loday: Homotopical excision, and \htmladdnormallink{Hurewicz theorems}{http://planetphysics.us/encyclopedia/ModuleAlgebraic.html}, for n-cubes of spaces, Proc. London Math. Soc., 54:(3), 176--192, (1987). \item R. Brown and J.-L. Loday: Van Kampen Theorems for \htmladdnormallink{diagrams}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} of spaces, Topology, 26: 311-337 (1987). \item R. Brown and G. H. Mosa: Double algebroids and crossed modules of algebroids, University of Wales-Bangor, Maths Preprint, 1986. \item R. Brown and C.B. Spencer: Double groupoids and crossed modules, Cahiers Top. G\'eom. Diff. 17 (1976), 343--362. \item Madalina (Ruxi) Buneci.: \emph{\htmladdnormallink{groupoid representations}{http://planetphysics.us/encyclopedia/GroupoidRepresentations.html}}., Ed. Mirton: Timisoara (2003). \item Allain Connes: \emph{\htmladdnormallink{noncommutative geometry}{http://planetphysics.us/encyclopedia/NoncommutativeGeometry4.html}}, Academic Press 1994. \end{enumerate}

\subsection{Non--Abelian Algebraic Topology and Higher Dimensional Algebra} \begin{enumerate} \item Ronald Brown: \htmladdnormallink{non--Abelian algebraic topology}{http://planetphysics.us/encyclopedia/ModuleAlgebraic.html}, vols. I and II. 2010. (in press: Springer): \htmladdnormallink{Nonabelian Algebraic Topology:filtered spaces, crossed complexes, cubical higher homotopy groupoids}{http://www.bangor.ac.uk/~mas010/rbrsbookb-e040310.pdf}

\item \htmladdnormallink{Higher Dimensional Algebra: An Introduction}{http://en.wikipedia.org/wiki/Higher_dimensional_algebra}

\item \htmladdnormallink{Higher Dimensional Algebra and Algebraic Topology., 282 pages, Feb. 10, 2010}{http://en.wikipedia.org/wiki/User:Bci2/Books/Higher_Dimensional_Algebra}

\end{enumerate}

\end{document}

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