Talk:PlanetPhysics/Index of Algebraic Topology

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\section{Foundations}

\subsection{Basic Definitions} \begin{itemize} \item open and closed sets \item map \item arrow \item \htmladdnormallink{morphism}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} \item path \item cycle \item \htmladdnormallink{boundary}{http://planetphysics.us/encyclopedia/PiecewiseLinear.html} \item torus, n-tori \item sphere, n-sphere \item \htmladdnormallink{square}{http://planetphysics.us/encyclopedia/PiecewiseLinear.html} \item cube \item dimension \item \htmladdnormallink{vector space}{http://planetphysics.us/encyclopedia/NormInducedByInnerProduct.html} \item CW-complex \item \htmladdnormallink{graph}{http://planetphysics.us/encyclopedia/Cod.html} \item \htmladdnormallink{simplicial complex}{http://planetphysics.us/encyclopedia/PiecewiseLinear.html} \item \htmladdnormallink{spin network}{http://planetphysics.us/encyclopedia/SimplicialCWComplex.html} \item \htmladdnormallink{topological}{http://planetphysics.us/encyclopedia/CoIntersections.html} \htmladdnormallink{dynamics}{http://planetphysics.us/encyclopedia/NewtonianMechanics.html} \item qualitative dynamics \item dense space \item \htmladdnormallink{spin foam}{http://planetphysics.us/encyclopedia/SimplicialCWComplex.html} \item map \item \htmladdnormallink{function}{http://planetphysics.us/encyclopedia/Bijective.html} \item analytical space \item \htmladdnormallink{identity}{http://planetphysics.us/encyclopedia/Cod.html} \item \htmladdnormallink{commutativity}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} \item associativity \item \htmladdnormallink{object}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} space \item source space \item target space \item \htmladdnormallink{thin square}{http://planetphysics.us/encyclopedia/Tree.html} \item topological space \item \htmladdnormallink{homeomorphism}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} \item \htmladdnormallink{topological groups}{http://planetphysics.us/encyclopedia/PolishGroup.html} \item \htmladdnormallink{Lie groups}{http://planetphysics.us/encyclopedia/BilinearMap.html} and \htmladdnormallink{Lie algebras}{http://planetphysics.us/encyclopedia/BilinearMap.html} \item graded lie algebras/Lie \htmladdnormallink{superalgebras}{http://planetphysics.us/encyclopedia/NewtonianMechanics.html} \item \htmladdnormallink{supergroups}{http://planetphysics.us/encyclopedia/Paragroups.html} \item \htmladdnormallink{groupoid}{http://planetphysics.us/encyclopedia/GroupoidHomomorphism2.html} \item \htmladdnormallink{groupoid homomorphism}{http://planetphysics.us/encyclopedia/GroupoidHomomorphism2.html} \item \htmladdnormallink{groupoid categories}{http://planetphysics.us/encyclopedia/GroupoidCategory3.html} \item compact space/Hausdorff space \item \htmladdnormallink{double groupoid}{http://planetphysics.us/encyclopedia/ThinEquivalence.html} \item \htmladdnormallink{double algebroid}{http://planetphysics.us/encyclopedia/GeneralizedSuperalgebras.html} \item \htmladdnormallink{Hamiltonian algebroid}{http://planetphysics.us/encyclopedia/HamiltonianAlgebroid3.html} \item Polish space \item \htmladdnormallink{Polish group}{http://planetphysics.us/encyclopedia/InvariantBorelSet.html} \item \htmladdnormallink{Polish G-space}{http://planetphysics.us/encyclopedia/PolishGSpace.html} \item \htmladdnormallink{homotopy}{http://planetphysics.us/encyclopedia/ThinEquivalence.html} \item \htmladdnormallink{homotopy groups}{http://planetphysics.us/encyclopedia/ExtendedHurewiczFundamentalTheorem.html} \item \htmladdnormallink{homotopy category}{http://planetphysics.us/encyclopedia/HomotopyCategory.html} \item homology cycle \item cohomology cocycle \item \htmladdnormallink{fundamental groups}{http://planetphysics.us/encyclopedia/HomotopyCategory.html} \item homotopy lemma and corollary \item topological groups \item \htmladdnormallink{fundamental groupoids}{http://planetphysics.us/encyclopedia/CubicalHigherHomotopyGroupoid.html} \item \htmladdnormallink{fundamental groupoid functor}{http://planetphysics.us/encyclopedia/QuantumFundamentalGroupoid.html} \item \htmladdnormallink{homotopy double groupoid of a Hausdorff space}{http://planetphysics.us/encyclopedia/ThinEquivalence.html} \item \htmladdnormallink{homology theory}{http://planetphysics.us/encyclopedia/CubicalHigherHomotopyGroupoid.html} \item axiomatic homology theory \item \htmladdnormallink{cohomology groups}{http://planetphysics.us/encyclopedia/CohomologyTheoryOnCWComplexes.html} \item cohomology groupoids \item \htmladdnormallink{cohomology theory}{http://planetphysics.us/encyclopedia/NoncommutativeGeometry4.html} \item cohomology \htmladdnormallink{theorems}{http://planetphysics.us/encyclopedia/Formula.html} \item topological \htmladdnormallink{categories}{http://planetphysics.us/encyclopedia/Cod.html} \end{itemize}

\subsection{Category theory concepts in algebraic topology} \begin{itemize} \item category \item \htmladdnormallink{topos}{http://planetphysics.us/encyclopedia/GrothendieckTopos.html} \item subspace \item subcategory \item automorphism \item \htmladdnormallink{commutative diagram}{http://planetphysics.us/encyclopedia/Commutativity.html} \item concrete category \item dual category \item \htmladdnormallink{duality principle}{http://planetphysics.us/encyclopedia/DualityAndTriality.html} \item endomorphism \item \htmladdnormallink{epimorphism}{http://planetphysics.us/encyclopedia/Epimorphism2.html} \item monic \item \htmladdnormallink{monomorphism}{http://planetphysics.us/encyclopedia/InjectiveMap.html} \item source \item \htmladdnormallink{pushout}{http://planetphysics.us/encyclopedia/Pushout.html} \item pullback \item cones and cocones \item limit and colimit \htmladdnormallink{functors}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} \item direct sum \item open covers \item isomorphism-closed subcategory \item locally finite category \item preimage of category \item product of categories \item \htmladdnormallink{types}{http://planetphysics.us/encyclopedia/Bijective.html} of morphisms \item wellpowered category \item null object \item zero object \item $\mathcal{U}$-small \item equalizer \item subobject \item \htmladdnormallink{quotient object}{http://planetphysics.us/encyclopedia/DenseSubcategory.html} \item categorical direct product \item categorical direct sum \item direct limit \item limiting cone \item complete category \item groupoid (category theoretic) \item \htmladdnormallink{enriched category}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} \item \htmladdnormallink{double category}{http://planetphysics.us/encyclopedia/HorizontalIdentities.html} \item \htmladdnormallink{2-category}{http://planetphysics.us/encyclopedia/2Category.html} \item \htmladdnormallink{n-category}{http://planetphysics.us/encyclopedia/InfinityGroupoid.html} \end{itemize}

\subsection{Fundamental Theorems} \begin{enumerate} \item \htmladdnormallink{Hurewicz theorem}{http://planetphysics.us/encyclopedia/ModuleAlgebraic.html} \item cohomology theorems \item homotopy lemma and corollary \item topological space approximation theorem \item Yoneda-Grothendieck Lemma \item \htmladdnormallink{Van Kampen theorem}{http://planetphysics.us/encyclopedia/VanKampenTheorems.html} \item Higher dimensional, \htmladdnormallink{generalized Van Kampen theorem}{http://planetphysics.us/encyclopedia/SingularComplexOfASpace.html} (HGvKT) \item properties of monomorphisms and epimorphisms \item properties of \htmladdnormallink{regular}{http://planetphysics.us/encyclopedia/CoIntersections.html} and extremal monomorphisms \item monomorphisms are pullback stable \item proof that an equalizer is a monomorphism \item categorical direct product is an inverse limit \item kernel is an inverse limit \item \htmladdnormallink{Grothendieck theorem}{http://planetphysics.us/encyclopedia/GrothendiecksTheorem.html} \end{enumerate}

\subsection{2-Categories and Supercategories} \begin{itemize} \item functor \item endofunctor \item category \htmladdnormallink{isomorphism}{http://planetphysics.us/encyclopedia/IsomorphicObjectsUnderAnIsomorphism.html} \item diagonal functor \item \htmladdnormallink{representable functors}{http://planetphysics.us/encyclopedia/CategoryOfLogicAlgebras.html} \item categorical \htmladdnormallink{representations}{http://planetphysics.us/encyclopedia/CategoricalGroupRepresentation.html} and supergroups \item forgetful functor \item \htmladdnormallink{identity functor}{http://planetphysics.us/encyclopedia/IdentityMorphism.html} \item isomorphism \item multifunctor \item \htmladdnormallink{natural transformation}{http://planetphysics.us/encyclopedia/VariableCategory2.html} \item essentially \htmladdnormallink{surjective}{http://planetphysics.us/encyclopedia/BCConjecture.html} \item faithful functor \item full functor \item \htmladdnormallink{adjoint functor}{http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html} \item \htmladdnormallink{natural equivalence}{http://planetphysics.us/encyclopedia/IsomorphismClass.html} \item equivalence of categories \item isomorphic categories \item universal properties \item representable functors \item Equivalent definition of a Representable Functor \item \htmladdnormallink{simplicial}{http://planetphysics.us/encyclopedia/PiecewiseLinear.html} objects \end{itemize}

\section{Additive Categories and Homology} \begin{enumerate} \item preadditive category \item \htmladdnormallink{additive category}{http://planetphysics.us/encyclopedia/DenseSubcategory.html} \item \htmladdnormallink{abelian category}{http://planetphysics.us/encyclopedia/AbelianCategory2.html} \item \htmladdnormallink{non-Abelian categories}{http://planetphysics.us/encyclopedia/AbelianCategory3.html} and \htmladdnormallink{non-Abelian algebraic topology}{http://planetphysics.us/encyclopedia/ModuleAlgebraic.html} \item supplemental axioms for an Abelian category \item exact sequence \item exact functor \item Grothendieck spectral sequence \item enough projectives \item enough \htmladdnormallink{injectives}{http://planetphysics.us/encyclopedia/BCConjecture.html} \item \htmladdnormallink{projective object}{http://planetphysics.us/encyclopedia/ProjectiveObject.html} \item injective object \item derived functor \item derived category \item \htmladdnormallink{algebraic K-theory}{http://planetphysics.us/encyclopedia/PAdicMeasure.html} \item examples of algebraic K-theory \htmladdnormallink{groups}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} \item Grothendieck group \item \htmladdnormallink{Grothendieck category}{http://planetphysics.us/encyclopedia/GrothendieckCategory.html} \item delta functor \item horseshoe lemma \item syzygy \item Ext \item Tor \item projective dimension \item 5-lemma \item proof of 5-lemma \item 9-lemma \item snake lemma \item proof of snake lemma \item chain homotopy \item chain homotopy equivalence \item chain map \item homology of a chain complex \item Leray spectral sequence \item spectral sequence \end{enumerate}

\section{Sheaves, Topoi, Generalizations} \begin{enumerate} \item \htmladdnormallink{presheaf}{http://planetphysics.us/encyclopedia/CommutativeRingWithUnit.html} \item sheaf \item sheafification \item presheaf of a topological basis \item stalk \item \'Etal\'e space \item resolution of a sheaf \item gerbes \item site \item small site on a scheme \item topos \item cosmos \item subobject classifier \item well-pointed topos \item \htmladdnormallink{power}{http://planetphysics.us/encyclopedia/Power.html} object \item natural numbers object \item Cartesian closed category \item exponential object \end{enumerate}

\subsection{Examples of Categories} \begin{enumerate} \item discrete category \item category example (arrow category) \item category associated to a partial order \item category of \htmladdnormallink{matrices}{http://planetphysics.us/encyclopedia/Matrix.html} \item Category of pseudomorphisms \item Category of intermorphisms \item examples of initial objects and terminal objects and zero objects \item category of sets \item monomorphisms of category of sets \item \htmladdnormallink{monoid}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} as a category \item comma category \item category of pointed topological spaces \item simplicial category \end{enumerate}

\subsection{Algebraic Categories} \begin{enumerate} \item \htmladdnormallink{algebra formed from a categories}{http://planetphysics.us/encyclopedia/AlgebraFormedFromACategory.html} \item \htmladdnormallink{monad}{http://planetphysics.us/encyclopedia/CategoricalGroupRepresentation.html} \item comonad \item monoidal category \item group object \item nerve \end{enumerate}

\subsection{Generalized Categorical Galois theory} \subsubsection{Categorical Galois theory and and Topological greoupoid category} \begin{itemize} \item category of \htmladdnormallink{topological groupoids}{http://planetphysics.us/encyclopedia/GroupoidHomomorphism2.html} \item \htmladdnormallink{fundamental groupoid functors}{http://planetphysics.us/encyclopedia/QuantumFundamentalGroupoid3.html} \end{itemize}

\subsection{Nonabelian Algebraic Topology (NAAT)} \subsection{Nonabelian Quantum Algebraic Topology (NA-QAT)} \begin{itemize} \item \emph{\htmladdnormallink{Clifford algebras}{http://planetphysics.us/encyclopedia/CliffordAlgebra.html}} \item \emph{\htmladdnormallink{Hopf algebras}{http://planetphysics.us/encyclopedia/QuantumGroup4.html}} \item \emph{Weak-Hopf algebras} \item \emph{Grasssmann-Hopf algebras} \item Hamiltonian algebroids \item \htmladdnormallink{Lie algebroids}{http://planetphysics.us/encyclopedia/LieAlgebroids.html} \item \htmladdnormallink{quantum algebroids}{http://planetphysics.us/encyclopedia/LongRangeCoupling.html} \item Quantum Groups and Quantum Compact Groups (\htmladdnormallink{QCGs}{http://planetphysics.us/encyclopedia/QuantumCompactGroupoids.html}) \item Quantum Groupoids \end{itemize} \subsubsection{Noncommutative Geometry (non-commutative geometry)}

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