PlanetPhysics/Topics in Algebraic Topology

Algebraic topology topics

Introduction

[[../ModuleAlgebraic/|algebraic topology]] (AT) utilizes [[../CoIntersections/|algebraic]] approaches to solve [[../CoIntersections/|topological]] problems, such as the [[../TrivialGroupoid/|classification]] of surfaces, proving [[../TrivialGroupoid/|duality]] [[../Formula/|theorems]] for [[../NoncommutativeGeometry4/|manifolds]] 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 [[../ThinEquivalence/|homotopy]], homology and [[../CohomologyTheoryOnCWComplexes/|cohomology groups]]. There are close connections between algebraic topology, Algebraic Geometry (AG), and [[../NAQAT2/|non-commutative geometry]] / NAAT. On the other hand, there are also close ties between algebraic geometry and number theory.

Background

Latin quote: ""Non multa sed multum

Outline

1. [[../CubicalHigherHomotopyGroupoid/|homotopy theory]] and [[../HomotopyCategory/|fundamental groups]] #Topology and [[../QuantumOperatorAlgebra5/|groupoids]]; van Kampen theorem
2. Homology and [[../NoncommutativeGeometry4/|cohomology theories]]
3. Duality
4. [[../CategoricalOntology/|category theory applications]] in algebraic topology
5. [[../IndexOfCategories/|indexes of category]], [[../TrivialGroupoid/|functors]] and [[../VariableCategory2/|natural transformations]]
6. Grothendieck's Descent theory
7. `[[../IsomorphismClass/|Anabelian Geometry]]' #Categorical Galois theory
8. [[../InfinityGroupoid/|higher dimensional algebra]] ([[../2Groupoid2/|HDA]])
9. [[../TriangulationMethodsForQuantizedSpacetimes2/|Quantum Algebraic Topology]] ([[../QuantumOperatorAlgebra5/|QAT]])
10. Quantum Geometry
11. Non-Abelian algebraic topology (NAAT)

Homotopy theory and fundamental groups

1. Homotopy
2. [[../SingularComplexOfASpace/|fundamental group]] of a space
3. Fundamental theorems
4. [[../VanKampenTheorems/|Van Kampen theorem]] #Whitehead [[../TrivialGroupoid/|groups]], torsion and towers
5. Postnikov towers

Topology and Groupoids

1. Topology definition, axioms and basic [[../PreciseIdea/|concepts]] #[[../CubicalHigherHomotopyGroupoid/|fundamental groupoid]] #[[../GroupoidHomomorphism2/|topological groupoid]] #van Kampen theorem for groupoids
2. Groupoid [[../Pushout/|pushout]] theorem
3. [[../WeakHomotopy/|double groupoids]] and [[../CubicalHigherHomotopyGroupoid/|crossed modules]]
4. new4

Homology theory

1. [[../ExtendedHurewiczFundamentalTheorem/|homology group]] #Homology sequence
2. Homology complex
3. new4

Cohomology theory

1. Cohomology group
2. Cohomology sequence
3. DeRham cohomology
4. new4

Duality in algebraic topology and category theory

1. Tanaka-Krein duality
2. Grothendieck duality
3. [[../TrivialGroupoid/|categorical duality]] #[[../DualityAndTriality/|tangled duality]] #DA5
4. DA6
5. DA7

Category theory applications

1. [[../AbelianCategory2/|abelian categories]]
2. Topological [[../Cod/|category]] #[[../QuantumFundamentalGroupoid/|fundamental groupoid functor]] #Categorical Galois theory
3. Non-Abelian algebraic topology
4. Group category
5. [[../GroupoidCategory3/|groupoid category]] #$𝒯op$ category
6. [[../GrothendieckTopos/|topos]] and topoi axioms
7. [[../ManyValuedLogicSubobjectClassifiers/|generalized toposes]] #Categorical logic and algebraic topology
8. [[../MetaTheorems/|meta-theorems]] #Duality between spaces and algebras

Index of categories

The following is a listing of categories relevant to algebraic topology:

1. Algebraic categories
2. Topological category
3. Category of sets, Set
4. Category of topological spaces
5. [[../CategoryOfRiemannianManifolds/|category of Riemannian manifolds]] #Category of CW-complexes
6. Category of Hausdorff spaces
7. [[../CategoryOfBorelSpaces/|category of Borel spaces]] #Category of CR-complexes
8. Category of [[../Cod/|graphs]] #Category of [[../SimplicialCWComplex/|spin networks]] #Category of groups
9. Galois category
10. Category of fundamental groups
11. Category of [[../InvariantBorelSet/|Polish groups]]
12. Groupoid category
13. [[../GroupoidCategory/|category of groupoids]] (or groupoid category)
14. [[../CategoryOfBorelGroupoids/|category of Borel groupoids]] #Category of fundamental groupoids
15. Category of functors (or [[../TrivialGroupoid/|functor category]])
16. Double groupoid category
17. [[../HorizontalIdentities/|double category]] #[[../CategoryOfHilbertSpaces/|category of Hilbert spaces]] #[[../CategoryOfQuantumAutomata/|category of quantum automata]] #[[../RCategory/|R-category]] #Category of [[../Algebroids/|algebroids]] #Category of [[../GeneralizedSuperalgebras/|double algebroids]]
18. Category of [[../ContinuousGroupoidHomomorphism/|dynamical systems]]

Index of functors

The following is a contributed listing of functors:

1. Covariant functors
2. Contravariant functors
3. [[../SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence/|adjoint functors]]
4. [[../PreadditiveFunctor/|preadditive functors]]
5. Additive functor
6. [[../CategoryOfLogicAlgebras/|representable functors]]
7. Fundamental groupoid functor
8. Forgetful functors
9. Grothendieck group functor
10. Exact functor
11. Multi-functor
12. [[../RightAdjointFunctor/|section functors]]
13. NT2
14. NT3

Index of natural transformations

The following is a contributed listing of natural transformations:

1. [[../IsomorphismClass/|natural equivalence]] #Natural transformations in a [[../2Category/|2-category]] #NT3
2. NT1
3. NT2
4. NT3

Grothendieck proposals

1. Esquisse d'un Programme

\item Pursuing Stacks

1. S2
2. S3
3. S4

Descent theory

1. D1
2. D2
3. D3
4. D4

Higher dimensional algebra (HDA)

1. Categorical groups
2. Double groupoids
3. Double algebroids
4. Bi-algebroids
5. $R$-algebroid
6. $2$-category
7. $n$-category
8. [[../SuperCategory6/|super-category]] #weak [[../InfinityGroupoid/|n-categories]] #Bi-dimensional Geometry
9. [[../NoncommutativeGeometry/|Noncommutative geometry]]
10. Higher-Homotopy theories
11. Higher-Homotopy Generalized van Kampen Theorem (HGvKT)
12. H1
13. H2
14. H3
15. H4

Axioms of cohomology theory

1. A1
2. A2
3. A3
4. A4
5. A5
6. A6
7. A7

Axioms of homology theory

1. A1

1. A2
2. A3
3. A4
4. A5
5. A6

Quantum algebraic topology (QAT)

(a). Quantum algebraic topology is described as the mathematical and physical study of \htmladdnormallink{general theories {http://planetphysics.us/encyclopedia/GeneralTheory.html} of quantum [[../TrivialGroupoid/|algebraic structures]] from the standpoint of algebraic topology, [[../TrivialGroupoid/|category theory]] and their [[../AbelianCategory3/|non-Abelian]] extensions in higher dimensional algebra and [[../SuperCategory6/|supercategories]]}

1. [[../Groupoid/|quantum operator algebras]] (such as: involution, *-algebras, or $*$-algebras, [[../CoordinateSpace/|von Neumann algebras]],

, JB- and JL- algebras, $C^{*}$ - or C*- algebras,

1. Quantum von Neumann algebra and subfactors; Jone's towers and subfactors
2. Kac-Moody and K-algebras
3. categorical groups
4. [[../QuantumOperatorAlgebra5/|Hopf algebras]], quantum Groups and [[../QuantumGroup4/|quantum group]] algebras
5. [[../WeakHopfAlgebra/|quantum groupoids]] and weak Hopf $C^{*}$-algebras
6. [[../GroupoidCConvolutionAlgebra/|groupoid C*-convolution algebras]] and *-convolution algebroids
7. [[../NonAbelianQuantumAlgebraicTopology3/|quantum spacetimes]] and [[../QuantumFundamentalGroupoid4/|quantum fundamental groupoids]]
8. Quantum double Algebras
9. [[../LQG2/|quantum gravity]], [[../Supersymmetry/|supersymmetries]], [[../AntiCommutationRelations/|supergravity]], [[../MathematicalFoundationsOfQuantumTheories/|superalgebras]] and graded `[[../BilinearMap/|Lie' algebras]] #Quantum [[../CategoryOfLogicAlgebras/|categorical algebra]] and higher--dimensional, Failed to parse (unknown function "\L"): {\displaystyle \L{}-M_n} - Toposes
10. Quantum R-categories, [[../RDiagram/|R-supercategories]] and [[../LongRangeCoupling/|spontaneous symmetry breaking]] #[[../NonAbelianQuantumAlgebraicTopology3/|Non-Abelian Quantum Algebraic Topology]] (NA-QAT): closely related to NAAT and HDA.

Quantum Geometry

1. [[../QuantumGeometry2/|Quantum Geometry overview]]
2. Quantum non-commutative geometry

Non-Abelian Algebraic Topology (NAAT)

1. [[../AbelianCategory3/|non-Abelian categories]]
2. [[../AbelianCategory3/|non-commutative]] groupoids (including non-Abelian groups)
3. Generalized van Kampen theorems
4. [[../NoncommutativeGeometry/|Noncommutative Geometry (NCG)]]
5. Non-commutative `spaces' of [[../Bijective/|functions]] #new4

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1. new1

1. new2
2. new3
3. new4

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1. new1
2. new2
3. new3
4. new4

References

Bibliography on Category theory, AT and QAT

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