topics in algebraic topology

1 Algebraic topology topics

1.1 Introduction

Algebraic topology (AT) utilizes algebraic approaches to solve topological problems,such as the classification of surfaces, proving duality theorems for manifolds andapproximation theorems for topological spaces. A central problem in algebraic topologyis to find algebraic invariants of topological spaces, which is usually carried out by meansof homotopy, homology and cohomology groups. There are close connections between algebraic topology,Algebraic Geometry (AG) (http://planetmath.org/AlgebraicGeometry), and Non-commutative Geometry/NAAT. On the other hand, there are also close ties between algebraic geometry and numbertheory.

1.2 Outline

1. Homotopy theory and fundamental groups

2. Topology and groupoids; van Kampen theorem (http://planetmath.org/VanKampensTheorem)

3. Homology and cohomology theories

4. Duality

5. Category theory applications in algebraic topology

6. Index of categories, functors and natural transformations

7. http://www.uclouvain.be/17501.htmlGrothendieck’s Descent theory

8. ‘Anabelian geometry’

9. Categorical Galois theory

10. Higher dimensional algebra (HDA)

11. Quantum algebraic topology (QAT)

12. Quantum Geometry

13. Non-Abelian algebraic topology (NAAT)

1.3 Homotopy theory and fundamental groups

1. Homotopy

2. Fundamental group of a space

3. Fundamental theorems

4. van Kampen theorem

5. Whitehead groups, torsion and towers

6. Postnikov towers

1.4 Topology and Groupoids

1. Topology definition, axioms and basic concepts

2. Fundamental groupoid

3. Topological groupoid

4. van Kampen theorem for groupoids

5. Groupoid pushout theorem

6. Double groupoids and crossed modules

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1.5 Homology theory

1. Homology group

2. Homology sequence

3. Homology complex

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1.6 Cohomology theory

1. Cohomology group

2. Cohomology sequence

3. DeRham cohomology

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1.7 Duality in algebraic topology and category theory

1. Tanaka-Krein duality

2. Grothendieck duality

3. Categorical duality

4. Tangled duality

5. DA5

6. DA6

7. DA7

1.8 Category theory applications

1. Abelian categories

2. Topological category

3. Fundamental groupoid functor

4. Categorical Galois theory

5. Non-Abelian algebraic topology

6. Group category

7. Groupoid category

8. $𝒯op$ category

9. Topos and topoi axioms

10. Generalized toposes

11. Categorical logic and algebraic topology

12. Meta-theorems

13. Duality between spaces and algebras

1.9 Index of categories

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

1. http://www.uclouvain.be/17501.htmlAlgebraic categories

2. Topological category

3. Category of sets, Set

4. Category of topological spaces

5. Category of Riemannian manifolds

6. Category of CW-complexes

7. Category of Hausdorff spaces

8. Category of Borel spaces

9. Category of CR-complexes

10. Category of graphs

11. Category of spin networks

12. Category of groups

13. Galois category

14. Category of fundamental groups

15. Category of Polish groups

16. Groupoid category

17. Category of groupoids (or groupoid category)

18. Category of Borel groupoids

19. Category of fundamental groupoids

20. Category of functors (or functor category)

21. Double groupoid category

22. Double category

23. Category of Hilbert spaces

24. Category of quantum automata

25. R-category

26. Category of algebroids

27. Category of double algebroids

28. Category of dynamical systems

1.10 Index of functors

The following is a contributed listing of functors:

1. Covariant functors

2. Contravariant functors

3. Adjoint functors

4. Preadditive functors

5. Additive functor

6. Representable functors

7. Fundamental groupoid functor

8. Forgetful functors

9. Grothendieck group functor

10. Exact functor

11. Multi-functor

12. Section functors

13. NT2

14. NT3

1.11 Index of natural transformations

The following is a contributed listing of natural transformations:

1. Natural equivalence

2. Natural transformations in a 2-category

3. NT3

4. NT1

5. NT2

6. NT3

1.12 Grothendieck proposals

1. Esquisse d’un Programme (http://planetmath.org/AlexSMathematicalHeritageEsquisseDunProgramme)

2. http://www.math.jussieu.fr/ leila/grothendieckcircle/stacks.psPursuing Stacks

3. S2

4. S3

5. S4

1.13 Descent theory

1. D1

2. D2

3. D3

4. D4

1.14 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. Super-category

9. weak n-categories

10. Bi-dimensional Geometry

11. Noncommutative geometry (http://planetmath.org/NoncommutativeGeometry)

12. Higher-Homotopy theories

13. Higher-Homotopy Generalized van Kampen Theorem (HGvKT) (http://planetmath.org/GeneralizedVanKampenTheoremsHigherDimensional)

14. H1

15. H2

16. H3

17. H4

1.14.1 Axioms of cohomology theory

1. A1

2. A2

3. A3

4. A4

5. A5

6. A6

7. A7

1.14.2 Axioms of homology theory

1. A1

2. A2

3. A3

4. A4

5. A5

6. A6

1.15 Quantum algebraic topology (QAT)

(a). Quantum algebraic topology is described as the mathematical and physical study of general theories of quantum algebraic structures from the standpoint of algebraic topology, category theory andtheir non-Abelian extensions in higher dimensional algebra and supercategories

1. Quantum operator algebras (such as: involution, *-algebras, or $*$-algebras, von Neumann algebras,, JB- and JL- algebras, $C^{*}$ - or C*- algebras,

2. Quantum von Neumann algebra and subfactors; Jone’s towers and subfactors

3. Kac-Moody and K-algebras

4. categorical groups

5. Hopf algebras, quantum Groups and quantum group algebras

6. Quantum groupoids and weak Hopf $C^{*}$-algebras

7. Groupoid C*-convolution algebras and *-convolution algebroids

8. Quantum spacetimes and quantum fundamental groupoids

9. Quantum double Algebras

10. Quantum gravity, supersymmetries, supergravity, superalgebras and graded ‘Lie’ algebras

11. Quantum categorical algebra and higher–dimensional, $Ł-M_n$- Toposes

12. Quantum R-categories, R-supercategories and spontaneous symmetry breaking

13. Non-Abelian Quantum algebraic topology (NA-QAT): closely related to NAAT and HDA.

1.16 Quantum Geometry

1. Quantum Geometry overview (http://planetmath.org/QuantumGeometry2)

2. Quantum non-commutative geometry

1.17 Non-Abelian Algebraic Topology (NAAT)

1. Non-Abelian categories

2. Non-commutative groupoids (including non-Abelian groups)

3. Generalized van Kampen theorems (http://planetmath.org/GeneralizedVanKampenTheoremsHigherDimensional)

4. Noncommutative Geometry (NCG) (http://planetmath.org/NoncommutativeGeometry)

5. Non-commutative ‘spaces’ of functions

6. http://planetphysics.org/encyclopedia/NonAbelianAlgebraicTopology5.htmlnon-Abelian Algebraic Topology

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1.21 References

http://planetmath.org/?op=getobj&from=objects&id=10746Bibliography on Category theory, AT and QAT

1.21.1 Textbooks and Expositions:

1. A http://planetmath.org/?op=getobj&from=books&id=172Textbook1

2. A http://planetmath.org/?op=getobj&from=books&id=156Textbook2

3. A http://planetmath.org/?op=getobj&from=books&id=159Textbook3

4. A http://planetmath.org/?op=getobj&from=books&id=160Textbook4

5. A http://planetmath.org/?op=getobj&from=books&id=153Textbook5

6. A http://planetmath.org/?op=getobj&from=lec&id=68Textbook6

7. A http://planetmath.org/?op=getobj&from=books&id=158Textbook7

8. A http://planetmath.org/?op=getobj&from=lec&id=75Textbook8

9. A http://planetmath.org/?op=getobj&from=lec&id=73Textbook9

10. A http://planetmath.org/?op=getobj&from=books&id=174Textbook10

11. A http://planetmath.org/?op=getobj&from=books&id=169Textbook11

12. A http://planetmath.org/?op=getobj&from=books&id=178Textbook12

13. A http://www.math.cornell.edu/ hatcher/VBKT/VB.pdfTextbook13

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