Algebraic topology topics

Introduction

Background

Latin quote: “Non multa sed multum”

Outline

1. homotopy theory and fundamental groups
2. Topology and groupoids; van Kampen theorem
3. Homology and cohomology theories
4. Duality
5. category theory applications in algebraic topology
6. indexes of category, functors and natural transformations
7. Grothendieck'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)

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

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
7. new4

Homology theory

1. homology group
2. Homology sequence
3. Homology complex
4. 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. categorical duality
4. tangled duality
5. DA5
6. DA6
7. DA7

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. category
9. topos and topoi axioms
10. generalized toposes
11. Categorical logic and algebraic topology
12. meta-theorems
13. Duality between spaces and algebras

Index of categories

1. Algebraic 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

Index 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

Index of natural transformations

1. Natural equivalence
2. Natural transformations in a 2-category
3. NT3
4. NT1
5. NT2
6. NT3

Grothendieck proposals

1. Esquisse d'un Programme
2. Pursuing Stacks
3. S2
4. S3
5. 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. -algebroid
6. -category
7. -category
8. super-category
9. weak n-categories
10. Bi-dimensional Geometry
11. Noncommutative geometry
12. Higher-Homotopy theories
13. Higher-Homotopy Generalized van Kampen Theorem (HGvKT)
14. H1
15. H2
16. H3
17. 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
2. A2
3. A3
4. A4
5. A5
6. A6

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 and their 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, - 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 -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, - 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.

Quantum Geometry

1. Quantum Geometry overview
2. Quantum non-commutative geometry

Non-Abelian Algebraic Topology (NAAT)

1. non-Abelian categories
2. non-commutative groupoids (including non-Abelian groups)
3. Generalized van Kampen theorems
4. Noncommutative Geometry (NCG)
5. Non-commutative `spaces' of functions
6. new4

12

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

13

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

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References

Bibliography on Category theory, AT and QAT

Textbooks and Expositions:

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