# 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 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 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 number theory.

### 1.2 Outline

1. 1.
2. 2.

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

3. 3.

Homology and cohomology theories

4. 4.

Duality

5. 5.

Category theory applications in algebraic topology

6. 6.
7. 7.

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

8. 8.

‘Anabelian geometry’

9. 9.

Categorical Galois theory

10. 10.
11. 11.

Quantum algebraic topology (QAT)

12. 12.
13. 13.

Non-Abelian algebraic topology (NAAT)

### 1.3 Homotopy theory and fundamental groups

1. 1.

Homotopy

2. 2.

Fundamental group of a space

3. 3.

Fundamental theorems

4. 4.

van Kampen theorem

5. 5.

6. 6.

Postnikov towers

### 1.4 Topology and Groupoids

1. 1.

Topology definition, axioms and basic concepts

2. 2.
3. 3.
4. 4.

van Kampen theorem for groupoids

5. 5.

Groupoid pushout theorem

6. 6.

Double groupoids and crossed modules

7. 7.

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### 1.5 Homology theory

1. 1.
2. 2.

Homology sequence

3. 3.

Homology complex

4. 4.

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### 1.6 Cohomology theory

1. 1.

Cohomology group

2. 2.

Cohomology sequence

3. 3.

DeRham cohomology

4. 4.

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### 1.7 Duality in algebraic topology and category theory

1. 1.

Tanaka-Krein duality

2. 2.

Grothendieck duality

3. 3.
4. 4.

Tangled duality

5. 5.

DA5

6. 6.

DA6

7. 7.

DA7

### 1.8 Category theory applications

1. 1.
2. 2.

Topological category

3. 3.

Fundamental groupoid functor

4. 4.

Categorical Galois theory

5. 5.

Non-Abelian algebraic topology

6. 6.

Group category

7. 7.

Groupoid category

8. 8.

$\mathcal{T}op$ category

9. 9.

Topos and topoi axioms

10. 10.

Generalized toposes

11. 11.

Categorical logic and algebraic topology

12. 12.
13. 13.

Duality between spaces and algebras

### 1.9 Index of categories

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

1. 1.

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

2. 2.

Topological category

3. 3.
4. 4.

Category of topological spaces

5. 5.

Category of Riemannian manifolds

6. 6.

Category of CW-complexes

7. 7.

Category of Hausdorff spaces

8. 8.
9. 9.

Category of CR-complexes

10. 10.

Category of graphs

11. 11.

Category of spin networks

12. 12.

Category of groups

13. 13.

Galois category

14. 14.

Category of fundamental groups

15. 15.

Category of Polish groups

16. 16.

Groupoid category

17. 17.

Category of groupoids (or groupoid category)

18. 18.
19. 19.

Category of fundamental groupoids

20. 20.

Category of functors (or functor category)

21. 21.

Double groupoid category

22. 22.
23. 23.

Category of Hilbert spaces

24. 24.

Category of quantum automata

25. 25.
26. 26.

Category of algebroids

27. 27.

Category of double algebroids

28. 28.

Category of dynamical systems

### 1.10 Index of functors

The following is a contributed listing of functors:

1. 1.
2. 2.
3. 3.
4. 4.
5. 5.
6. 6.
7. 7.

Fundamental groupoid functor

8. 8.
9. 9.
10. 10.
11. 11.

Multi-functor

12. 12.
13. 13.

NT2

14. 14.

NT3

### 1.11 Index of natural transformations

The following is a contributed listing of natural transformations:

1. 1.
2. 2.

Natural transformations in a 2-category

3. 3.

NT3

4. 4.

NT1

5. 5.

NT2

6. 6.

NT3

### 1.12 Grothendieck proposals

1. 1.

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

2. 2.

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

3. 3.

S2

4. 4.

S3

5. 5.

S4

1. 1.

D1

2. 2.

D2

3. 3.

D3

4. 4.

D4

### 1.14 Higher dimensional algebra (HDA)

1. 1.

Categorical groups

2. 2.

Double groupoids

3. 3.

Double algebroids

4. 4.

Bi-algebroids

5. 5.

$R$-algebroid

6. 6.

$2$-category

7. 7.

$n$-category

8. 8.
9. 9.

weak n-categories

10. 10.

Bi-dimensional Geometry

11. 11.

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

12. 12.

Higher-Homotopy theories

13. 13.

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

14. 14.

H1

15. 15.

H2

16. 16.

H3

17. 17.

H4

1. 1.

A1

2. 2.

A2

3. 3.

A3

4. 4.

A4

5. 5.

A5

6. 6.

A6

7. 7.

A7

1. 1.

A1

2. 2.

A2

3. 3.

A3

4. 4.

A4

5. 5.

A5

6. 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 and their non-Abelian extensions in higher dimensional algebra and supercategories

1. 1.

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

2. 2.

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

3. 3.

Kac-Moody and K-algebras

4. 4.

categorical groups

5. 5.

Hopf algebras, quantum Groups and quantum group algebras

6. 6.

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

7. 7.

Groupoid C*-convolution algebras and *-convolution algebroids

8. 8.

Quantum spacetimes and quantum fundamental groupoids

9. 9.

Quantum double Algebras

10. 10.

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

11. 11.

Quantum categorical algebra and higher–dimensional, $\L{}-M_{n}$- Toposes

12. 12.

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

13. 13.

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

### 1.16 Quantum Geometry

1. 1.

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

2. 2.

Quantum non-commutative geometry

### 1.17 Non-Abelian Algebraic Topology (NAAT)

1. 1.

Non-Abelian categories

2. 2.

Non-commutative groupoids (including non-Abelian groups)

3. 3.

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

4. 4.

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

5. 5.

Non-commutative ‘spaces’ of functions

6. 6.

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

1. 1.

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2. 2.

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3. 3.

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4. 4.

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1. 1.

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2. 2.

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3. 3.

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4. 4.

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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. 1.

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

2. 2.

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

3. 3.

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

4. 4.

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

5. 5.

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

6. 6.

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

7. 7.

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

8. 8.

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

9. 9.

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

10. 10.

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

11. 11.

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

12. 12.

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

13. 13.

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

14. 14.

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 Title topics in algebraic topology Canonical name TopicsInAlgebraicTopology Date of creation 2013-03-22 18:23:42 Last modified on 2013-03-22 18:23:42 Owner bci1 (20947) Last modified by bci1 (20947) Numerical id 82 Author bci1 (20947) Entry type Topic Classification msc 57R19 Classification msc 57N65 Classification msc 11F23 Classification msc 11E72 Classification msc 18-00 Classification msc 55N30 Classification msc 55N15 Classification msc 55N99 Classification msc 55N40 Classification msc 55N20 Classification msc 55-01 Synonym category theory Synonym algebraic geometry Synonym topology and groupoids Related topic Category Related topic Functor Related topic GrothendieckGroup Related topic NoncommutativeGeometry Related topic TopologicalGroup2 Related topic Naturaltransformation Related topic AlgebraicTopology Related topic AlgebraicGeometry Related topic CategoricalOntologyABibliographyOfCategoryTheory Related topic IndexOfCategories Related topic OverviewOfTheContentOfPlanetMath Related topic CategoryOf