topics in algebraic topology

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.

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.html

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)

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

7.

new4

1.5

Homology theory

1.

Homology group

2.

Homology

sequence

3.

Homology complex

4.

new4

1.6

Cohomology theory

1.

Cohomology group

2.

Cohomology sequence

3.

DeRham cohomology

4.

new4

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.

𝒯

⁢

o

⁢

p

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.html

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

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.ps

Pursuing 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 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,

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.html

non-Abelian Algebraic Topology

1.18

12

1.

new1

2.

new2

3.

new3

4.

new4

1.19

13

1.

new1

2.

new2

3.

new3

4.

new4

1.20

14

1.21

References

http://planetmath.org/?op=getobj&from=objects&id=10746

Bibliography on Category theory, AT and QAT

1.21.1

Textbooks and Expositions:

1.

A

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

Textbook1

2.

A

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

Textbook2

3.

A

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

Textbook3

4.

A

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

Textbook4

5.

A

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

Textbook5

6.

A

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

Textbook6

7.

A

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

Textbook7

8.

A

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

Textbook8

9.

A

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

Textbook9

10.

A

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

Textbook10

11.

A

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

Textbook11

12.

A

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

Textbook12

13.

A

http://www.math.cornell.edu/ hatcher/VBKT/VB.pdf

Textbook13

14.

new1

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

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AlgebraicGeometry

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CategoricalOntologyABibliographyOfCategoryTheory

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OverviewOfTheContentOfPlanetMath

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