# PlanetPhysics/Algebraic Topology
## Algebraic topology
### 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), non-commutative geometry and, of course, its most recent development-- non-Abelian Algebraic Topology (NAAT). On the other hand, there are also close ties between algebraic geometry and number theory.

### Outline

* Homotopy theory and fundamental groups #Topology and groupoids; van Kampen theorem
* Homology and cohomology theories
* Duality
* category theory applications in algebraic topology
* indexes of category, functors and natural transformations
* Grothendieck's Descent theory
* `Anabelian Geometry' #Categorical Galois theory
* higher dimensional algebra (HDA)
* Non-Abelian Quantum Algebraic Topology (NAQAT)
* Quantum Geometry
* Non-Abelian algebraic topology (NAAT)

### Homotopy theory and fundamental groups

* Homotopy
* Fundamental group of a space
* Fundamental theorems
* Van Kampen theorem #Whitehead groups, torsion and towers
* Postnikov towers

### Topology and Groupoids

* Topology definition, axioms and basic concepts #Fundamental groupoid
* topological groupoid #van Kampen theorem for groupoids
* Groupoid pushout theorem
* double groupoids and crossed modules
* new4

### Homology theory

* homology group #Homology sequence
* Homology complex
* new4

### Cohomology theory

* Cohomology group
* Cohomology sequence
* DeRham cohomology
* new4

### Duality in algebraic topology and category theory

* Tanaka-Krein duality
* Grothendieck duality
* categorical duality #tangled duality #DA5
* DA6
* DA7

### Category theory applications

* abelian categories
* Topological category #Fundamental groupoid functor
* Categorical Galois theory
* Non-Abelian algebraic topology
* Group category
* groupoid category #${\mathcal {T}}op$ category
* topos and topoi axioms
* generalized toposes #Categorical logic and algebraic topology
* meta-theorems #Duality between spaces and algebras

### Index of categories

The following is a listing of categories relevant to algebraic topology:
* Algebraic categories
* Topological category
* Category of sets, Set
* Category of topological spaces
* category of Riemannian manifolds #Category of CW-complexes
* Category of Hausdorff spaces
* category of Borel spaces #Category of CR-complexes
* Category of graphs #Category of spin networks #Category of groups
* Galois category
* Category of fundamental groups
* Category of Polish groups
* Groupoid category
* category of groupoids (or groupoid category)
* category of Borel groupoids #Category of fundamental groupoids
* Category of functors (or functor category)
* Double groupoid category
* double category #category of Hilbert spaces #category of quantum automata #R-category #Category of algebroids #Category of double algebroids
* Category of dynamical systems

### Index of functors

The following is a contributed listing of functors:
* Covariant functors
* Contravariant functors
* adjoint functors
* preadditive functors
* Additive functor
* representable functors
* Fundamental groupoid functor
* Forgetful functors
* Grothendieck group functor
* Exact functor
* Multi-functor
* section functors
* NT2
* NT3

### Index of natural transformations

The following is a contributed listing of natural transformations:
* natural equivalence #Natural transformations in a 2-category #NT3
* NT1
* NT2
* NT3

### Grothendieck proposals

* Esquisse d'un Programme
\item Pursuing Stacks
* S2
* S3
* S4

### Descent theory

* D1
* D2
* D3
* D4

### Higher dimensional algebra (HDA)

* Categorical groups
* Double groupoids
* Double algebroids
* Bi-algebroids
* $R$-algebroid
* $2$-category
* $n$-category
* super-category #weak n-categories #Bi-dimensional Geometry
* Noncommutative geometry
* Higher-Homotopy theories
* Higher-Homotopy Generalized van Kampen Theorem (HGvKT)
* H1
* H2
* H3
* H4

#### Axioms of cohomology theory

* A1
* A2
* A3
* A4
* A5
* A6
* A7

#### Axioms of homology theory

* A1
* A2
* A3
* A4
* A5
* A6

### Non-Abelian Algebraic Topology (NAAT)

* An overview of Nonabelian Algebraic Topology
* non-Abelian categories
* non-commutative groupoids (including non-Abelian groups)
* Generalized van Kampen theorems
* Noncommutative Geometry (NCG)
* Non-commutative `spaces' of functions #Non-Abelian Algebraic Topology textbook

#### References for NAAT

* [1] M. Alp and C. D. Wensley, XMod, Crossed modules and Cat1--groups: a GAP4 package,(2004) (http://www.maths.bangor.ac.uk/chda/)
* [2] R. Brown, Elements of Modern Topology, McGraw Hill, Maidenhead, 1968. second edition as Topology: a geometric account of general topology, homotopy types, and the fundamental groupoid, Ellis Horwood, Chichester (1988) 460 pp.
* [3] R. Brown, `Higher dimensional group theory'
* [4] R. Brown.`crossed complexes and homotopy groupoids as non commutative tools for higher dimensional local--to--global problems', Proceedings of the fields Institute Workshop on Categorical Structures for Descent and Galois Theory, Hopf Algebras and Semiabelian Categories, September 23--28, 2002, Contemp. Math. (2004). (to appear), UWB Math Preprint
02.26.pdf (30 pp.)
* [5] R. Brown and P. J. Higgins, On the connection between the second relative homotopy groups of some related spaces, Proc.London Math. Soc., (3) 36 (1978) 193--212.
* [6] R. Brown and R. Sivera, `Nonabelian algebraic topology', (in preparation) Part I is downloadable from
(http://www.bangor.ac.uk/~mas010/nonab-a-t.html)
* [7] R. Brown and C. B. Spencer, Double groupoids and crossed modules, Cahiers Top. G'/eom.Diff., 17 (1976) 343--362.
* [8] R. Brown and C. D.Wensley, `computation and homotopical applications of induced crossed modules', J. Symbolic Computation, 35 (2003) 59--72.
* [9] The GAP Group, 2004, GAP --Groups, algorithms, and programming, version 4.4 , Technical report, (http://www.gap-system.org)
* [10] A. Grothendieck, `Pursuing stacks', 600p, 1983, distributed from Bangor. Now being edited by G. Maltsiniotis for the SMF.
* [11] P. J. Higgins, 1971, Categories and Groupoids,
Van Nostrand, New York. Reprint Series, Theory and Appl. Categories (to appear).
* [12] V. Sharko, 1993, Functions on manifolds: algebraic and topological aspects, number 131 in Translations of Mathematical Monographs, American Mathematical Society.
* new1
* new2
* new3
* new4

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* new1
* new2
* new3
* new4