# PlanetPhysics/Index of Algebraic Geometry

This is a contributed entry in progress

## Index of Algebraic Geometry

Algebraic Geometry (AG), and non-commutative geometry/. On the other hand, there are also close ties between algebraic geometry and number theory.

### Disciplines in algebraic geometry

* Birational geometry, Dedekind \htmladdnormallink{domains {http://planetphysics.us/encyclopedia/Bijective.html} and Riemann-Roch theorem}
* Homology and cohomology theories
* Algebraic groups: Lie groups, matrix group schemes,group machines, linear groups, generalizing Lie groups, representation theory
* Abelian varieties
* Arithmetic algebraic geometry
* duality #category theory applications in algebraic geometry
* indexes of category, functors and natural transformations
* Grothendieck's Descent theory
* `Anabelian Geometry' #Categorical Galois theory
* higher dimensional algebra (HDA)
* Quantum Algebraic Topology (QAT)
* Quantum Geometry
* computer algebra systems; an example is: explicit projective resolutions for finitely-generated modules over suitable rings

### Cohomology

Cohomology is an essential theory in the study of complex manifolds. computations in cohomology studies of complex manifolds in algebraic geometry utilize similar computations to those of cohomology theory in algebraic topology: spectral sequences, excision, the Mayer-Vietoris sequence, etc.
* cohomology groups are defined and then cohomology functors associate Abelian groups to sheaves on a scheme; one may view such Abelian groups them as cohomology with coefficients in a scheme.
* Cohomology functors
* fundamental cohomology theorems
* A basic type of cohomology for schemes is the sheaf cohomology
* Whitehead groups, torsion and towers
* xyz

### Seminars on Algebraic Geometry and Topos Theory (SGA)

* SGA1
* SGA2
* SGA3
* SGA4
* SGA5
* SGA6
* SGA7

### Algebraic varieties and the GAGA principle

* new1x
* new2y
* new3z

### Cohomology theory

* Cohomology group
* Cohomology sequence
* DeRham cohomology
* new4

### Homology theory

* homology group #Homology sequence
* Homology complex
* 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

### Examples 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

### Grothendieck proposals

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

### Descent theory

* D1
* D2
* D3

### Higher Dimensional Algebraic Geometry (HDAG)

* Categorical groups and supergroup algebras
* Double groupoid varieties
* Double algebroids
* Bi-algebroids
* $R$-algebroid
* $2$-category
* $n$-category
* super-category #weak n-categories of algebraic varieties
* Bi-dimensional Algebraic Geometry
* Anabelian Geometry
* Noncommutative geometry
* Higher-homology/cohomology theories
* H1
* H2
* H3
* H4

#### Axioms of cohomology theory

* A1
* A2
* A3

#### Axioms of homology theory

* A1
* A2
* A3

### Quantum algebraic topology (QAT)

(a). Quantum algebraic topology is described as the mathematical and physical study of \htmladdnormallink{general theories {http://planetphysics.us/encyclopedia/GeneralTheory.html} of quantum algebraic structures from the standpoint of algebraic topology, category theory and their non-Abelian extensions in higher dimensional algebra and supercategories}
* quantum operator algebras (such as: involution, *-algebras, or $*$-algebras, von Neumann algebras,
, JB- and JL- algebras, $C^{*}$ - or C*- algebras,
* Quantum von Neumann algebra and subfactors; Jone's towers and subfactors
* Kac-Moody and K-algebras
* categorical groups
* Hopf algebras, quantum Groups and quantum group algebras
* quantum groupoids and weak Hopf $C^{*}$-algebras
* groupoid C*-convolution algebras and *-convolution algebroids
* quantum spacetimes and quantum fundamental groupoids
* Quantum double Algebras
* quantum gravity, supersymmetries, supergravity, superalgebras and graded `Lie' algebras #Quantum categorical algebra and higher--dimensional, Failed to parse (unknown function "\L"): {\displaystyle \L-M_n} - Toposes
* Quantum R-categories, R-supercategories and spontaneous symmetry breaking #Non-Abelian Quantum Algebraic Topology (NA-QAT): closely related to NAAT and HDA.

### Quantum Geometry

* Quantum Geometry overview
* Quantum non-commutative geometry

### 2x

* new1x
* new2y

### 13

* new1x
* new2y

### Textbooks and bibliograpies

Bibliography on Category theory, AT and QAT

#### Textbooks and Expositions:

* A Textbook1
* A Textbook2
* A Textbook3
* A Textbook4
* A Textbook5
* A Textbook6
* A Textbook7
* A Textbook8
* A Textbook9
* A Textbook10
* A Textbook11
* A Textbook12
* A Textbook13
* new1x