topics in algebraic topology
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), and non-commutative geometry/. On the other hand, there are also close ties between algebraic geometry and number
theory.
Latin quote: “Non multa sed multum”
- 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)
- Quantum Algebraic Topology
(QAT)
- Quantum Geometry
- Non-Abelian algebraic topology (NAAT)
- Homotopy
- fundamental group
of a space
- Fundamental theorems
- van Kampen theorem
- Whitehead groups, torsion and towers
- Postnikov towers
- Topology definition, axioms and basic concepts
- fundamental groupoid
- topological groupoid
- van Kampen theorem for groupoids
- Groupoid pushout theorem
- double groupoids
and crossed modules
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- homology group
- Homology sequence
- Homology complex
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- Cohomology group
- Cohomology sequence
- DeRham cohomology
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- Tanaka-Krein duality
- Grothendieck duality
- categorical duality
- tangled duality
- DA5
- DA6
- DA7
- abelian categories
- Topological category
- fundamental groupoid functor
- Categorical Galois theory
- Non-Abelian algebraic topology
- Group category
- groupoid category
-
category
- topos
and topoi axioms
- generalized toposes
- Categorical logic and algebraic topology
- meta-theorems
- Duality between spaces and algebras
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
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
The following is a contributed listing of natural transformations:
- Natural equivalence
- Natural transformations in a 2-category
- NT3
- NT1
- NT2
- NT3
- Esquisse d'un Programme
- Pursuing Stacks
- S2
- S3
- S4
- D1
- D2
- D3
- D4
- Categorical groups
- Double groupoids
- Double algebroids
- Bi-algebroids
-algebroid
-category
-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
- A1
- A2
- A3
- A4
- A5
- A6
- A7
- A1
- A2
- A3
- A4
- A5
- A6
(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
- quantum operator algebras
(such as: involution, *-algebras, or
-algebras, von Neumann algebras,
, JB- and JL- algebras,
- 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
-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,
- 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 overview
- Quantum non-commutative geometry
- non-Abelian categories
- non-commutative
groupoids (including non-Abelian groups)
- Generalized van Kampen theorems
- Noncommutative Geometry (NCG)
- Non-commutative `spaces' of functions
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Bibliography on Category theory, AT and QAT
- 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
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Contributors to this entry (in most recent order):
As of this snapshot date, this entry was owned by bci1.