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index of algebraic topology (Topic)

Foundations

Basic Definitions

Category theory concepts in algebraic topology

Fundamental Theorems

  1. Hurewicz theorem
  2. cohomology theorems
  3. homotopy lemma and corollary
  4. topological space approximation theorem
  5. Yoneda-Grothendieck Lemma
  6. van Kampen theorem
  7. Higher dimensional, generalized Van Kampen theorem (HGvKT)
  8. properties of monomorphisms and epimorphisms
  9. properties of regular and extremal monomorphisms
  10. monomorphisms are pullback stable
  11. proof that an equalizer is a monomorphism
  12. categorical direct product is an inverse limit
  13. kernel is an inverse limit
  14. Grothendieck theorem

2-Categories and Supercategories

Additive Categories and Homology

  1. preadditive category
  2. additive category
  3. abelian category
  4. non-Abelian categories and non-Abelian algebraic topology
  5. supplemental axioms for an Abelian category
  6. exact sequence
  7. exact functor
  8. Grothendieck spectral sequence
  9. enough projectives
  10. enough injectives
  11. projective object
  12. injective object
  13. derived functor
  14. derived category
  15. algebraic K-theory
  16. examples of algebraic K-theory groups
  17. Grothendieck group
  18. Grothendieck category
  19. delta functor
  20. horseshoe lemma
  21. syzygy
  22. Ext
  23. Tor
  24. projective dimension
  25. 5-lemma
  26. proof of 5-lemma
  27. 9-lemma
  28. snake lemma
  29. proof of snake lemma
  30. chain homotopy
  31. chain homotopy equivalence
  32. chain map
  33. homology of a chain complex
  34. Leray spectral sequence
  35. spectral sequence

Sheaves, Topoi, Generalizations

  1. presheaf
  2. sheaf
  3. sheafification
  4. presheaf of a topological basis
  5. stalk
  6. Étalé space
  7. resolution of a sheaf
  8. gerbes
  9. site
  10. small site on a scheme
  11. topos
  12. cosmos
  13. subobject classifier
  14. well-pointed topos
  15. power object
  16. natural numbers object
  17. Cartesian closed category
  18. exponential object

Examples of Categories

  1. discrete category
  2. category example (arrow category)
  3. category associated to a partial order
  4. category of matrices
  5. Category of pseudomorphisms
  6. Category of intermorphisms
  7. examples of initial objects and terminal objects and zero objects
  8. category of sets
  9. monomorphisms of category of sets
  10. monoid as a category
  11. comma category
  12. category of pointed topological spaces
  13. simplicial category

Algebraic Categories

  1. algebra formed from a categories
  2. monad
  3. comonad
  4. monoidal category
  5. group object
  6. nerve

Generalized Categorical Galois theory

Categorical Galois theory and and Topological greoupoid category

Nonabelian Algebraic Topology (NAAT)

Nonabelian Quantum Algebraic Topology (NA-QAT)

Noncommutative Geometry (non-commutative geometry)



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See Also: category, categorical algebra, double groupoid with connection, axioms of topoi, axioms of metacategories and supercategories, algebraic quantum field theories (AQFT), groupoid, Polish G-space, Lie algebra, quantum field theories (QFT), topos axioms, axiomatic theories of metacategories and supercategories

Also defines:  HGvkT
Keywords:  index, algebraic topology

Cross-references: QCGs, quantum algebroids, Lie algebroids, Hopf algebras, Clifford algebras, fundamental groupoid functors, topological groupoids, monad, algebra formed from a categories, monoid, matrices, power, presheaf, Grothendieck category, groups, algebraic K-theory, projective object, injectives, non-Abelian algebraic topology, non-Abelian categories, abelian category, additive category, simplicial, adjoint functor, surjective, natural transformation, identity functor, representations, representable functors, isomorphism, Grothendieck theorem, regular, generalized Van Kampen theorem, Hurewicz theorem, n-category, 2-category, double category, enriched category, quotient object, types, functors, pushout, monomorphism, epimorphism, duality principle, commutative diagram, topos, categories, theorems, cohomology theory, cohomology groups, homology theory, homotopy double groupoid of a Hausdorff space, fundamental groupoid functor, fundamental groupoids, fundamental groups, homotopy category, homotopy groups, homotopy, Polish G-space, Hamiltonian algebroid, double algebroid, double groupoid, groupoid categories, groupoid homomorphism, groupoid, supergroups, superalgebras, Lie algebras, Lie groups, topological groups, homeomorphism, thin square, object, commutativity, identity, function, spin foam, dynamics, topological, spin network, simplicial complex, graph, vector space, square, boundary, morphism

This is version 5 of index of algebraic topology, born on 2009-05-02, modified 2009-05-02.
Object id is 720, canonical name is IndexOfAlgebraicTopology.
Accessed 502 times total.

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Physics Classification: 00. (GENERAL)
 02. (Mathematical methods in physics)
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