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PlanetPhysics/Index of Algebraic Topology

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Foundations

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Basic Definitions

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Category theory concepts in algebraic topology

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Fundamental Theorems

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

Grothendieck theorem

2-Categories and Supercategories

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Additive Categories and Homology

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

Sheaves, Topoi, Generalizations

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

Examples of Categories

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

Algebraic Categories

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

Generalized Categorical Galois theory

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Categorical Galois theory and and Topological greoupoid category

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Nonabelian Algebraic Topology (NAAT)

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Nonabelian Quantum Algebraic Topology (NA-QAT)

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Noncommutative Geometry (non-commutative geometry)

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