index of category theory

Index of Category Theory

Foundamental Concepts and Axioms

Basic Definitions

  1. category theory
  2. object
  3. identity
  4. arrow
  5. morphism
  6. composition law
  7. commutativity
  8. associativity
  9. axioms of category theory
  10. ETAC
  11. triples
  12. Abelian group
  13. group
  14. groupoid
  15. semigroup
  16. monoid
  17. ring
  18. diagram
  19. precategory
  20. category
  21. alternative definition of category
  22. subcategory
  23. small category
  24. category of sets ($Set$, $Ens$)
  25. automorphism
  26. commutative diagram
  27. concrete category
  28. dual category
  29. duality principle
  30. endomorphism
  31. epi
  32. monic
  33. extremal monomorphism
  34. source
  35. sink
  36. initial source
  37. final sink
  38. isomorphism-closed subcategory
  39. locally finite category
  40. preimage of category
  41. product of categories
  42. types of morphisms
  43. wellpowered category
  44. zero object
  45. $\mathcal{U}$-small
  46. equalizer
  47. subobject
  48. quotient object
  49. direct product
  50. direct sum
  51. pullback
  52. pushout
  53. direct limit
  54. limiting cone
  55. cocone and colimit functor
  56. complete category
  57. groupoid (category theoretic)

2-Categories and Generalizations

  1. functor
  2. autofunctor
  3. category isomorphism
  4. diagonal functor
  5. endofunctor
  6. forgetful functor
  7. identity functor
  8. isomorphism
  9. multifunctor
  10. natural transformation
  11. surjective maps
  12. univalent functors
  13. faithful functor
  14. full functor
  15. natural equivalence
  16. adjoint functor
  17. equivalence of categories
  18. isomorphic categories
  19. universal property
  20. representable functor
  21. Equivalent definition of a Representable Functor
  22. simplicial object

Fundamental Theorems

Examples of Categories

  1. discrete category
  2. category example (arrow category)
  3. category of sets
  4. category of Abelian groups
  5. category of topological spaces
  6. category of pointed topological spaces
  7. simplicial category
  8. category of groupoids
  9. category associated to a partial order
  10. category of matrices
  11. Category of pseudomorphisms
  12. Category of intermorphisms
  13. examples of initial objects and terminal objects and zero objects
  14. monoid as a category
  15. comma category
  16. enriched category
  17. algebraic category
  18. Logic category
  19. quantum logic category
  20. quantum topos

Algebraic categories

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

Additive Categories and Homology

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

Sheaves, Topoi, and the like

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

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