PlanetPhysics/Index of Category Theory

Index of Category Theory

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Foundamental Concepts and Axioms

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Basic Definitions

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  1. category theory #object #identity #arrow
  2. morphism #composition law #commutativity #associativity
  3. axioms of category theory
  4. ETAC #triples
  5. Abelian group #group #groupoid #semigroup #monoid #ring
  6. diagram #precategory
  7. category #alternative definition of category #subcategory
  8. small category #category of sets ( ,  )
  9. automorphism
  10. commutative diagram #concrete category
  11. dual category
  12. duality principle #endomorphism
  13. epi
  14. monic
  15. extremal monomorphism #source
  16. sink
  17. initial source
  18. final sink
  19. isomorphism-closed subcategory
  20. locally finite category
  21. preimage of category
  22. product of categories
  23. types of morphisms
  24. wellpowered category
  25. zero object
  26.  -small
  27. equalizer
  28. subobject
  29. quotient object #direct product
  30. direct sum
  31. pullback
  32. pushout #direct limit
  33. limiting cone
  34. cocone and colimit functor #complete category
  35. groupoid (category theoretic)

2-Categories and Generalizations

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

Fundamental Theorems

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  • Yoneda-Grothendieck lemma
  • properties of monomorphisms and epimorphisms
  • properties of regular and extremal monomorphisms
  • monomorphisms are pullback stable
  • proof that an equalizer is a monomorphism
  • categorical direct product is an inverse limit
  • kernel is an inverse limit

Examples of Categories

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  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 #category associated to a partial order
  9. category of matrices #Category of pseudomorphisms
  10. Category of intermorphisms
  11. examples of initial objects and terminal objects and zero objects
  12. monoid as a category
  13. comma category
  14. enriched category #algebraic category #Logic category
  15. quantum logic category

quantum topos

Algebraic categories

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

Additive Categories and Homology

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  1. preadditive category
  2. additive category #abelian category #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
  11. Grothendieck group
  12. delta functor
  13. horseshoe lemma
  14. syzygy
  15. Ext
  16. Tor
  17. projective dimension
  18. 5-lemma
  19. proof of 5-lemma
  20. 9-lemma
  21. snake lemma
  22. proof of snake lemma
  23. chain homotopy #chain homotopy equivalence
  24. chain map
  25. homology of a chain complex
  26. Leray spectral sequence
  27. spectral sequence

Sheaves, Topoi, and the like

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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. site
  8. small site on a scheme
  9. topos #cosmos
  10. Heyting logic algebra
  11. subobject classifier
  12. well-pointed topos
  13. power object
  14. exponential object
  15. Cartesian closed category
  16. natural numbers object

{\mathbf This is a contributed entry in progress...}