1 Index of Category Theory

1.1 Foundamental Concepts and Axioms

1.1.1 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. 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)

1.2 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

1.3 Fundamental Theorems

• 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

1.4 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

1.5 Algebraic categories

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

2 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

3 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

This is a contributed entry in progress...