PlanetPhysics/Index of Category Theory

Index of Category Theory

Foundamental Concepts and Axioms

Basic Definitions

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 ($Set$, $Ens$)
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. ${\mathcal {U}}$-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

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

• 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

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

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

Additive Categories and Homology

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

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...}