PlanetPhysics/Index of Category Theory

Index of Category Theory

Foundamental Concepts and Axioms

Basic Definitions

1. [[../TrivialGroupoid/|category theory]] #[[../TrivialGroupoid/|object]] #[[../Cod/|identity]] #arrow
2. [[../TrivialGroupoid/|morphism]] #[[../Identity2/|composition law]] #[[../TrivialGroupoid/|commutativity]] #associativity
3. axioms of category theory
4. [[../ETACAxioms/|ETAC]] #triples
5. [[../TrivialGroupoid/|Abelian group]] #[[../TrivialGroupoid/|group]] #[[../GroupoidHomomorphism2/|groupoid]] #[[../TrivialGroupoid/|semigroup]] #[[../TrivialGroupoid/|monoid]] #ring
6. [[../TrivialGroupoid/|diagram]] #precategory
7. [[../Cod/|category]] #[[../AlternativeDefinitionOfCategory/|alternative definition of category]] #subcategory
8. [[../Cod/|small category]] #category of sets ($Set$, $Ens$)
9. automorphism
10. [[../Commutativity/|commutative diagram]] #concrete category
11. dual category
12. [[../DualityAndTriality/|duality principle]] #endomorphism
13. epi
14. monic
15. extremal [[../InjectiveMap/|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. [[../Bijective/|types]] of morphisms
24. wellpowered category
25. zero object
26. $𝒰$-small
27. equalizer
28. subobject
29. [[../DenseSubcategory/|quotient object]] #direct product
30. direct sum
31. pullback
32. [[../Pushout/|pushout]] #direct limit
33. limiting cone
34. cocone and colimit [[../TrivialGroupoid/|functor]] #complete category
35. groupoid (category theoretic)

2-Categories and Generalizations

1. functor
2. autofunctor
3. category [[../IsomorphicObjectsUnderAnIsomorphism/|isomorphism]] #diagonal functor
4. endofunctor
5. forgetful functor
6. [[../IdentityMorphism/|identity functor]] #isomorphism
7. multifunctor
8. [[../VariableCategory2/|natural transformation]] #[[../BCConjecture/|surjective]] maps
9. univalent functors
10. faithful functor
11. full functor
12. [[../IsomorphismClass/|natural equivalence]] #[[../SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence/|adjoint functor]] #equivalence of categories
13. isomorphic categories
14. universal property
15. [[../CategoryOfLogicAlgebras/|representable functor]] #Equivalent definition of a Representable Functor
16. [[../PiecewiseLinear/|simplicial]] object

Fundamental Theorems

• Yoneda-Grothendieck lemma
• properties of monomorphisms and [[../IsomorphicObjectsUnderAnIsomorphism/|epimorphisms]]
• properties of [[../CoIntersections/|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 [[../CoIntersections/|topological]] spaces
6. category of pointed topological spaces
7. simplicial category
8. [[../GroupoidCategory/|category of groupoids]] #category associated to a partial order
9. category of [[../Matrix/|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. [[../TrivialGroupoid/|enriched category]] #[[../CategoryOfLogicAlgebras/|algebraic category]] #Logic category
15. [[../LQG2/|quantum logic]] category

[[../QuantumCategories/|quantum topos]]

Algebraic categories

1. [[../AlgebraFormedFromACategory/|algebra formed from a category]] #[[../CategoricalGroupRepresentation/|monad]] #comonad
2. monoidal category
3. group object
4. nerve
5. gerbs

Additive Categories and Homology

1. preadditive category
2. [[../DenseSubcategory/|additive category]] #[[../AbelianCategory2/|abelian category]] #supplemental axioms for an Abelian category
3. exact sequence
4. exact functor
5. Grothendieck spectral sequence
6. enough projectives
7. enough [[../BCConjecture/|injectives]] #[[../ProjectiveObject/|projective object]] #injective object
8. derived functor
9. derived category
10. [[../PAdicMeasure/|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 [[../ThinEquivalence/|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. [[../CommutativeRingWithUnit/|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. [[../GrothendieckTopos/|topos]] #cosmos
10. Heyting logic algebra
11. subobject classifier
12. well-pointed topos
13. [[../Power/|power]] object
14. exponential object
15. Cartesian closed category
16. natural numbers object

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