PlanetPhysics/Index of Algebraic Topology

Foundations

Basic Definitions

• open and closed sets
• map
• arrow
• [[../TrivialGroupoid/|morphism]]
• path
• cycle
• [[../PiecewiseLinear/|boundary]]
• torus, n-tori
• sphere, n-sphere
• [[../PiecewiseLinear/|square]]
• cube
• dimension
• [[../NormInducedByInnerProduct/|vector space]]
• CW-complex
• [[../Cod/|graph]]
• [[../PiecewiseLinear/|simplicial complex]]
• [[../SimplicialCWComplex/|spin network]]
• [[../CoIntersections/|topological]] [[../NewtonianMechanics/|dynamics]]
• qualitative dynamics
• dense space
• [[../SimplicialCWComplex/|spin foam]]
• map
• [[../Bijective/|function]]
• analytical space
• [[../Cod/|identity]]
• [[../TrivialGroupoid/|commutativity]]
• associativity
• [[../TrivialGroupoid/|object]] space
• source space
• target space
• [[../Tree/|thin square]]
• topological space
• [[../TrivialGroupoid/|homeomorphism]]
• [[../PolishGroup/|topological groups]]
• [[../BilinearMap/|Lie groups]] and [[../BilinearMap/|Lie algebras]]
• graded lie algebras/Lie [[../NewtonianMechanics/|superalgebras]]
• [[../Paragroups/|supergroups]]
• [[../GroupoidHomomorphism2/|groupoid]]
• [[../GroupoidHomomorphism2/|groupoid homomorphism]]
• [[../GroupoidCategory3/|groupoid categories]]
• compact space/Hausdorff space
• [[../ThinEquivalence/|double groupoid]]
• [[../GeneralizedSuperalgebras/|double algebroid]]
• [[../HamiltonianAlgebroid3/|Hamiltonian algebroid]]
• Polish space
• [[../InvariantBorelSet/|Polish group]]
• [[../PolishGSpace/|Polish G-space]]
• [[../ThinEquivalence/|homotopy]]
• [[../ExtendedHurewiczFundamentalTheorem/|homotopy groups]]
• [[../HomotopyCategory/|homotopy category]]
• homology cycle
• cohomology cocycle
• [[../HomotopyCategory/|fundamental groups]]
• homotopy lemma and corollary
• topological groups
• [[../CubicalHigherHomotopyGroupoid/|fundamental groupoids]]
• [[../QuantumFundamentalGroupoid/|fundamental groupoid functor]]
• [[../ThinEquivalence/|homotopy double groupoid of a Hausdorff space]]
• [[../CubicalHigherHomotopyGroupoid/|homology theory]]
• axiomatic homology theory
• [[../CohomologyTheoryOnCWComplexes/|cohomology groups]]
• cohomology groupoids [[../NoncommutativeGeometry4/|cohomology theory]] cohomology [[../Formula/|theorems]] topological [[../Cod/|categories]]

Category theory concepts in algebraic topology

• category
• [[../GrothendieckTopos/|topos]]
• subspace
• subcategory
• automorphism
• [[../Commutativity/|commutative diagram]]
• concrete category
• dual category
• [[../DualityAndTriality/|duality principle]]
• endomorphism
• [[../Epimorphism2/|epimorphism]]
• monic
• [[../InjectiveMap/|monomorphism]]
• source
• [[../Pushout/|pushout]]
• pullback
• cones and cocones
• limit and colimit [[../TrivialGroupoid/|functors]]
• direct sum
• open covers
• isomorphism-closed subcategory
• locally finite category
• preimage of category
• product of categories
• [[../Bijective/|types]] of morphisms
• wellpowered category
• null object
• zero object
• $𝒰$-small
• equalizer
• subobject
• [[../DenseSubcategory/|quotient object]]
• categorical direct product
• categorical direct sum
• direct limit
• limiting cone
• complete category
• groupoid (category theoretic) [[../TrivialGroupoid/|enriched category]] [[../HorizontalIdentities/|double category]] [[../2Category/|2-category]] [[../InfinityGroupoid/|n-category]]

Fundamental Theorems

1. [[../ModuleAlgebraic/|Hurewicz theorem]] #cohomology theorems
2. homotopy lemma and corollary
3. topological space approximation theorem
4. Yoneda-Grothendieck Lemma
5. [[../VanKampenTheorems/|Van Kampen theorem]] #Higher dimensional, [[../SingularComplexOfASpace/|generalized Van Kampen theorem]] (HGvKT)
6. properties of monomorphisms and epimorphisms
7. properties of [[../CoIntersections/|regular]] and extremal monomorphisms
8. monomorphisms are pullback stable
9. proof that an equalizer is a monomorphism
10. categorical direct product is an inverse limit
11. kernel is an inverse limit

[[../GrothendiecksTheorem/|Grothendieck theorem]]

2-Categories and Supercategories

• functor
• endofunctor
• category [[../IsomorphicObjectsUnderAnIsomorphism/|isomorphism]]
• diagonal functor
• [[../CategoryOfLogicAlgebras/|representable functors]]
• categorical [[../CategoricalGroupRepresentation/|representations]] and supergroups
• forgetful functor
• [[../IdentityMorphism/|identity functor]]
• isomorphism
• multifunctor
• [[../VariableCategory2/|natural transformation]]
• essentially [[../BCConjecture/|surjective]]
• faithful functor
• full functor
• [[../SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence/|adjoint functor]]
• [[../IsomorphismClass/|natural equivalence]]
• equivalence of categories
• isomorphic categories
• universal properties
• representable functors
• Equivalent definition of a Representable Functor
• [[../PiecewiseLinear/|simplicial]] objects

Additive Categories and Homology

1. preadditive category
2. [[../DenseSubcategory/|additive category]] #[[../AbelianCategory2/|abelian category]] #[[../AbelianCategory3/|non-Abelian categories]] and [[../ModuleAlgebraic/|non-Abelian algebraic topology]] #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 [[../TrivialGroupoid/|groups]] #Grothendieck group
11. [[../GrothendieckCategory/|Grothendieck category]] #delta functor
12. horseshoe lemma
13. syzygy
14. Ext
15. Tor
16. projective dimension
17. 5-lemma
18. proof of 5-lemma
19. 9-lemma
20. snake lemma
21. proof of snake lemma
22. chain homotopy
23. chain homotopy equivalence
24. chain map
25. homology of a chain complex
26. Leray spectral sequence
27. spectral sequence

Sheaves, Topoi, Generalizations

1. [[../CommutativeRingWithUnit/|presheaf]] #sheaf
2. sheafification
3. presheaf of a topological basis
4. stalk
5. \'Etal\'e space
6. resolution of a sheaf
7. gerbes
8. site
9. small site on a scheme
10. topos
11. cosmos
12. subobject classifier
13. well-pointed topos
14. [[../Power/|power]] object
15. natural numbers object
16. Cartesian closed category
17. exponential object

Examples of Categories

1. discrete category
2. category example (arrow category)
3. category associated to a partial order
4. category of [[../Matrix/|matrices]] #Category of pseudomorphisms
5. Category of intermorphisms
6. examples of initial objects and terminal objects and zero objects
7. category of sets
8. monomorphisms of category of sets
9. [[../TrivialGroupoid/|monoid]] as a category
10. comma category
11. category of pointed topological spaces
12. simplicial category

Algebraic Categories

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

Generalized Categorical Galois theory

Categorical Galois theory and and Topological greoupoid category

• category of [[../GroupoidHomomorphism2/|topological groupoids]] [[../QuantumFundamentalGroupoid3/|fundamental groupoid functors]]

Nonabelian Quantum Algebraic Topology (NA-QAT)

• [[../CliffordAlgebra/|Clifford algebras]]
• [[../QuantumGroup4/|Hopf algebras]]
• Weak-Hopf algebras
• Grasssmann-Hopf algebras
• Hamiltonian algebroids
• [[../LieAlgebroids/|Lie algebroids]]
• [[../LongRangeCoupling/|quantum algebroids]]
• Quantum Groups and Quantum Compact Groups ([[../QuantumCompactGroupoids/|QCGs]])
• Quantum Groupoids