PlanetPhysics/Index of Algebraic Topology

Foundations

Basic Definitions

• open and closed sets
• map
• arrow
• morphism
• path
• cycle
• boundary
• torus, n-tori
• sphere, n-sphere
• square
• cube
• dimension
• vector space
• CW-complex
• graph
• simplicial complex
• spin network
• topological dynamics
• qualitative dynamics
• dense space
• spin foam
• map
• function
• analytical space
• identity
• commutativity
• associativity
• object space
• source space
• target space
• thin square
• topological space
• homeomorphism
• topological groups
• Lie groups and Lie algebras
• graded lie algebras/Lie superalgebras
• supergroups
• groupoid
• groupoid homomorphism
• groupoid categories
• compact space/Hausdorff space
• double groupoid
• double algebroid
• Hamiltonian algebroid
• Polish space
• Polish group
• Polish G-space
• homotopy
• homotopy groups
• homotopy category
• homology cycle
• cohomology cocycle
• fundamental groups
• homotopy lemma and corollary
• topological groups
• fundamental groupoids
• fundamental groupoid functor
• homotopy double groupoid of a Hausdorff space
• homology theory
• axiomatic homology theory
• cohomology groups
• cohomology groupoids cohomology theory cohomology theorems topological categories

Category theory concepts in algebraic topology

• category
• topos
• subspace
• subcategory
• automorphism
• commutative diagram
• concrete category
• dual category
• duality principle
• endomorphism
• epimorphism
• monic
• monomorphism
• source
• pushout
• pullback
• cones and cocones
• limit and colimit functors
• direct sum
• open covers
• isomorphism-closed subcategory
• locally finite category
• preimage of category
• product of categories
• types of morphisms
• wellpowered category
• null object
• zero object
• ${\mathcal {U}}$-small
• equalizer
• subobject
• quotient object
• categorical direct product
• categorical direct sum
• direct limit
• limiting cone
• complete category
• groupoid (category theoretic) enriched category double category 2-category n-category

Fundamental Theorems

1. Hurewicz theorem #cohomology theorems
2. homotopy lemma and corollary
3. topological space approximation theorem
4. Yoneda-Grothendieck Lemma
5. Van Kampen theorem #Higher dimensional, generalized Van Kampen theorem (HGvKT)
6. properties of monomorphisms and epimorphisms
7. properties of 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

Grothendieck theorem

2-Categories and Supercategories

• functor
• endofunctor
• category isomorphism
• diagonal functor
• representable functors
• categorical representations and supergroups
• forgetful functor
• identity functor
• isomorphism
• multifunctor
• natural transformation
• essentially surjective
• faithful functor
• full functor
• adjoint functor
• natural equivalence
• equivalence of categories
• isomorphic categories
• universal properties
• representable functors
• Equivalent definition of a Representable Functor
• simplicial objects

Additive Categories and Homology

1. preadditive category
2. additive category #abelian category #non-Abelian categories and 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 injectives #projective object #injective object
8. derived functor
9. derived category
10. algebraic K-theory #examples of algebraic K-theory groups #Grothendieck group
11. 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. 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 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 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. monoid as a category
10. comma category
11. category of pointed topological spaces
12. simplicial category

Algebraic Categories

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

Generalized Categorical Galois theory

Categorical Galois theory and and Topological greoupoid category

• category of topological groupoids fundamental groupoid functors

Nonabelian Quantum Algebraic Topology (NA-QAT)

• Clifford algebras
• Hopf algebras
• Weak-Hopf algebras
• Grasssmann-Hopf algebras
• Hamiltonian algebroids
• Lie algebroids
• quantum algebroids
• Quantum Groups and Quantum Compact Groups (QCGs)
• Quantum Groupoids