PlanetPhysics/Index of Algebraic Topology
Appearance
Foundations
[edit | edit source]Basic Definitions
[edit | edit source]- 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
[edit | edit source]- 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
- -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
[edit | edit source]- Hurewicz theorem #cohomology theorems
- homotopy lemma and corollary
- topological space approximation theorem
- Yoneda-Grothendieck Lemma
- Van Kampen theorem #Higher dimensional, generalized Van Kampen theorem (HGvKT)
- 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
2-Categories and Supercategories
[edit | edit source]- 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
[edit | edit source]- preadditive category
- additive category #abelian category #non-Abelian categories and non-Abelian algebraic topology #supplemental axioms for an Abelian category
- exact sequence
- exact functor
- Grothendieck spectral sequence
- enough projectives
- enough injectives #projective object #injective object
- derived functor
- derived category
- algebraic K-theory #examples of algebraic K-theory groups #Grothendieck group
- Grothendieck category #delta functor
- horseshoe lemma
- syzygy
- Ext
- Tor
- projective dimension
- 5-lemma
- proof of 5-lemma
- 9-lemma
- snake lemma
- proof of snake lemma
- chain homotopy
- chain homotopy equivalence
- chain map
- homology of a chain complex
- Leray spectral sequence
- spectral sequence
Sheaves, Topoi, Generalizations
[edit | edit source]- presheaf #sheaf
- sheafification
- presheaf of a topological basis
- stalk
- \'Etal\'e space
- resolution of a sheaf
- gerbes
- site
- small site on a scheme
- topos
- cosmos
- subobject classifier
- well-pointed topos
- power object
- natural numbers object
- Cartesian closed category
- exponential object
Examples of Categories
[edit | edit source]- discrete category
- category example (arrow category)
- category associated to a partial order
- category of matrices #Category of pseudomorphisms
- Category of intermorphisms
- examples of initial objects and terminal objects and zero objects
- category of sets
- monomorphisms of category of sets
- monoid as a category
- comma category
- category of pointed topological spaces
- simplicial category
Algebraic Categories
[edit | edit source]- algebra formed from a categories #monad #comonad
- monoidal category
- group object
- nerve
Generalized Categorical Galois theory
[edit | edit source]Categorical Galois theory and and Topological greoupoid category
[edit | edit source]- category of topological groupoids fundamental groupoid functors
Nonabelian Algebraic Topology (NAAT)
[edit | edit source]Nonabelian Quantum Algebraic Topology (NA-QAT)
[edit | edit source]- 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