PlanetPhysics/Index of Algebraic Topology

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
Grothendieck theorem
 * 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

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
 * Ext
 * Tor
 * 1) projective dimension
 * 2) 5-lemma
 * 3) proof of 5-lemma
 * 4) 9-lemma
 * 5) snake lemma
 * 6) proof of snake lemma
 * 7) chain homotopy
 * 8) chain homotopy equivalence
 * 9) chain map
 * 10) homology of a chain complex
 * 11) Leray spectral sequence
 * 12) 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

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