# 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

* 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
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

* 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

* 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

* 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

* algebra formed from a categories #monad #comonad
* monoidal category
* group object
* 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