# PlanetPhysics/Index of Category Theory
## Index of Category Theory
### Foundamental Concepts and Axioms
#### Basic Definitions

* category theory #object #identity #arrow
* morphism #composition law #commutativity #associativity
* axioms of category theory
* ETAC #triples
* Abelian group #group #groupoid #semigroup #monoid #ring
* diagram #precategory
* category #alternative definition of category #subcategory
* small category #category of sets ($Set$, $Ens$)
* automorphism
* commutative diagram #concrete category
* dual category
* duality principle #endomorphism
* epi
* monic
* extremal monomorphism #source
* sink
* initial source
* final sink
* isomorphism-closed subcategory
* locally finite category
* preimage of category
* product of categories
* types of morphisms
* wellpowered category
* zero object
* ${\mathcal {U}}$-small
* equalizer
* subobject
* quotient object #direct product
* direct sum
* pullback
* pushout #direct limit
* limiting cone
* cocone and colimit functor #complete category
* groupoid (category theoretic)

### 2-Categories and Generalizations

* functor
* autofunctor
* category isomorphism #diagonal functor
* endofunctor
* forgetful functor
* identity functor #isomorphism
* multifunctor
* natural transformation #surjective maps
* univalent functors
* faithful functor
* full functor
* natural equivalence #adjoint functor #equivalence of categories
* isomorphic categories
* universal property
* representable functor #Equivalent definition of a Representable Functor
* simplicial object

### Fundamental Theorems

* Yoneda-Grothendieck lemma
* 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

### Examples of Categories

* discrete category
* category example (arrow category)
* category of sets
* category of Abelian groups
* category of topological spaces
* category of pointed topological spaces
* simplicial category
* category of groupoids #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
* monoid as a category
* comma category
* enriched category #algebraic category #Logic category
* quantum logic category
quantum topos

### Algebraic categories

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

## Additive Categories and Homology

* preadditive category
* additive category #abelian category #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
* 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, and the like

* presheaf #sheaf
* sheafification
* presheaf of a topological basis
* stalk
* \'Etal\'e space
* resolution of a sheaf
* site
* small site on a scheme
* topos #cosmos
* Heyting logic algebra
* subobject classifier
* well-pointed topos
* power object
* exponential object
* Cartesian closed category
* natural numbers object
{\mathbf This is a contributed entry in progress...}