PlanetPhysics/Index of Category Theory

Basic Definitions

 * 1) category theory #object #identity #arrow
 * 2) morphism #composition law #commutativity #associativity
 * 3) axioms of category theory
 * 4) ETAC #triples
 * 5) Abelian group #group #groupoid #semigroup #monoid #ring
 * 6) diagram #precategory
 * 7) category #alternative definition of category #subcategory
 * 8) small category #category of sets ($$Set$$, $$Ens$$)
 * 9) automorphism
 * 10) commutative diagram #concrete category
 * 11) dual category
 * 12) duality principle #endomorphism
 * epi
 * 1) monic
 * 2) extremal monomorphism #source
 * 3) sink
 * 4) initial source
 * 5) final sink
 * 6) isomorphism-closed subcategory
 * 7) locally finite category
 * 8) preimage of category
 * 9) product of categories
 * 10) types of morphisms
 * 11) wellpowered category
 * 12) zero object
 * 13) $$\mathcal{U}$$-small
 * 14) equalizer
 * 15) subobject
 * 16) quotient object #direct product
 * 17) direct sum
 * 18) pullback
 * 19) pushout #direct limit
 * 20) limiting cone
 * 21) cocone and colimit functor #complete category
 * 22) groupoid (category theoretic)

2-Categories and Generalizations

 * 1) functor
 * 2) autofunctor
 * 3) category isomorphism #diagonal functor
 * 4) endofunctor
 * 5) forgetful functor
 * 6) identity functor #isomorphism
 * 7) multifunctor
 * 8) natural transformation #surjective maps
 * 9) univalent functors
 * 10) faithful functor
 * 11) full functor
 * 12) natural equivalence #adjoint functor #equivalence of categories
 * 13) isomorphic categories
 * 14) universal property
 * 15) representable functor #Equivalent definition of a Representable Functor
 * 16) 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
quantum topos
 * 1) discrete category
 * 2) category example (arrow category)
 * 3) category of sets
 * 4) category of Abelian groups
 * 5) category of topological spaces
 * 6) category of pointed topological spaces
 * 7) simplicial category
 * 8) category of groupoids #category associated to a partial order
 * 9) category of matrices #Category of pseudomorphisms
 * 10) Category of intermorphisms
 * 11) examples of initial objects and terminal objects and zero objects
 * 12) monoid as a category
 * 13) comma category
 * 14) enriched category #algebraic category #Logic category
 * 15) quantum logic category

Algebraic categories

 * 1) algebra formed from a category #monad #comonad
 * 2) monoidal category
 * 3) group object
 * 4) nerve
 * 5) gerbs

Additive Categories and Homology

 * 1) preadditive category
 * 2) additive category #abelian category #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
 * 11) Grothendieck group
 * 12) delta functor
 * 13) horseshoe lemma
 * 14) 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 #chain homotopy equivalence
 * 8) chain map
 * 9) homology of a chain complex
 * 10) Leray spectral sequence
 * 11) spectral sequence

Sheaves, Topoi, and the like

 * 1) presheaf #sheaf
 * 2) sheafification
 * 3) presheaf of a topological basis
 * 4) stalk
 * 5) \'Etal\'e space
 * 6) resolution of a sheaf
 * 7) site
 * 8) small site on a scheme
 * 9) topos #cosmos
 * 10) Heyting logic algebra
 * 11) subobject classifier
 * 12) well-pointed topos
 * 13) power object
 * 14) exponential object
 * 15) Cartesian closed category
 * 16) natural numbers object

{\mathbf This is a contributed entry in progress...}