Outline of category theory

The following outline is provided as an overview of and guide to category theory:

Essence of category theory

• Category
• Functor
• Natural transformation

Branches of category theory

• Homological algebra
• Diagram chasing
• Topos theory
• Enriched category theory
• Higher category theory
• Categorical logic
• Applied category theory

Specific categories

• Category of sets
• Concrete category
• Category of small categories
• Category of vector spaces
• Category of graded vector spaces
• Category of chain complexes
• Category of finite dimensional Hilbert spaces
• Category of sets and relations
• Category of topological spaces
• Category of metric spaces
• Category of preordered sets
• Category of groups
• Category of abelian groups
• Category of rings
• Category of magmas

Objects

• Initial object
• Terminal object
• Zero object
• Subobject
• Group object
• Magma object
• Natural number object
• Exponential object

Morphisms

• Epimorphism
• Monomorphism
• Zero morphism
• Normal morphism
• Dual (category theory)
• Groupoid
• Image (category theory)
• Coimage
• Commutative diagram
• Cartesian morphism
• Slice category

Functors

• Isomorphism of categories
• Natural transformation
• Equivalence of categories
• Subcategory
• Faithful functor
• Full functor
• Forgetful functor
• Representable functor
• Functor category
• Adjoint functors
• Galois connection
• Pontryagin duality
• Affine scheme
• Monad (category theory)
• Comonad
• Combinatorial species
• Exact functor
• Derived functor
• Dominant functor
• Enriched functor
• Kan extension of a functor
• Hom functor
• Yoneda lemma

Limits

• Product (category theory)
• Equaliser (mathematics)
• Kernel (category theory)
• Pullback (category theory)/fiber product
• Inverse limit
• Pro-finite group
• Colimit
• Coproduct
• Coequalizer
• Cokernel
• Pushout (category theory)
• Direct limit
• Biproduct
• Direct sum

Additive structure

• Preadditive category
• Additive category
• Pre-Abelian category
• Abelian category
• Exact sequence
• Exact functor
• Snake lemma
• Nine lemma
• Five lemma
• Short five lemma
• Mitchell's embedding theorem
• Injective cogenerator
• Derived category
• Triangulated category
• Model category
• 2-category

Dagger categories

• Dagger symmetric monoidal category
• Dagger compact category
• Strongly ribbon category

Monoidal categories

• Closed monoidal category
• Braided monoidal category
• Symmetric monoidal category

Structure

• Semigroupoid
• Comma category
• Localization of a category
• Enriched category
• Bicategory

Topoi, toposes

• Sheaf
• Gluing axiom
• Descent (category theory)
• Grothendieck topology
• Introduction to topos theory
• Subobject classifier
• Pointless topology
• Heyting algebra

History of category theory

• History of category theory

Persons influential in the field of category theory

Category theory scholars

• Saunders Mac Lane
• Samuel Eilenberg
• Max Kelly
• William Lawvere
• André Joyal

See also

• Abstract nonsense
• Glossary of category theory