Outline of category theory

The following outline is provided as an overview of and guide to category theory, the area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows (also called morphisms, although this term also has a specific, non category-theoretical sense), where these collections satisfy certain basic conditions. Many significant areas of mathematics can be formalised as categories, and the use of category theory allows many intricate and subtle mathematical results in these fields to be stated, and proved, in a much simpler way than without the use of categories.

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 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
• Yoneda lemma
• 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

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

Cartesian closed category

• Topos
• Category of small categories

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

• v
• t
• e

Category theory

Key concepts

Key concepts Category Abelian Additive Concrete Pre-abelian Preadditive Bicategory Adjoint functors CCC Commutative diagram End Exponential Functor Kan extension Morphism Natural transformation Universal property Universal constructions Limits Terminal objects Products Equalizers Kernels Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels and quotients Pushout Direct limit Algebraic categories Sets Relations Magmas Groups Abelian groups Rings (Fields) Modules (Vector spaces) Constructions on categories Free category Functor category Kleisli category Opposite category Quotient category Product category Comma category Subcategory

Higher category theory

Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model category Simplex category String diagram Topos n-categories Weak n-categories Bicategory (pseudofunctor) Tricategory Tetracategory Kan complex ∞-groupoid ∞-topos Strict n-categories 2-category (2-functor) 3-category Categorified concepts 2-group 2-ring En-ring (Traced)(Symmetric) monoidal category n-group n-monoid

• Category
• Outline
• Glossary