Outline of category theory explained

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

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

See main article: Functor.

• • • • • • • • • • • • • • • • • • • • • • • • • • 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

See main article: Limit (category theory).

• • • • • • • • • • • • • • • • • • • • • 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

See main article: Dagger category.

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

Monoidal categories

See main article: Monoidal category.

• • 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