The following outline is provided as an overview of and guide to category theory: Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the mid-20th century in their foundational work on algebraic topology. Category theory can be used in most areas of mathematics. In particular, many constructions of new mathematical objects from previous ones that appear similarly in several contexts are conveniently expressed and unified in terms of categories. Examples include quotient spaces, direct products, completion, and duality. Many areas of computer science also rely on category theory, such as functional programming and semantics.
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