Outline of category theory

In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from sets and functions to objects linked in diagrams by morphisms or arrows. The study of categories is an attempt to axiomatically capture what is commonly found in various classes of related mathematical structures by relating them to the structure-preserving functions between them. A systematic study of category theory then allows us to prove general results about any of these types of mathematical structures from the axioms of a category.

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

Essence of category theory

Main article: Category theory

• Category –
• Functor –
• Natural transformation –

Branches of category theory

• Homological algebra –
• Diagram chasing –
• Topos theory –

Specific categories

• Category of sets –

• Concrete category –

• Category of vector spaces –

• Category of graded vector spaces –

• 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 –
• Category of medial magmas –

Objects

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

Morphisms

Main article: morphism

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

Functors

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 –
• Enriched functor –
• Kan extension of a functor –
• Hom functor –

Limits

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 –
• Bicategory –

Dagger categories

Main article: Dagger category

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

Monoidal categories

Main article: monoidal category

• Closed monoidal category –
• Braided monoidal category –

Structure

Main article: Structure (category theory)

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

Topoi, toposes

Main article: Topos

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

History of category theory

Main article: History of category theory

See also

• Abstract nonsense –
• Homological algebra –
• Glossary of category theory –