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, 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 –
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 –
- Category of medial 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 –
- Groupoid –
- Image –
- 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 –
- Comonad –
- Combinatorial species –
- Exact functor –
- Derived functor –
- Enriched functor –
- Kan extension of a functor –
- Hom functor –
Limits
- Product –
- Equaliser –
- Kernel –
- Pullback /fiber product –
- Inverse limit –
- *Pro-finite group –
- Colimit –
- *Coproduct –
- *Coequalizer –
- *Cokernel –
- *Pushout –
- *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 –
- 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