 Outline of category theory

The following outline is provided as an overview of and guide to category theory:
Category theory – 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 categorytheoretical 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
Main article: Category theoryBranches of category theory
 Homological algebra –
 Diagram chasing –
 Topos theory –
 Enriched category theory –
Specific categories

 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 –

 Profinite group –
 Colimit –
Additive structure
 Injective cogenerator –
 Derived category –
 Triangulated category –
 Model category –
 2category –
 Bicategory –
Dagger categories
Main article: Dagger categoryMonoidal categories
Main article: monoidal categoryCartesian closed category
Structure
Main article: Structure (category theory)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
Persons influential in the field of category theory
Category theory scholars
See also
References
External links
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