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
editBranches of category theory
editSpecific categories
edit- Category of sets
- Category of 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
editMorphisms
editFunctors
edit- Isomorphism of categories
- Natural transformation
- Equivalence of categories
- Subcategory
- Faithful functor
- Full functor
- Forgetful functor
- Yoneda lemma
- Representable functor
- Functor category
- Adjoint functors
- Monad (category theory)
- Comonad
- Combinatorial species
- Exact functor
- Derived functor
- Dominant functor
- Enriched functor
- Kan extension of a functor
- Hom functor
