Outline of category theory
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
[edit]Branches of category theory
[edit]- Homological algebra
- Diagram chasing
- Topos theory
- Enriched category theory
- Higher category theory
- Categorical logic
- Applied category theory
Specific categories
[edit]- Category of sets
- Category of small categories
- 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
[edit]- Initial object
- Terminal object
- Zero object
- Subobject
- Group object
- Magma object
- Natural number object
- Exponential object
Morphisms
[edit]- Epimorphism
- Monomorphism
- Zero morphism
- Normal morphism
- Dual (category theory)
- Groupoid
- Image (category theory)
- Coimage
- Commutative diagram
- Cartesian morphism
- Slice category
Functors
[edit]- Isomorphism of categories
- Natural transformation
- Equivalence of categories
- Subcategory
- Faithful functor
- Full functor
- Forgetful functor
- 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
Limits
[edit]- Product (category theory)
- Equaliser (mathematics)
- Kernel (category theory)
- Pullback (category theory)/fiber product
- Inverse limit
- Colimit
- Biproduct
Additive structure
[edit]- Preadditive category
- Additive category
- Pre-Abelian category
- Abelian category
- Injective cogenerator
- Derived category
- Triangulated category
- Model category
- 2-category
Dagger categories
[edit]Monoidal categories
[edit]Structure
[edit]Topoi, toposes
[edit]- Sheaf
- Gluing axiom
- Descent (category theory)
- Grothendieck topology
- Introduction to topos theory
- Subobject classifier
- Pointless topology
- Heyting algebra
History of category theory
[edit]Persons influential in the field of category theory
[edit]Category theory scholars
[edit]See also
[edit]
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