Outline of category theory
Template:Short description 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 theoryEdit
Branches of category theoryEdit
- Homological algebra
- Diagram chasing
- Topos theory
- Enriched category theory
- Higher category theory
- Categorical logic
- Applied category theory
Specific categoriesEdit
- 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
ObjectsEdit
- Initial object
- Terminal object
- Zero object
- Subobject
- Group object
- Magma object
- Natural number object
- Exponential object
MorphismsEdit
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- Epimorphism
- Monomorphism
- Zero morphism
- Normal morphism
- Dual (category theory)
- Groupoid
- Image (category theory)
- Coimage
- Commutative diagram
- Cartesian morphism
- Slice category
FunctorsEdit
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- 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
LimitsEdit
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- Product (category theory)
- Equaliser (mathematics)
- Kernel (category theory)
- Pullback (category theory)/fiber product
- Inverse limit
- Colimit
- Biproduct
Additive structureEdit
- Preadditive category
- Additive category
- Pre-Abelian category
- Abelian category
- Injective cogenerator
- Derived category
- Triangulated category
- Model category
- 2-category
Dagger categoriesEdit
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Monoidal categoriesEdit
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Cartesian closed categoryEdit
StructureEdit
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Topoi, toposesEdit
- Sheaf
- Gluing axiom
- Descent (category theory)
- Grothendieck topology
- Introduction to topos theory
- Subobject classifier
- Pointless topology
- Heyting algebra