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 theory
Branches of category theory
- Homological algebra
- Diagram chasing
- Topos theory
- Enriched category theory
- Higher category theory
- Categorical logic
- Applied category theory
Specific categories
- 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
- Initial object
- Terminal object
- Zero object
- Subobject
- Group object
- Magma object
- Natural number object
- Exponential object
Morphisms
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- Epimorphism
- Monomorphism
- Zero morphism
- Normal morphism
- Dual (category theory)
- Groupoid
- Image (category theory)
- Coimage
- Commutative diagram
- Cartesian morphism
- Slice category
Functors
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- 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
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- Product (category theory)
- Equaliser (mathematics)
- Kernel (category theory)
- Pullback (category theory)/fiber product
- Inverse limit
- Colimit
- Biproduct
Additive structure
- Preadditive category
- Additive category
- Pre-Abelian category
- Abelian category
- Injective cogenerator
- Derived category
- Triangulated category
- Model category
- 2-category
Dagger categories
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Monoidal categories
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Structure
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Topoi, toposes
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- Sheaf
- Gluing axiom
- Descent (category theory)
- Grothendieck topology
- Introduction to topos theory
- Subobject classifier
- Pointless topology
- Heyting algebra