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The following outline is provided as an overview of and guide to category theory:
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Essence of category theory
- Category
- Functor
- Natural transformation
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
- Concrete category
- Category of small categories
- Category of vector spaces
- Category of graded 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
- Epimorphism
- Monomorphism
- Zero morphism
- Normal morphism
- Dual (category theory)
- Groupoid
- Image (category theory)
- Coimage
- Commutative diagram
- Cartesian morphism
- Slice category
Functors
- Isomorphism of categories
- Natural transformation
- Equivalence of categories
- Subcategory
- Faithful functor
- Full functor
- Forgetful functor
- Representable functor
- Functor category
- Adjoint functors
- Galois connection
- Pontryagin duality
- Affine scheme
- Monad (category theory)
- Comonad
- Combinatorial species
- Exact functor
- Derived functor
- Dominant functor
- Enriched functor
- Kan extension of a functor
- Hom functor
- Yoneda lemma
Limits
- Product (category theory)
- Equaliser (mathematics)
- Kernel (category theory)
- Pullback (category theory)/fiber product
- Inverse limit
- Pro-finite group
- Colimit
- Coproduct
- Coequalizer
- Cokernel
- Pushout (category theory)
- Direct limit
- Biproduct
- Direct sum
Additive structure
- Preadditive category
- Additive category
- Pre-Abelian category
- Abelian category
- Exact sequence
- Exact functor
- Snake lemma
- Nine lemma
- Five lemma
- Short five lemma
- Mitchell's embedding theorem
- Injective cogenerator
- Derived category
- Triangulated category
- Model category
- 2-category
Dagger categories
- Dagger symmetric monoidal category
- Dagger compact category
- Strongly ribbon category
Monoidal categories
- Closed monoidal category
- Braided monoidal category
- Symmetric monoidal category
Structure
- Semigroupoid
- Comma category
- Localization of a category
- Enriched category
- Bicategory
Topoi, toposes
- Sheaf
- Gluing axiom
- Descent (category theory)
- Grothendieck topology
- Introduction to topos theory
- Subobject classifier
- Pointless topology
- Heyting algebra
History of category theory
- History of category theory
Persons influential in the field of category theory
Category theory scholars
- Saunders Mac Lane
- Samuel Eilenberg
- Max Kelly
- William Lawvere
- André Joyal
See also
- Abstract nonsense
- Glossary of category theory