This is a list of category theory topics, by Wikipedia page.

Specific categories

• Category of sets
  • Concrete category
• Category of vector spaces
  • Category of graded vector spaces
• Category of topological spaces
• Category of metric spaces
• Category of preordered sets
• Category of groups
• Category of abelian groups
• Category of commutative rings
• Category of magmas
• Category of medial magmas

Objects

• Initial object
• Terminal object
• Zero object
• Subobject
• Group object
• Magma object
• Natural number object

Morphisms

• 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
• Yoneda lemma
• Representable functor
• Functor category
• Adjoint functors
  • Galois connection
  • Pontryagin duality
  • Affine scheme
• Monad (category theory)
• Comonad
• Combinatorial species
• Exact functor
• Derived functor
• Enriched functor

Limits

• Limit (category theory)
  • Product (category theory)
  • Equalizer
  • Kernel (category theory)
  • Pullback, 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

Monoidal category

• Identity object

Cartesian closed category

Structure (category theory)

• Semigroupoid
• Comma category
• Localization of a category
• Enriched category
• Bicategory

Toposes

• Sheaf
• Gluing axiom
• Descent (category theory)
• Grothendieck topology
• Introduction to topos theory
• Subobject classifier
• Pointless topology
• Heyting algebra

See also: abstract nonsense, homological algebra