Category theory; homological algebra Categories, functors, natural transformations Concept
Category
Summary
A category is made of objects, of morphisms (arrows) between pairs of objects, and of a composition of morphisms that is associative and has identities. It is the basic notion of category theory. Sets with functions, groups with group homomorphisms, and topological spaces with continuous maps are categories. A category can also be a single structure: a monoid is a category with one object. A preorder is a category with at most one morphism between any two objects.
Where this leads
94 pages build on this one.
- Quotient category
- Forgetful functor
- dg category
- Well-powered category
- Representable functor
- Kleisli category
- Opposite category
- Monoidal category
- Bicategory
- Pullback pasting lemma
- Product
- Nerve of a category
- Diagram
- Density theorem
- 2-category
- Initial and terminal objects
- Equalizer
- Multicategory
- F-algebra
- Profunctor
- Injective object
- Category of elements
- Grothendieck construction
- Subcategory
- Elementary theory of the category of sets
- Eilenberg–Moore category
- Generator
- Zero object
- Cone
- Category of posets
- Karoubi envelope
- Yoneda lemma
- Commutative diagram
- Lawvere theory
- Topos
- Presheaf
- Concrete category
- Galois category
- Yoneda embedding
- Complete category
- Free category
- Subobject
- Section and retraction
- Sketch
- Lifting property
- Exponential object
- Enough projectives
- Solution set condition
- Coequalizer
- End and coend
- Kernel pair
- Existence of limits from products and equalizers
- Product category
- Homotopy category of chain complexes
- Functor
- Localization of a category
- Adjoint functors
- Preadditive category
- Pullback
- Functor of points
- Weak equivalence
- Zero morphism
- Groupoid
- Subobject classifier
- Epimorphism
- Category theory
- Grothendieck topology
- Cartesian closed category
- Model category
- Finitely complete category
- Accessible category
- Hom functor
- Monomorphism
- Factorization system
- Direct limit
- Filtered category
- Simplicial object
- Universal property
- Limit
- Coproduct
- Sieve
- Inverse limit
- Colimit
- Enriched category
- Skeleton of a category
- Morphism
- Full subcategory
- Projective object
- Site
- Slice category
- Duality in category theory
- Pushout
- Functor category
- Category of small categories
Further reading
About this page
| Tag | A2DE66937D |
|---|---|
| URI | m2://A2DE66937D |
| Kind | concept |
| Writing | W0 stub |
| MSC | 18A05, 18A15 |
| Same as | |
| Changed | 2 November 2026 |
Markdown · JSON · Edit on GitHub
What this page needs for W1
- a source with a locator (0 of 1)
The writing levels say how complete the text of a page is. They do not measure its truth.
What links here
Special cases
- Functor category
- Quotient category
- Subcategory
- Kleisli category
- Groupoid
- Concrete category
- Monoidal category
- Lawvere theory
- Free category
- Well-powered category
Used by
- Quotient category
- Forgetful functor
- dg category
- Well-powered category
- Representable functor
- Kleisli category
- Opposite category
- Monoidal category
- Bicategory
- Pullback pasting lemma
- Product
- Nerve of a category
- Diagram
- Density theorem
- 2-category
- Initial and terminal objects
- Equalizer
- Multicategory
- F-algebra
- Profunctor
- Injective object
- Category of elements
- Grothendieck construction
- Subcategory
- Elementary theory of the category of sets
- Eilenberg–Moore category
- Generator
- Zero object
- Cone
- Category of posets
- Karoubi envelope
- Yoneda lemma
- Commutative diagram
- Lawvere theory
- Topos
- Presheaf
- Concrete category
- Galois category
- Yoneda embedding
- Complete category
- Free category
- Subobject
- Section and retraction
- Sketch
- Lifting property
- Exponential object
- Enough projectives
- Solution set condition
- Coequalizer
- End and coend
- Kernel pair
- Existence of limits from products and equalizers
- Product category
- Homotopy category of chain complexes
- Functor
- Localization of a category
- Adjoint functors
- Preadditive category
- Pullback
- Weak equivalence
- Zero morphism
- Groupoid
- Subobject classifier
- Epimorphism
- Category theory
- Grothendieck topology
- Cartesian closed category
- Model category
- Finitely complete category
- Accessible category
- Hom functor
- Monomorphism
- Factorization system
- Direct limit
- Filtered category
- Simplicial object
- Universal property
- Limit
- Coproduct
- Sieve
- Inverse limit
- Colimit
- Enriched category
- Skeleton of a category
- Morphism
- Full subcategory
- Projective object
- Site
- Slice category
- Duality in category theory
- Pushout
- Functor category
- Category of small categories
Instances
- Category of groups
- Category of rings
- Simplex category
- Category of topological spaces
- Category of sets
- Category of small categories