Category theory; homological algebra Categories, functors, natural transformations Concept
Functor
Summary
A functor is a map between two categories: it sends objects to objects and morphisms to morphisms, and it respects composition and identities. Functors are how one field of mathematics is translated into another, as when the fundamental group turns spaces and continuous maps into groups and homomorphisms. Because functors preserve composition, they send isomorphisms to isomorphisms, which makes them a source of invariants. Functors between two categories are compared by natural transformations.
Where this leads
57 pages build on this one.
- Quotient category
- Forgetful functor
- Contravariant functor
- Full and faithful functors
- Representable functor
- Monoidal category
- Bicategory
- Final functor
- Nerve of a category
- Splitting lemma
- Diagram
- Bernstein–Gelfand–Ponomarev reflection functors
- 2-category
- Freyd–Mitchell embedding theorem
- F-algebra
- Profunctor
- Category of elements
- Grothendieck construction
- Moduli space
- Essentially surjective functor
- Natural transformation
- Lawvere theory
- Kan extension
- Presheaf
- Concrete category
- Galois category
- Monad
- Yoneda embedding
- Free category
- Sketch
- Fibred category
- Solution set condition
- Additive functor
- Equivalence of categories
- End and coend
- Four lemma
- Localization of a category
- Adjoint functors
- Functor of points
- Category theory
- Preservation of limits
- Monoidal functor
- Pseudofunctor
- Comonad
- Lambek's lemma
- Hom functor
- F-coalgebra
- Simplicial object
- Universal property
- Comma category
- Five lemma
- Fully faithful essentially surjective functor is an equivalence
- Skeleton of a category
- Universal coalgebra
- Anti-equivalence of affine schemes and rings
- Functor category
- Category of small categories
Further reading
About this page
| Tag | 28NQ0KRHD3 |
|---|---|
| URI | m2://28NQ0KRHD3 |
| Kind | concept |
| Writing | W0 stub |
| MSC | 18A05, 18A22 |
| 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
Used by
- Quotient category
- Forgetful functor
- Contravariant functor
- Full and faithful functors
- Representable functor
- Monoidal category
- Bicategory
- Final functor
- Nerve of a category
- Diagram
- Bernstein–Gelfand–Ponomarev reflection functors
- 2-category
- Freyd–Mitchell embedding theorem
- F-algebra
- Profunctor
- Category of elements
- Grothendieck construction
- Moduli space
- Essentially surjective functor
- Natural transformation
- Lawvere theory
- Kan extension
- Presheaf
- Concrete category
- Galois category
- Monad
- Yoneda embedding
- Free category
- Sketch
- Fibred category
- Solution set condition
- Additive functor
- Equivalence of categories
- End and coend
- Localization of a category
- Adjoint functors
- Functor of points
- Category theory
- Preservation of limits
- Monoidal functor
- Pseudofunctor
- Comonad
- Lambek's lemma
- Hom functor
- F-coalgebra
- Simplicial object
- Universal property
- Comma category
- Fully faithful essentially surjective functor is an equivalence
- Universal coalgebra
- Anti-equivalence of affine schemes and rings
- Functor category
- Category of small categories
Special cases
- Yoneda embedding
- Presheaf
- Diagram
- Additive functor
- Representable functor
- Free functor
- Monoidal functor
- Contravariant functor
- Hom functor
- Forgetful functor