Functor
A map on objects and morphisms preserving sources, targets, identities, and composition. A contravariant functor reverses arrows, equivalently a functor from the opposite category.
Find a concept
REFERENCE SHEET
Functors, naturality & Yoneda
Transport structure between categories and compare those transports coherently.
Before this Categories & morphisms
G(f)∘ηX = ηY∘F(f), f:X→Y
Definitions
Natural transformation
For F,G:C→D, a family ηX:F(X)→G(X) satisfying the displayed naturality equation for every morphism f.
Representable functor
A covariant set-valued functor is representable if naturally isomorphic to HomC(A,−) for some object A. Assume hom-collections are sets.
Key theorems
Yoneda lemma, covariant form
For a locally small category C and F:C→Set, natural transformations HomC(A,−)→F correspond bijectively to elements of F(A), by η↦ηA(idA).
Naturality of induced invariants
The fundamental group construction is a functor from based spaces and based continuous maps to groups and homomorphisms.
Notable applications
Coordinate-free constructions
Naturality expresses compatibility with every allowed change of object, rather than a lucky choice of coordinates.
Recovering objects from maps
Yoneda formalizes how an object is determined, up to isomorphism, by its pattern of maps to or from other objects.
Work it out
Try before revealing01 / Construct
For a vector space V, define jV:V→V** by jV(v)(ℓ)=ℓ(v). Show compatibility with a linear map T:V→W.
Hint
Evaluate T**jV(v) on an arbitrary ℓ∈W*.
Solution
(T**jV(v))(ℓ)=jV(v)(ℓ∘T)=ℓ(Tv)=jW(Tv)(ℓ). Hence T**∘jV=jW∘T. In finite dimension jV is an isomorphism.
02 / Identify
Under Yoneda, what determines a natural transformation Hom(A,−)→F?
Hint
Look at the identity of A.
Solution
The element a=ηA(idA)∈F(A). Conversely, for u:A→X, define ηX(u)=F(u)(a); functoriality gives naturality.
Connect the ideas
Build on this
These sheets use this topic as a prerequisite.
Go deeper
Further reading and full-length treatments: