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Representable functors
Functors naturally isomorphic to Hom(A, −) or Hom(−, A): the forgetful functor on groups (represented by ℤ), the power-set functor (represented by a two-element set), and representing objects as solutions of universal problems.
math.cat.representable-functors
Se basa en
Requisitos previos obligatorios, hasta el fondo: 17
Aprenderás a
- Show that the forgetful functor Grp → Set is represented by ℤ, and that the contravariant power-set functor on Set is represented by {0, 1}.
- Prove that a representing object, together with its universal element, is unique up to unique isomorphism.
- Express products, free objects and quotients as representing objects of suitable functors.
Definición
- A functor F : C → Set is representable if F ≅ Hom(A, −) for some object A; a functor F : Cᵒᵖ → Set is representable if F ≅ Hom(−, A).
- A universal element of F : C → Set is a pair (A, x) with x ∈ F(A) such that for every object B and y ∈ F(B) there is a unique f : A → B with F(f)(x) = y.
Teorema
- The forgetful functor Grp → Set is represented by ℤ, with universal element 1.
- The contravariant power-set functor is isomorphic to Hom(−, {0, 1}): subsets correspond to characteristic functions.
- Representations of F correspond to universal elements of F.
Errores frecuentes
- Thinking every functor to Set is representable.
- Confusing covariant with contravariant representability.
Conduce a
Descubrimientos opcionales
- Algebraic geometry Grothendieck's functor of points views a scheme X through the functor Hom(−, X); many moduli problems ask whether a given functor is representable.
- Elementary topos theory In a topos, the functor sending an object to its set of subobjects is represented by the subobject classifier Ω, generalising the representation of power sets by {0, 1}.
Lecturas complementarias
- Emily Riehl, Category Theory in Context (2016)
- Tom Leinster, Basic Category Theory (2014)