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The Lumenary
本科高年级Category Theory计划中

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

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前置知识

Natural transformations

17 项必需的前置知识（一直追溯到底）

你将学会

  • 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.

定义

  • 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.

定理

  • 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.

常见误解

  • Thinking every functor to Set is representable.
  • Confusing covariant with contravariant representability.

通向

The Yoneda lemma

可选发现

  • 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}.

延伸阅读

  • Emily Riehl, Category Theory in Context (2016)
  • Tom Leinster, Basic Category Theory (2014)