Representable functors

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عدد المتطلبات السابقة الإلزامية (حتى الأساس): 17

ستتعلّم

• 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.

تعريف

• 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.

مبرهنة

• 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.

مفاهيم خاطئة شائعة

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

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اكتشافات اختيارية

• 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}.

قراءات إضافية

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