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Fall 2016 - Problem 8

category theory

Prove that if a functor F ⁣:C→Set\func{\mathcal{F}}{\cat{C}}{\Set} has a left adjoint functor, then F\mathcal{F} is representable.
Solution.

Let L ⁣:Set→C\func{L}{\Set}{\cat{C}} be its left adjoint so that for any X∈SetX \in \Set and C∈CC \in \cat{C}, we have

Hom⁡C(L(X),C)≃Hom⁡Set(X,F(C)).\Hom_{\cat{C}}\p{L\p{X}, C} \simeq \Hom_{\Set}\p{X, \mathcal{F}\p{C}}.

Set X={∗}X = \set{*}, which gives

Hom⁡C(L({∗}),C)≃Hom⁡Set({∗},F(C))≃F(C),\Hom_{\cat{C}}\p{L\p{\set{*}}, C} \simeq \Hom_{\Set}\p{\set{*}, \mathcal{F}\p{C}} \simeq \mathcal{F}\p{C},

so FF is represented by L({∗})L\p{\set{*}}.