Yoneda lemma

• given a functor \(F: \mathbf{Set} \to \mathbf{Set}\) and a set \(\mathbf{S}\)

• there is an isomorphism \(F(S) \cong \mathbf{Set}^{\mathbf{Set}} (y^{S}, F)\)

  • RHS: homset of natural transformation \(y^{S} \to F\)
  • RHS: \(y^{S}\) is the representable functor \(S\)
  • moreover, the isomorphism is natural in both \(S\) and \(F\)

• it characters maps out of representables

  • for an arbitray functor \(F: \mathbf{Set} \to \mathbf{Set}\), natural transformationf \(y^{S} \to F\) are in natural correspondence with elements of \(F(S)\)

• even in a totally arbitrary cartesian category whose objects are not sets of any kind, we can still reason about them as if they were

  • at lease when it comes to pairing elements and applying functions