representable functor

• on cateogry \(\mathbf{Set}\)

  • representable functors \(y^{S}: \mathbf{Set} \to \mathbf{Set}\), where \(S\) is a set
    • for each set \(X\), it sends it to the homset of morphisms from \(S\) to \(X\), \(X^{S}:= \mathbf{Set}(S, X)\)
    • for a function \(h: X \to Y\), it sends it to \(h^{S}: X^{S} \to Y^{S}\)
    • we say \(S\) is the representing set of \(y^{S}\)
    • \(y\) stands for Yoneda; we also call \(y^{S}\) pure powers
    • representable functors are working on any category \(\mathcal{C} \to \mathbf{Set}\), here we only care about \(\mathcal{C}\) being \(\mathbf{Set}\)
      • more general definition in Category Theory for the Sciences:
        • a functor represented by \(c \in \mathcal{C}\), we say the functor \(Y_{c}\) sends every object \(d \in \mathcal{C}\) to \(\text{Hom}_{\mathcal{C}}(c, -)\). It acts similarily on morphisms \(d \to d'\)
      • so, given a category \(\mathcal{C}\) and an object \(c \in \text{Ob}(\mathcal{C})\), we get a representable functor \(Y_{\mathcal{C}}\)

• maps between representable functors are natural transformations

  • Proposition: for any function \(f: R \to S\), there is an induced natural transformation \(y^{f}: y^{S} \to y^{R}\), where on any set \(X\), its \(X\)-component ... is giving by ...
    • proof: shows the naturality square commutes
    • :
      • R and S are set
      • \(y^{S}\) is the representable functor of S
      • \(y^{R}\) is the representable functor of R
  • the representable functor and natural transformation lives in a larger category \(\mathbf{Set}^{\mathbf{Set}}\)

• \(\mathbf{Set}^{\mathbf{Set}}\)

  • objects: functors \(\mathbf{Set} \to \mathbf{Set}\)
  • morphisms: natural transformation between them