Exercise 3-73

Currying was an adjunction between functors in Set, but the example only discussed what the functors did to objects.

  1. Given a morphism \(X \xrightarrow{f}Y\), what morphism should \(X \times B \xrightarrow{-\times B}Y\times B\) return?

  2. Given a morphism \(X \xrightarrow{f}Y\), what morphism should \(X^ B \xrightarrow{(-)^B}Y^B\) return?

  3. Consider \(\mathbb{N}\times \mathbb{N}\xrightarrow{+}\mathbb{N}\). Currying \(+\), we get a certain function \(\mathbb{N}\xrightarrow{p}\mathbb{N}^\mathbb{N}\). What is \(p(3)\)?

Solution(1)

  1. This morphism maps \((x,b)\mapsto (f(x),b)\)

  2. This morphism takes in a function \(B \xrightarrow{bx} X\) and composes with f to give \(B \xrightarrow{bx;f} Y\)

  3. It takes a number and returns a function which adds three to that number.