Exercise 4-4

Suppose we have the preorders \(X:=\boxed{monoid\rightarrow category \leftarrow preorder}, Y:=\boxed{nothing \rightarrow thisBook}\)

  1. Draw the Hasse diagram for the preorder \(X^{op} \times Y\)

  2. Write a profunctor \(X \overset{\phi}\nrightarrow Y\)

Solution(1)

  2. Let \(\phi((M,B))=\phi((P,B))=True\) else \(False\)

    • The preimage of true being an upper set is a consequence of monotone maps to Bool. The domain orders combinations by feasibility (\(x\leq y\) means x is easier than y), and the preimage being an upper set says that if my aunt can explain \(x\) given \(y\), then she can do something easier than \(x\) given \(y\) and can explain \(x\) with something with more explanatory power than \(y\).