Exercise 4 - Chapter 4

Fredrick Eisele

Suppose we have the preorders $$ \mathcal{X} := \left[ \text{monoid} \rightarrow \text{category} \leftarrow \text{preorder} \right] $$ $$ \mathcal{Y} := \left[ \text{nothing} \rightarrow \text{this book} \right] $$

1. Draw the Hasse diagram for the preorder \( \mathcal{X}^{op} \times \mathcal{Y} \).
2. Write down a profunctor \( \Lambda : \mathcal{X} \nrightarrow \mathcal{Y} \) and, reading \( \Lambda(x, y) = \text{true} \) as “my uncle can explain \(x\) given \(y\)”, give an interpretation of the fact that the preimage of \( \text{true} \) forms an upper set in \( \mathcal{X}^{op} \times \mathcal{Y} \).

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Definition 4.2

Let \( \mathcal{X} = (X, \le_X ) \text{ and } \mathcal{Y} = (Y, \le_Y ) \) be preorders. A feasibility relation for \( \mathcal{X} \) given \( \mathcal{Y} \) is a monotone map $$ \tag{4.3} \Phi : \mathcal{X}^{op} \times \mathcal{Y} \rightarrow \textbf{Bool} $$ We denote this by \( \Phi : \mathcal{X} \nrightarrow \mathcal{Y} \). Given \( x \in X \text{ and } y \in Y \), if \( \Phi(x, y) = \text{true} \) we say \(x\) can be obtained given \(y\).

Fredrick Eisele

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

Fredrick Eisele

2) Write down a profunctor \( \Lambda : \mathcal{X} \nrightarrow \mathcal{Y} \) and, reading \( \Lambda(x, y) = \text{true} \) as “my uncle can explain \(x\) given \(y\)”, give an interpretation of the fact that the preimage of \( \text{true} \) forms an upper set in \( \mathcal{X}^{op} \times \mathcal{Y} \).

I am not entirely sure what it means to "write down a profunctor" but I will make a guess.

$$ \begin{array}{c c | c} x & y & p \\ \text{monoid} & \text{this book} & \text{true} \\ \text{monoid} & \text{nothing} & \text{false} \\ \text{preorder} & \text{this book} & \text{true} \\ \text{preorder} & \text{nothing} & \text{false} \\ \text{category} & \text{this book} & \text{true} \\ \text{category} & \text{nothing} & \text{false} \end{array} $$ My uncle can explain categories, monoids and preorders if he has studied the book.

As will be seen later this can also be expressed as...

$$ \begin{array}{c c | c} \Lambda & \text{this book} & \text{nothing} \\ \text{monoid} & \text{true} & \text{false} \\ \text{preorder} & \text{true} & \text{false} \\ \text{category} & \text{true} & \text{false} \end{array} $$ Another way to think about this is to convert the \( \le \) into "better". On the consumer side, consuming less it "better", while on the producing side more is "better". Thus consuming less and/or producing more are both "better". Converting the consuming side means \( X_{op} \).