Let's start trying to understand enriched profunctors. We'll start with a very nice special case: 'feasibility relations'.
**Definition.** Suppose \\( (X, \le_X) \\) and \\( (Y, \le_Y) \\) are preorders. Then a **feasibility relation** from \\(X\\) to \\(Y\\) is a monotone function
\[ \Phi : X^{\text{op}} \times Y \to \mathbf{Bool} .\]
If \\(\Phi\\) is a feasibility relation from \\(X\\) to \\(Y\\) we write \\( \Phi : X\nrightarrow Y \\). If \\( \Phi(x,y) = \text{true}\\), we say **\\(x\\) can be obtained given \\(y\\)**.
The idea is that we use elements of \\( X\\) to describe 'requirements' - things you want - and elements of \\(Y\\) to describe 'resources' - things you have. A feasibility relation \\( \Phi : X \nrightarrow Y \\) says when what you want can be obtained from what you have! And the fact that it's _monotone_ makes a lot of sense.
In fact:
**Theorem.** A function \\(\Phi : X^{\text{op}} \times Y \to \mathbf{Bool}\\) is a feasibility relation if and only if:
1. If \\(\Phi(x,y) = \text{true}\\) and \\(x' \le_X x\\) then \\(\Phi(x',y) = \text{true}\\).
2. If \\(\Phi(x,y) = \text{true}\\) and \\(y \le_Y y'\\) then \\(\Phi(x,y') = \text{true}\\).
Translating this into English, we see it makes perfect sense:
1. If what you want can be obtained from the resources you have, and then you change your mind and want _less_, you can still obtain what you want.
2. If what you want can be obtained from the resources you have, and then you acquire _more_ resources, you can still obtain what you want.
But let's prove the theorem. This is mainly a nice review of various concepts.
**Proof.** First, remember that \\(\textbf{Bool}\\) is the preorder with two elements \\(\text{true}\\) and \\(\text{false}\\), with
\[ \text{false} \le \text{true} . \]
We can read \\(\le\\) here as 'implies'.
Second, remember that \\(X^{\text{op}}\\) is the **opposite** of the preorder \\(X\\), with the definition of \\(\le\\) turned around:
\[ x \le_{X^{\text{op}}} x' \text{ if and only if } x' \le_X x \]
Third, remember that \\( X^{\text{op}} \times Y \\) is the **product** of the preorders \\(X^{\text{op}}\\) and \\(Y\\). So, its elements are pairs \\( (x,y) \\) with \\(x \in X\\) and \\(y \in Y\\), and we define a concept of \\(\le\\) on these pairs by
\[ (x,y) \le (x',y') \text{ if and only if } x' \le_X x \text{ and } y \le_Y y' .\]
Note how I said \\(x' \le x\\). This looks backwards, but it's not one of my usual typos! It works this way because of the \\(\text{op}\\) in
\\( X^{\text{op}} \times Y \\).
Now we're ready to see exactly what a feasibility relation, that is a monotone function \\(\Phi : X^{\text{op}} \times Y \to \mathbf{Bool}\\), really is. It's a function that obeys
\[ (x,y) \le (x',y') \text{ implies } \Phi(x,y) \le \Phi(x',y') \]
or in other words
\[ \text{ if } x' \le_X x \text{ and } y \le_Y y', \text{ then } \Phi(x,y) \text{ implies } \Phi(x',y') \]
Translating this into English to see what this means, it says:
> If you can get what you want from the resources you have, and then you change your mind and want _less_, and also go out and get _more_ resources, then you can still get what you want.
Here "less" really means "less than or equal to", and "more" really means "greater than or equal to" - English is not very good at saying these things quickly! So, the process of wanting less and getting more resources can always be broken into two steps:
1. wanting less but keeping the resources the same, and then
2. getting more resources but wanting the same thing.
So, this condition
\[ \text{ if } x' \le_X x \text{ and } y \le_Y y', \text{ then } \Phi(x,y) \text{ implies } \Phi(x',y') \]
is equivalent to _the combination of both_ these conditions:
1. If \\(x' \le_X x \\), then \\( \Phi(x,y) \\) implies \\(\Phi(x',y)\\).
2. If \\(y \le y'\\), then \\( \Phi(x,y) \\) implies \\( \Phi(x,y')\\).
But here's another equivalent way to say these two things:
1. If \\(\Phi(x,y) = \text{true}\\) and \\(x' \le_X x\\) then \\(\Phi(x',y) = \text{true}\\).
2. If \\(\Phi(x,y) = \text{true}\\) and \\(y \le_X y'\\) then \\(\Phi(x,y') = \text{true}\\).
Logic! Ain't it great? \\( \qquad \blacksquare \\)
Next time I'll show you how to draw pictures of feasibility relations, and look at some examples. We've already drawn pictures of preorders, or at least posets: they're called **[Hasse diagrams](https://en.wikipedia.org/wiki/Hasse_diagram)**, and they look like a bunch of dots, one for each element of our poset \\(X\\), and a bunch of arrows, enough so that \\(x \le y\\) whenever there's a path of arrows leading from \\(x \\) to \\( y \\). So, to draw a feasibility
relation \\(\Phi : X \nrightarrow Y\\), we'll draw two Hasse diagrams and some extra arrows to say when \\( \Phi(x,y) = \text{true}\\).
Finally, two puzzles:
**Puzzle 169.** I gave a verbal argument for how we can break up any inequality \\( (x,y) \le (x',y') \\) in \\(X^{\text{op}} \times Y\\) into two other inequalities. Can you write this out in a purely mathematical way?
**Puzzle 170.** What if we have a morphism in a product of categories \\( \mathcal{C} \times \mathcal{D}\\). Can we always write it as a composite of two morphisms, copying the procedure in my verbal argument and Puzzle 169? How does this work?
**[To read other lectures go here.](http://www.azimuthproject.org/azimuth/show/Applied+Category+Theory#Chapter_4)**