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Fong and Spivak start out by explaining posets, which is short for "partially ordered sets". Whenever you have a set of things and a reasonable way deciding when anything in that set is "bigger" than some other thing, or "more expensive", or "taller", or "heavier", or "better" in any well-defined sense, or... anything like that, you've got a poset. When \(y\) is bigger than \(x\) we write \(x \le y\). (You can also write \(y \ge x\), of course.)
What do I mean by "reasonable"? We demand that the \(\le\) relation obey these rules:
reflexivity: \(x \le x\)
transitivity \(x \le y\) and \(y \le z\) imply \(x \le z\).
A set with a relation obeying these rules is called a preorder.
This is a fundamental concept! After all, humans are always busy trying to compare things and see what's better. So, we'll start by studying preorders.
But I can't resist revealing a secret trick that Fong and Spivak are playing on you here. Why in the world should a book on applied category theory start by discussing preorders? Why not start by discussing categories?
The answer: a preorder is a specially simple kind of category. A category, as you may have heard, has a bunch of 'objects' \(x,y,z,\dots\) and 'morphisms' between them. A morphism from \(x\) to \(y\) is written \(f : x \to y\). You can 'compose' a morphism from \(f : x \to y\) with a morphism from \(g: y \to z\) and get a morphism \(gf : x \to z\). Every object \(x\) has an 'identity' morphism \(1_x : x \to x\). And a few simple rules must hold. We'll get into them later.
But a category with at most one morphism from any object \(x\) to any object \(y\) is really just a preorder! If there's a morphism from \(x\) to \(y\) we simply write \(x \le y\). We don't need to give the morphism a name because there's at most one from \(x\) to \(y\).
So, the study of preorders is a baby version of category theory, where everything gets much easier! And when Fong and Spivak are teaching you about preorders, they're sneakily getting you used to categories. Then, when they introduce categories explicitly, you can always fall back on preorders as examples.
I've posted four puzzles on preorders here. Look at them! I just answered Puzzle 3. Puzzle 4 has millions of answers - come up with another! Also look at Puzzle 5 here. And people who already know the definition of a category, and want to ponder how preorders are a special case of categories, should try Puzzles 6 and 7 here.