Question 1.6 - Preorders


A preorder is a category such that, for every two objects \(a,b \), there is at most one morphism \(a \to b \). That is, there either is or is not a morphism from \(a \) to \(b \), but there are never two morphisms \(a \) to \(b \). If there is a morphism \(a \to b \), we write \(a \leq b \); if there is not a morphism \(a \to b \), we don’t. For example, there is a preorder \(P \) whose objects are the positive integers \(Ob(P) = \mathbb{N}_{≥1} \) and whose hom-sets are given by \(P(a,b) := \{x \in N | x∗a = b \} \) This is a preorder because either \(P(a,b) \) is empty (if \(b \) is not divisible by \(a \)) or contains exactly one element.


(a) What is the identity on 12?

(b) Show that if \(x : a \to b \) and \(y : b \to c \) are morphisms, then there is a morphism \(y◦x \) to serve as their composite.

(c) Would it have worked just as well to take \(P \) to have all of \(\mathbb{N} \) as objects, rather than just the positive integers?


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