_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.

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(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?

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.

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(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?