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# Question 1.6 - Preorders

edited June 2020

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?

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1.
edited January 2020

a) 1 (1*12 = 12)

b) xa = b and yb = c, thus yxa = c

c) no, we would have more than one morphism from 0 -> 0

Comment Source:a) 1 (1*12 = 12) b) x*a = b and y*b = c, thus y*x*a = c c) no, we would have more than one morphism from 0 -> 0
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2.

If a =2 and b = 3 what is x?

Comment Source:If a =2 and b = 3 what is x?
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3.
edited January 2020

@JimStuttard there is no x, so there is no morphism from 2 to 3

Comment Source:@JimStuttard there is no x, so there is no morphism from 2 to 3
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4.

@FabricioOlivetti Duh. Tnx. My bad, somehow missed the divisibility part of the definition.

Comment Source:@FabricioOlivetti Duh. Tnx. My bad, somehow missed the divisibility part of the definition.