Category of Ordered Sets is Category
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Theorem
Let $\mathbf{OrdSet}$ be the category of ordered sets.
Then $\mathbf{OrdSet}$ is a metacategory.
Proof
Let us verify the axioms $(C1)$ up to $(C3)$ for a metacategory.
For any two increasing mappings their composition (in the usual set theoretic sense) is again increasing by Composite of Increasing Mappings is Increasing.
For any set $X$, we have the identity mapping $\operatorname{id}_X$.
By Identity Mapping is Left Identity and Identity Mapping is Right Identity that this is the identity morphism for $X$.
That it is increasing follows from Identity Mapping is Increasing.
Finally by Composition of Mappings is Associative, the associative property is satisfied.
Hence $\mathbf{OrdSet}$ is a metacategory.
$\blacksquare$
Sources
- 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 1.4.3$