Definition:Category of Ordered Sets
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Definition
The category of ordered sets, denoted $\mathbf{OrdSet}$, is the metacategory with:
Objects: | ordered sets | |
Morphisms: | increasing mappings | |
Composition: | composition of mappings | |
Identity morphisms: | identity mappings |
Note
The reason to call $\mathbf{OrdSet}$ a metacategory is foundational; allowing it to be a category would bring us to axiomatic troubles.
Also known as
Similar to some sources referring to an ordered set as a poset, $\mathbf{OrdSet}$ is also referred to as the category of posets, and consequently denoted $\mathbf{Pos}$.
Also see
- Results about the category of ordered sets can be found here.
Sources
- 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 1.4.3$