Definition:Category of Ordered Sets

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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


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.