# 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): $\S 1.4.3$