# Category Theory/Categories of ordered sets

**Definition (category of preordered sets)**:

The **category of preordered sets** is the category whose objects are given by the preordered sets and whose morphisms are given by the order homomorphisms.

**Definition (category of posets)**:

The **category of posets** is the category whose objects are the posets and whose morphisms are the order homomorphisms.

Directly from the definitions, we have:

**Proposition (posets form full subcategory of preordered sets)**:

is a full subcategory of .

**Proposition (products in the category of preordered sets)**:

Let be a family of preordered sets. Then a product of this family in the category is given by the set together with the product order, where the projections are given by the functions

- .

**Proof:** The functions are order homomorphisms, because if and , then by definition. Thus, we have a cone. Moreover, if the order homomorphisms define another cone, then as for the set product, the function given by

is the unique function from to such that for all , and it is an order homomorphism, because if , then for all we have .

**Proposition (coproducts in the category of preordered sets)**:

Let be a family of preordered sets. Then a coproduct of this family in the category is given by the parallel composition of the , where the inclusions are given by the functions

- .

**Proof:** The are order homomorphisms by definition of the parallel order, so that we do have a cocone. Suppose now that the maps define another cocone in the category . Then the unique function such that for all is given by

as in set theory, and it is an order homomorphism because if , then by def. of the parallel order and consequently .

**Proposition (products and coproducts in the category of posets)**:

In the category , products and coproducts are given by the respective products and coproducts in the category .

**Proof:** This follows since limits and colimits are preserved when restricting to a full subcategory.