Category Theory/Categories of ordered sets

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. 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: posets form full subcategory of preordered sets is a full subcategory of . 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 . 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 . 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 . 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 . 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 . This follows since limits and colimits are preserved when restricting to a full subcategory.