A preorder on a set is a reflexive and transitive relation, generally written . A preordered set, or proset, is a set equipped with a preorder. (This should not be confused with a pro-set, i.e. a pro-object in Set.)
Equivalently, a proset is a (strict) thin category: a strict category such that for any pair of objects , there is at most one morphism from to . In other words, it’s a (strict) category enriched over the cartesian monoidal category of truth values.
Any preordered set is equivalent to a poset. This is a special case of the theorem that every category has a skeleton, but (if you define ‘equivalence’ weakly enough) this case does not require the axiom of choice.
In set-theoretic foundations, a preordered set is the same as a thin category (a category in which any two parallel morphisms are equal), and it is partially ordered just when it is skeletal. Thus, asking for a preordered set to be partially ordered may seem evil. However, as remarked above, a thin category always has a skeleton which is a poset; so working with posets up to isomorphism is the same as working with preordered sets up to equivalence. In other words, if and , so that and are isomorphic, we may as well say that they are equal (since they are isomorphic in only one way).
Another way to say this is that the nerve simplicial set of a preorder, which is necessarily a Segal space or category object in an (infinity,1)-category in , is in addition a complete Segal space or genuine category object in if the preorder is in fact a partial order. For more on this perspective see at Segal space – Examples – In Set.
If we distinguish between isomorphism and equality of elements in a preordered set (hence considering preordered sets up to isomorphism, rather than up to equivalence), then this is equivalent to considering the corresponding thin category to also be a strict category. When treated in this sense, preordered sets are not equivalent to posets.
On the other hand, in non-set-theoretic foundations where not every category need have an underlying set (i.e. need not be a strict category in any canonical way) — such as homotopy type theory or preset theories — a preordered set defined as “a set with a relation …” is automatically a strict category, with a notion of equality of objects coming from the given set. By contrast, in this case a thin category (as opposed to a more general category) does have a canonical structure of strict category in which equality of objects means isomorphism, but not every strict thin category is canonical in this sense. In this case, partially ordered sets correspond to thin categories (with canonical strict-category structures), while preordered sets correspond to thin categories with arbitrary strict-category structures.
This appears as (Rosolini, prop. 2.1).
This appears as (Rosolini, corollary. 2.3).
Cauchy completion or preorders is discussed in