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(0,1)-category

(0,1)-topos

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Idea

A preorder (also sometimes called a quasi-order, especially if one works with $\lt$ instead of $\leq$) is like a partial order, but without the “antisymmetry” requirement that $x \leq y$ and $y \leq x$ implies $x = y$.

By interpreting the relation $\leq$ as the existence of a unique morphism, preorders may be regarded as certain categories (namely, thin categories). This category is sometimes called the preorder category associated to a preorder; see below for details.

Definition

A preorder on a set $S$ is a reflexive and transitive relation, generally written $\leq$. 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 $x, y$, there is at most one morphism from $x$ to $y$. The existence of such a morphism corresponds to the truth of the relation $x\leq y$. In other words, it’s a (strict) category enriched over the cartesian monoidal category of truth values.

In homotopy type theory

A preorder $A$ in homotopy type theory consists of the following:

• A homotopy type $A_0$, whose terms $a:A_0$ we write as $a:A$.
• For every $a, b:A$, a proposition $p: a \leq_A b$
• For every $a:A$, a proposition $refl_a: a \leq_A a$ called reflexivity
• For every $a, b, c:A$, a function
$(a \leq_A b) \rightarrow (b \leq_A c) \rightarrow (a \leq_A c)$

called transitivity, and denoted infix as an implication by $p \implies q \implies p \wedge q$

• For every $a,b:A$, and $p: a \leq b$ we have $p \iff refl_a \wedge p$ and $p \iff p \wedge refl_b$
• For every $a,b,c,d:A$ and $p:a\leq b$, $q:b\leq c$, $r:c\leq d$, we have $p \wedge (q \wedge r) \iff (p \wedge q) \wedge r$.

A proposition $p:a \leq b$ is an equivalence relation if there is a proposition $q:b \leq a$ such that $p \wedge q \iff refl_a$ and $q \wedge p \iff refl_b$. We write $a ~_A b$ for the type of such equivalence relation, which is also a proposition.

The only relationship between the equivalence relation $a ~_A b$ as defined and the identity type $Id_A(a, b)$ for $a,b:A$ is that there exists a function

$idtoeq_{a,b}: Id_A(a,b) \rightarrow a ~_A b$

A preordered set or proset is a preorder such that the type $A_0$ is 0-truncated, and a partial order is a preorder such that for every $a,b:A$, the function $idtoeq_{a,b}$ is a homotopy equivalence. A partially ordered set or poset is a preordered set and a partial order. Every set is the core of a proset.

Properties

Relation to partial orders

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.

If the foundations have quotient sets, then every preorder has a quotient set equivalent to a poset. Let $(P, \leq)$ be a preorder. Define the equivalence relation $\sim$ on $P$ for all elements $a, b \in P$ as $a \sim b := (a \leq b) \wedge (b \leq a)$. Then the quotient set $P / \sim$ is a poset. This is the 0-truncated version of the fact that because every precategory has a core pregroupoid and every pregroupoid has a Rezk completion into a groupoid, every precategory has a Rezk completion into a category.

Relation to thin categories

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 to break the principle of equivalence of category theory. 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 $x \le y$ and $y \le x$, so that $x$ and $y$ 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 $Set \hookrightarrow \infty Grpd$, is in addition a complete Segal space or genuine category object in $Set$ 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 $\leq$ …” 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.

Preorder reflection

The 2-category of preorders (more precisely, that of thin categories) is reflective in Cat. The reflector preserves the objects and declares $x \leq y$ if there exists an arrow from $x$ to $y$.

Cauchy completion

Proposition

Internal to any regular category every poset is a Cauchy complete category.

This appears as (Rosolini, prop. 2.1).

Proposition

Internal to any exact category the Cauchy completion of any preorder exists and is its poset reflection?.

This appears as (Rosolini, corollary. 2.3).

References

Cauchy completion for preorders is discussed in

• G. Rosolini, A note on Cauchy completeness for preorders (pdf)

Last revised on May 7, 2021 at 13:47:37. See the history of this page for a list of all contributions to it.