Rel, bicategory of relations, allegory
left and right euclidean;
extensional, well-founded relations.
A comparison or cotransitive relation or weakly linear relation on a set $A$ is a (binary) relation $\sim$ on $A$ such that in every pair of related elements, any other element is related to one of the original elements in the same order as the original pair:
which generalises from $3$ to any (finite, positive) number of elements. To include the case where $n = 1$, we must explicitly state that the relation is irreflexive.
Alternatively, this is the same condition as
Comparisons are most often studied in constructive mathematics. In particular, the relation $\lt$ on the (located Dedekind) real numbers is an irreflexive comparison, even though its negation $\geq$ is not constructively total. (Indeed, $\lt$ is a pseudo-order, even though $\geq$ is not constructively a total order.)
A comparison is a cartesian monoidal semicategory enriched on the co-Heyting algebra $TV^\op$, where $TV$ is the Heyting algebra of truth values.
In dependent type theory, there are two different notions of a cotransitive or weakly linear relation
This distinction could also be made in set theory: Given a proposition $P$, $P$ can be made into a subsingleton set by considering the subset $\{\top \vert P\} \coloneqq \{p \in \{\top\} \vert (p = \top) \wedge P\}$ of the singleton $\{\top\}$. Let $[A]$ denote the support of the set $A$, and let $A \uplus B$ denote the disjoint union of sets $A$ and $B$. Then
Wikipedia, Apartness relation
Wikipedia, Pseudo-order
Univalent Foundations Project, Homotopy Type Theory – Univalent Foundations of Mathematics (2013)
Auke Booij, Analysis in univalent type theory (2020) $[$etheses:10411, pdf$]$
Last revised on September 25, 2024 at 22:24:36. See the history of this page for a list of all contributions to it.