Dagger Preorder

definition example

A Preorder $(P,\le )$ is a dagger preorder if the Identity Function ${\mathrm{id}}_P$ is a Monotone Map $(P,\le )\to (P,{\le }^{\mathrm{op}})$ — that is, for all $p,q$: $p\le q$ implies $q\le p$. The name comes from linear algebra (dagger categories).

Sources: 7 Sketches Example 1.72, Exercise 1.73, Remark 1.74.

In a dagger preorder $\le$ is symmetric, reflexive and transitive: dagger preorders are exactly equivalence relations. A skeletal dagger preorder (one that is also a Partial Order) is a Discrete Preorder and hence “can be identified with” its underlying Set (7S Exercise 1.73).

Remark 1.74 explains “can be identified with”: any $A$ gives a unique $B$ and vice versa, with both round-trips returning where they started — the notion later made precise as Isomorphism of the corresponding objects, or Equivalence of Categories. Every Codiscrete Preorder is a dagger preorder; as categories, dagger preorders are thin groupoids.