Relation Liftings on Preorders and Posets

TL;DR

This paper extends the theory of relation liftings from sets to preorders and posets, showing how preservation conditions adapt and applying the results to Moss's coalgebraic structures.

Contribution

It generalizes relation lifting results from sets to enriched categories like preorders and posets, replacing weak pullbacks with exact lax squares.

Findings

Relation liftings extend to preorders and posets.

Preservation of weak pullbacks becomes preservation of exact lax squares.

Application to Moss's coalgebraic structures on posets.

Abstract

The category Rel(Set) of sets and relations can be described as a category of spans and as the Kleisli category for the powerset monad. A set-functor can be lifted to a functor on Rel(Set) iff it preserves weak pullbacks. We show that these results extend to the enriched setting, if we replace sets by posets or preorders. Preservation of weak pullbacks becomes preservation of exact lax squares. As an application we present Moss's coalgebraic over posets.