Homomorphisms of Coalgebras from Predicate Liftings

Abstract

This paper assumes the concept of a predicate lifting from coalgebraic modal logic, and associates with every set Λ of predicate liftings for a set functor T a category $\mathbb{C}^\Lambda_T$ of T-coalgebras and socalled Λ-homomorphisms. From this construction, some natural constructions on models such as products of models and submodels can be defined. A relationship with simulations of coalgebras arising from lax extensions is established, and the main technical result gives a condition under which the category $\mathbb{C}^\Lambda_T$ is both complete and cocomplete.