Coalgebras and Approximation

Motivated by a new approach in the categorical semantics of linear logic, we investigate some specific categories of coalgebras. They all arise from the canonical comonad that one has on a category of algebras. We obtain a very simple model of linear logic where linear formulas are complete lattices and intuitionistic formulas are just sets. Also, in another, domain theoretic example, we give a new characterization of continuous posets (where every elemtent is join of elements way below) as coalgebras. And finally we describe a related example where categories in which every object is coproduct of indecomposables, are coalgebras. Approximation is the key ingredient of all these coalgebras.