Types and coalgebraic structure

Abstract.

We relate weak limit preservation properties of coalgebraic type functors F to structure theoretic properties of the class $\mathcal{S}et_F$ of all F-coalgebras. In particular, we give coalgebraic characterizations for the condition that F weakly preserves pullbacks, kernel pairs or preimages. We also describe regular monos and epis. In case that |F(1)| ≠ 1 we show that F preserves preimages iff $\mathcal{H}\mathcal{S}(\mathcal{K}) = \mathcal{S}\mathcal{H}(\mathcal{K})$ for every class $\mathcal{K}$ of F-coalgebras. The case |F(1)| = 1 is left as an open problem.