A simplification functor for coalgebras (Q2469042)

Summary: For an arbitrary-type functor \(F\), the notion of split coalgebras, that is, coalgebras for which the canonical projections onto the simple factor split, generalizes the well-known notion of simple coalgebras. In case \(F\) weakly preserves kernels, the passage from a coalgebra to its simple factor is functorial. This is the simplification functor. It is left adjoint to the inclusion of the subcategory of simple coalgebras into the category \(\text{Set}_{F}\) of \(F\)-coalgebras, making it an epireflective one. If a product of split coalgebras exists, then this is split and preserved by the simplification functor. In particular, if a product of simple coalgebras exists, this is simple too.