A simplification functor for coalgebras
Maurice Kianpi, Célestin Nkuimi Jugnia · International Journal of Mathematics and Mathematical Sciences · 2006
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 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.