A simplification functor for coalgebras

• Main Authors: Maurice Kianpi, Celestin Nkuimi Jugnia
• Format: Article
• Language: English
• Published: Hindawi Limited 2006-01-01
• Series: International Journal of Mathematics and Mathematical Sciences
• Online Access: http://dx.doi.org/10.1155/IJMMS/2006/56786

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 SetF 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.