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International Journal of Mathematics and Mathematical Sciences
Volume 2006, Article ID 56786, 9 pages
http://dx.doi.org/10.1155/IJMMS/2006/56786

A simplification functor for coalgebras

Laboratory of Algebra, Department of Mathematics, Faculty of Science, University of Yaounde 1, P.O. Box 812, Yaounde, Cameroon

Received 22 July 2005; Revised 21 June 2006; Accepted 5 July 2006

Copyright © 2006 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.