WEAK HOMOMORPHISMS AND GRAPH COALGEBRAS

When coalgebras are used to model mathematical structures, such as graphs or topological spaces, standard coalgebra homomorphisms may be too strict. Relaxations of the coalgebra homomorphism concept, in either the upper or lower direction, then yield appropriate maps between the mathematical structures. There are both gains and losses of coalgebraic properties. The main examples of the paper consider coalgebras for the powerset functor, as used in the modeling of graphs. The lower morphisms yield a bicomplete category, while it is shown that the category of upper morphisms is not cocomplete.