Towards coalgebraic behaviourism

In this paper we show that it is possible to model observable behaviour of coalgebras independently from their internal dynamics, but within the general framework of representing behaviour by a map into a “final” coalgebra. In the first part of the paper we characterise Set-endofunctors F with the property that bisimilarity of elements of F-coalgebras coincides with having the same observable behaviour. We show that such functors have the final coalgebra of a rather simple nature, and preserve some weak pullbacks. We also show that this is the case if and only if F-bisimilarity corresponds to logical equivalence in the finitary fragment of the coalgebraic logic. In the second part of the paper, we present a construction of a “final” coalgebra that captures the observable behaviour of F-coalgebras. We keep the word “final” quoted since the object we are going to construct need not belong to the original category. The construction is carried out for arbitrary Set-endofunctor F, throughout the construction we remain in Set, but the price to pay is the introduction of new morphisms. The paper concludes with a hint to a possible application to modelling weak bisimilarity for coalgebras.