A comonadic account of behavioural covarieties of coalgebras

2005 Mathematical Structures in Computer Science  
A class K of coalgebras for an endofunctor T : Set → Set is a behavioural covariety if it is closed under disjoint unions and images of bisimulation relations (hence closed under images and domains of coalgebraic morphisms, including subcoalgebras). K may be thought of as the class of all coalgebras that satisfy some computationally significant property. In any logical system suitable for specifying properties of state-transition systems in the Hennessy-Milner style, each formula will define a
more » ... lass of models that is a behavioural covariety. Assume that the forgetful functor on T -coalgebras has a right adjoint, providing for the construction of cofree coalgebras, and let G T be the comonad arising from this adjunction. Then we show that behavioural covarieties K are (isomorphic to) the Eilenberg-Moore categories of coalgebras for certain comonads G K naturally associated with G T . These are called pure subcomonads of G T , and a categorical characterization of them is given, involving a pullback condition on the naturality squares of a transformation from G K to G T . We show that there is a bijective correspondence between behavioural covarieties of T -coalgebras and isomorphism classes of pure subcomonads of G T .
doi:10.1017/s096012950400458x fatcat:g5aflvvy4rbkhoa45l7d3dnina