Covarieties of Coalgebras: Comonads and Coequations

Coalgebras provide effective models of data structures and state-transition systems. A virtual covariety is a class of coalgebras closed under coproducts, images of coalgebraic morphisms, and subcoalgebras defined by split equalisers. A covariety has the stronger property of closure under all subcoalgebras, and is behavioural if it is closed under domains of morphisms, or equivalently under images of bisimulations. There are many computationally interesting properties that define classes of these kinds. We identify conditions on the underlying category of a comonad G which ensure that there is an exact correspondence between (behavioural/virtual) covarieties of G-coalgebras and subcomonads of G defined by comonad morphisms to G with natural categorical properties. We also relate this analysis to notions of coequationally defined classes of coalgebras.