ON SIMPLE AND EXTENSIONAL COALGEBRAS BEYOND Set

The notions of simplicity and extensionality for coalgebras are generalized beyond Set and a study of the internal dynamics of extensional coalgebras yields the notion of unitary coalgebras. In some cases, unitary coalgebras include extensional ones. Under some reasonable circumstances, both notions of simplicity and extensionality agree. Other extensions of simplicity and extensionality are *-simplicity and *-extensionality, respectively. These are obtained via orthogonality. Combining these generalizations with the notion of splitting yields some new characterizations of both simple and extensional coalgebras. Imposing some assumptions on the base category, orthogonality also helps to functorially relate every coalgebra to a simple quotient and to an extensional strong quotient. The functors thereby obtained are called simplification and extensionalization and they are left adjoint to the inclusion of the subcategory of simple coalgebras and to that of extensional coalgebras making them epi-reflective and strong epi-reflective, respectively. These functors help to construct some well-behaved classes of coalgebras. Moreover, in some important cases, the simplification functor helps to show that products of nonempty families of simple coalgebras always exist and are simple too. This yields a third functor which relates every coalgebra to the complete lattice of subcoalgebras of its simple quotient.