On Varieties and Covarieties in a Category

A concept of equation morphism is introduced for every endofuctor F of a cocomplete category C. Equationally defined classes of F–algebras for which free algebras exist are called varieties. Every variety is proved to be monadic over C, and conversely, every monadic category is equivalent to a variety. And the Birkhoff Variety Theorem is proved for “Set–like” categories. By dualizing, we arrive at a concept of coequation such that covarieties, i.e., coequationally specified classes of coalgebras with cofree objects, precisely correspond to comonadic categories. Natural examples of covarieties are presented.