Varieties of F-algebras with respect to an endofunctor F on an arbitrary cocomplete category C are defined as equational classes admitting free algebras. They are shown to correspond precisely to the monadic categories over C. Under suitable assumptions satisfied in particular by any endofunctor on Set and Setop the Birkhoff Variety Theorem holds. By dualization, covarieties over complete categories C are introduced, which then correspond to the comonadic categories over C, and allow for a characterization in dual terms of the Birkhoff Variety Theorem. Moreover, the well known conditions of accessibilitly and boundedness for Set-functors F, sufficient for the existence of cofree F-coalgebras, are shown to be equivalent. ©2001 Published by Elsevier Science B.V.
Adámek, J., & Porst, H. E. (2001). From varieties of algebras to covarieties of coalgebras. In Electronic Notes in Theoretical Computer Science (Vol. 44, pp. 27–46). Elsevier. https://doi.org/10.1016/S1571-0661(04)80901-0