Descent in categories of (co)algebras (Q1770313)

A morphism \(p: A'\to A\) of a category \({\mathcal A}\) is called an effective descent morphism whenever the pullback functor \(p^*: {\mathcal A}/A\to{\mathcal A}/A'\) is monadic. For an given endofunctor \(\Gamma: {\mathcal X}\to{\mathcal X}\) of a category \({\mathcal X}\), a \(\Gamma\)-algebra (resp. \(\Gamma\)-coalgebra) is a pair \((X,\alpha)\) of an object \(X\) of \({\mathcal X}\) and a morphism \(\alpha: \Gamma(X)\to X\) (resp. \(\alpha: X\to \Gamma(X)\)). With their natural morphisms, they constitute the category \({\mathcal X}^\Gamma\) (resp. \({\mathcal X}_\Gamma\)) of \(\Gamma\)-algebras (resp. \(\Gamma\)-coalgebras). In this paper, conditions are given under which a morphism of \(\Gamma\)-algebras (resp. \(\Gamma\)-coalgebras) is effective for descent.