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Descent in categories of (co)algebras. (English) Zbl 1081.18001
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.

MSC:
18A20 Epimorphisms, monomorphisms, special classes of morphisms, null morphisms
18A30 Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.)
18C15 Monads (= standard construction, triple or triad), algebras for monads, homology and derived functors for monads
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