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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
##### Keywords:
effective descent; algebra; coalgebra
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