# Descent in monoidal categories

## Bachuki Mesablishvili

We consider a symmetric monoidal closed category $V = (V, \otimes, I, [-,-])$ together with a regular injective object $Q$ such that the functor $[-, Q] : \to V^{op}$ is comonadic and prove that in such a category, as in the monoidal category of abelian groups, a morphism of commutative monoids is an effective descent morphism for modules if and only if it is a pure monomorphism. Examples of this kind of monoidal categories are elementary toposes considered as cartesian closed monoidal categories, the module categories over a commutative ring object in a Grothendieck topos and Barr's star-autonomous categories.

Keywords: symmetric monoidal categories, effective descent morphisms, pure morphisms

2010 MSC: 18A20, 18D10, 18D35

Theory and Applications of Categories, Vol. 27, 2012, No. 10, pp 210-221.

Published 2012-10-09.

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