Descent in monoidal categories

Abstract

We consider a symmetric monoidal closed category V = (V, ⊕,I,[-,-]) together with a regular injective object Q such that the functor [-,Q]: V → 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 mod- ules if and only if it is a pure monomorphism. Examples of this kind of monoidal cat- egories 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. © Diana Rodelo and Tim Van der Linden, 2012.