#
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.

http://www.tac.mta.ca/tac/volumes/27/10/27-10.dvi

http://www.tac.mta.ca/tac/volumes/27/10/27-10.ps

http://www.tac.mta.ca/tac/volumes/27/10/27-10.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/27/10/27-10.dvi

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/27/10/27-10.ps

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