An indecomposable nonlocally finitely generated Grothendieck category with simple objects

A Grothendieck category C is said to be locally finitely generated if the subobject lattice of every object in C is compactly generated, or equivalently, if C possesses a family of finitely generated generators. Every nonzero locally finitely generated Grothendieck category possesses simple objects. We shall call a Grothendieck category C indecomposable if C is not equivalent to a product of nonzero Grothendieck categories C1×C2. In this paper an example of an indecomposable nonlocally finitely generated Grothendieck category possessing simple objects is constructed, answering in the negative a sharper form of a question posed by Albu, Iosif, and Teply in [T. Albu, M. Iosif, M.L. Teply, Dual Krull dimension and quotient finite dimensionality, J. Algebra 284 (2005) 52–79].