Morita duality for Grothendieck categories (Q5906584)

The aim of the paper is to define and investigate a general concept of Morita duality for Grothendieck categories. The first section develops a unified theory of linear topology and linear compactness in complete abelian categories. In the second section the general concept of Morita duality between two Grothendieck categories \({\mathcal C}_ 1\) and \({\mathcal C}_ 2\) is defined as being a pair \((D_ 1,D_ 2)\) of additive contravariant equivalence functors between finitely closed generating full subcategories \({\mathcal S}_ 1\) and \({\mathcal S}_ 2\) of \({\mathcal C}_ 1\) and \({\mathcal C}_ 2\), respectively. In this case one says that \({\mathcal C}_ 1\) (and \({\mathcal C}_ 2)\) possess a Morita duality. It is proved that a Grothendieck category \({\mathcal C}\) has a Morita duality iff \({\mathcal C}\) is generated by its discrete linearly compact objects. A special case of the Morita duality theory developed by the authors is the Morita duality theory in the sense of Colby and Fuller, which is discussed in the third section. The last section deals with the connection of the Morita duality introduced in this paper with Lefschetz duality of vector spaces, Pontrjagin duality of abelian groups, Oberst's duality for Grothendieck categories, Morita duality for rings with identity and also only with local units.