Grothendieck categories of enriched functors
Let \(\mathcal V\) be a closed symmetric monoidal category, \(\mathcal C\) a \(\mathcal V\)-enriched category. The authors study localizations in the category of enriched functors \([\mathcal C,\mathcal V]\), and exhibits an identification between complexes of generalized modules \(Ch(\mathcal C_R)\) and enriched functor \([\text{mod} R,Ch(\text{Mod} R)]\). The notion of ``unital algebraic almost stable homotopy category'' is defined, providing a strict expansion of the class of ``unital algebraic stable homotopy category'' of Hovey-Palmieri-Strickland, by showing that the derived category of generalized modules \(\mathcal D(\mathcal C_R)\) over a commutative ring \(R\) is a unital algebraic almost stable homotopy category that is not a unital algebraic stable homotopy category. In particular, it is shown that \(\text{Ch}(\mathcal C_R)\) is a closed symmetric monoidal category and the derived category \(D(\mathcal C_R)\) is closed symmetric monoidal, and also a compactly generated triangulated category with a duality on compact objects, but the compact objects are strongly dualizable. This satisfies all axioms of Hovey-Palmieri-Strickland's unital algebraic stable homotopy theory apart from the strongly dualizable property on compact objects.
scientific article; zbMATH DE number 6537412