Tilting Theory and Functor Categories III. The Maps Category

In this paper we continue the project of generalizing tilting theory to the category of contravariant functors $Mod(C)$, from a skeletally small preadditive category $C$ to the category of abelian groups. We introduced the notion of a a generalized tilting category $T$, and extended Happel's theorem to $Mod(C)$. We proved that there is an equivalence of triangulated categories $D^b (Mod(C))\cong Db (Mod(T))$. In the case of dualizing varieties, we proved a version of Happel's theorem for the categories of finitely presented functors. We also proved in this paper, that there exists a relation between covariantly finite coresolving categories, and generalized tilting categories. Extending theorems for artin algebras. In this article we consider the category of maps, and relate tilting categories in the category of functors, with relative tilting in the category of maps. Of special interest is the category $mod(mod \Lambda)$ with $\Lambda$ an artin algebra.