More on injectivity in locally presentable categories (Q2783423)

Let \(\mathbb{K}\) be a locally \(\lambda\)-presentable category, \({\mathcal M}\) a set of morphisms of \(\mathbb{K}\) with \(\lambda\)-presentable domains and codomains, and \({\mathcal M}\text{-Inj}\) the class of objects \(X\) of \(\mathbb{K}\) which are injective with respect to all morphisms \(h\) of \({\mathcal M}\), i.e., such that \(\Hom_\mathbb{K} (h,X)\) are surjective maps. Such classes of objects, called \(\lambda \)-injectivity classes, are characterized in this paper as being precisely the classes of objects which are closed under products, \(\lambda\)-filtered colimits and \(\lambda\)-pure subobjects. This is a sharpening of a previous result of the authors. It is somewhat extended to \(\lambda\)-cone-injectivity classes with the help of a new rather technical concept of strong closedness under \(\lambda\)-pure subobjects. Applications, examples and counter-examples are given.