Mal'cev conditions revisited
A full subcategory \({\mathcal L}\) of a locally presentable category \({\mathcal K}\) is a cosieve if for every \({\mathcal K}\)-morphism \(f:L\to K\) from \(L\in {\mathcal L}\) it follows \(K\in {\mathcal L}\). A cosieve \({\mathcal L}\) is set generated if there exists a set \({\mathcal X}\) of objects from \({\mathcal L}\) such that \(K\in {\mathcal L}\) exactly when there exist \(X\in {\mathcal X}\) and a \({\mathcal K}\)-morphism from \(X\) into \(K\). If \({\mathcal X}\) consists of exactly one object then \({\mathcal L}\) is principal. The Vopěnka principle says that there exists no large discrete full subcategory of a locally presentable category and a semiweak Vopěnka principle says that for no locally presentable category \({\mathcal K}\) there exists a family \(\{A_i\mid i\) is an ordinal\} of \({\mathcal K}\)-objects such that \(\Hom_{{\mathcal K}}(A_i,A_j)\neq\emptyset\) if and only if \(i\geq j\). It is proved that for a cosieve \({\mathcal L}\) of a locally presentable category are equivalent: \({\mathcal L}\) is set generated; \({\mathcal L}\) is closed under \(\lambda\)-pure subobjects for some regular cardinal \(\lambda\); \({\mathcal L}\) is an accessible category. The Vopěnka principle is equivalent with the fact that every cosieve of a locally presentable category is set generated and the semiweak Vopěnka principle is equivalent to the fact that every cosieve of a locally presentable category closed under products is principal. Set generated cosieves of locally presentable categories are investigated. Let \({\mathcal T}h\) be the category of algebraic theories then \({\mathcal T}h\) is locally presentable and the consequences for cosieves of \({\mathcal T}h\) are presented and their relation to Mal'cev conditions is discussed.
scientific article; zbMATH DE number 5577701
scientific article