Objects with closed diagonals
Let \({\mathfrak C}\) be a category with finite products, \({\mathfrak B}^ a \)subcategory, M a class of monomorphisms and \(\Delta\) (X): \(X\to X\times X\). The authors study conditions such that a diagonal property of the type \(X\in {\mathfrak B}\Leftrightarrow \Delta (X)\in M\) holds. For example: Let \({\mathfrak A}\) be a reflective subcategory, r its reflector, k(T,X) the canonical morphism r(T\(\times X)\to r(T)\times r(X)\), M the class of equalizers of any two morphisms into an object of \({\mathfrak A}\) and \({\mathfrak B}\) the category of all subobjects of objects of \({\mathfrak A}\). Then, if k(T,X) is mono for all \(t\in {\mathfrak C}\), X belongs to \({\mathfrak B}\) if and only if \(\Delta\) (X)\(\in M\).
scientific article; zbMATH DE number 4061466