Constant morphisms and constant subcategories
Let \({\mathcal K}\) be a category with an \(({\mathcal E}, {\mathcal M})\)-factorization system and let \({\mathcal C}\) be a subcategory of \({\mathcal K}\) closed under \({\mathcal M}\)-subobjects and \({\mathcal E} \)-quotients. A morphism \(f : A \to B\) of \({\mathcal K}\) is called constant if \(f = g \circ h\) where \(h : A \to C\) for a \(C \in{\mathcal C}\). We say that a full subcategory \({\mathcal A}\) of \({\mathcal K}\) is right-constant if there exists a class \({\mathcal B}\) of \({\mathcal K}\)-objects such that any \({\mathcal K}\)-morphism \(f : B \to A\) is constant whenever \(B \in {\mathcal B}\) and \(A \in {\mathcal A}\). The aim of this paper is to give characterizations of right-constant subcategories of \({\mathcal K}\). For example, a right-constant category is \({\mathcal E}\)-reflective if and only if \({\mathcal C}\) is an \({\mathcal E}\)-reflective subcategory of \({\mathcal K}\), or if \({\mathcal K}\) is \({\mathcal M}\)-complete and \({\mathcal M}\)-wellpowered then a reflective subcategory \({\mathcal A}\) is right-constant just when it is strongly upwards-closed (i.e., if any terminal \({\mathcal A}\)-fan belongs to \({\mathcal A})\). Hence for an \({\mathcal M}\)-complete, \({\mathcal M}\)-wellpowered, \({\mathcal E}\)-cocomplete category \({\mathcal K}\) and its reflective subcategory \({\mathcal C}\), a subcategory \({\mathcal A}\) is right-constant if and only if \({\mathcal A}\) is reflective and strongly upwards-closed. Many illustrating examples are presented.
\(({\mathcal E}, {\mathcal M})\)- factorization
scientific article; zbMATH DE number 758377