  The notion of an $\mathcal{M}$-coextensive object is introduced in an
arbitrary category $\mathbb{C}$, where $\mathcal{M}$ is a distinguished class
of morphisms from $\mathbb{C}$. This notion allows for a categorical treatment
of the strict refinement property in universal algebra, and highlights its
connection with extensivity in the sense of Carboni, Lack and Walters. If
$\mathcal{M}$ is the class of all product projections in a variety of algebras
$\mathbb{C}$, then the $\mathcal{M}$-coextensive (or projection-coextensive)
objects in $\mathbb{C}$ turn out to be precisely those algebras which have the
strict refinement property. If $\mathcal{M}$ is the class of surjective
homomorphisms in the variety, then the $\mathcal{M}$-coextensive objects are
precisely those algebras which have directly-decomposable (or factorable)
congruences. In exact Mal'tsev categories, every centerless object with global
support has the strict refinement property. We will also show that in every
exact majority category, every object with global support has the strict
refinement property.
