Well-supported object

An object of a regular category is well-supported when its support of an object in a regular category is the terminal object, equivalently when $A\to 1$ is a regular epimorphism. Finite products of well-supported objects are well-supported, by pullback and composition stability of regular epimorphisms. A regular epimorphism has the same support as its codomain, and an object with a point $1\to A$ is well-supported.