Definition 7.11 (Support, totally supported, capital).

• (a)
  By the support of an object $A$ in a regular category, we mean the image of $A\to 1$. We say $A$ is well-supported if $A\to 1$ is a cover.
• (b)
  We say a regular category $C$ is totally supported if every object is well-supported. We say $C$ is almost totally supported if every object is either well-supported or a strict initial object, where we cann an object $0$ strict if every $A\to 0$ is an isomorphism. (Given finite limits, a strict object is initial since for any $A$ there exists $0\stackrel{{\pi }^{-1}}{\to }0\times A\stackrel{{\pi }_2}{\to }A$, and the equaliser of any pair $0\underset{}{\overset{}{\rightrightarrows }}A$ is $a$).
• (c)
  We say a regular category $C$ is capital if its terminal object $1$ is a detector, i.e. $C(1,\bullet )$ reflects isomorphisms.