Almost total support for a regular category

A regular category is almost totally supported if every object is a well-supported object or a strict initial object. The case in which every object is well-supported is allowed. There can be at most one strict initial object up to isomorphism. In the Category of sets, these two cases distinguish nonempty sets from the empty set. A meet-semilattice regarded as a category satisfies the condition precisely when it has at most two objects up to isomorphism.

For a regular category that is a small category, a regular functor to the Category of sets that reflects isomorphisms exists exactly when the category has almost total support for a regular category. Necessity follows because a nonempty image forces the object's support to be terminal, while an empty image makes all projections $A\times X\to A$ invertible and makes all equalizers of maps out of $A$ invertible. This proves that $A$ is strict initial. For sufficiency, use the capitalization theorem to map conservatively and regularly into a capital regular category, then take global sections of a capital regular category. Well-supported objects retain points; proper strict initial objects retain empty point sets. A bijection between nonempty point sets makes the image subobject entire and makes the kernel-pair diagonal entire, by well-pointedness. Thus the image arrow is both a regular epimorphism and a monomorphism, and is invertible. The capitalization theorem is a substantive existence input; this argument does not infer arbitrary large-category representability from small-category hypotheses.