= Almost total support for a regular category
= Almost totally supported regular category
{synonym}
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.
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