complemented subobject

A subobject given by a monomorphism $i: S \to X$ in a coherent category is **complemented** if it has a complement: a subobject $\tilde{i}: \tilde{S} \to X$ such that $S \cap \tilde{S}$ is the initial object and $S \cup \tilde{S} = X$. Every subobject is complemented in a Boolean category.

In constructive mathematics, a complemented subobject in Set is called a decidable subset; in classical mathematics, every subset is decidable. Indeed, the law of excluded middle may be taken to say precisely that every subset of the point is complemented.

More generally, if every subobject of the terminal object of a well-pointed coherent category $C$ is complemented, then every subobject in $C$ is complemented.

Revised on December 11, 2009 10:20:42
by Toby Bartels
(173.60.119.197)