nLab complemented subobject

Definition

A subobject given by a monomorphism $i: S \to X$ in a coherent category is called complemented (also: decidable) if it has a complement: a subobject $\tilde{i} \colon \tilde{S} \to X$ such that

1. their intersection is the initial object

   $$S \cap \tilde{S} \;\simeq\; \emptyset$$

2. their union is all of $X$

   $$S \cup \tilde{S} \simeq X$$

Examples

In constructive mathematics, a complemented subobject in Set is called a decidable subset.

Related entries

• constructive model structure on simplicial sets

Properties

Every subobject is complemented in a Boolean category.

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.

References

• Nicola Gambino, Christian Sattler, Karol Szumiło, Section 1.1 of The constructive Kan-Quillen model structure: two new proofs (arXiv:1907.05394)