nLab
complemented subobject

Contents

Context

Category theory

Constructivism, Realizability, Computability

Contents

Definition

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

  1. their intersection is the initial object

    SS˜ S \cap \tilde{S} \;\simeq\; \emptyset
  2. their union is all of XX

    SS˜X S \cup \tilde{S} \simeq X

Examples

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

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 CC is complemented, then every subobject in CC is complemented.

References

Last revised on July 15, 2019 at 06:45:08. See the history of this page for a list of all contributions to it.