# nLab complemented subobject

Contents

### Context

category theory

#### Constructivism, Realizability, Computability

intuitionistic mathematics

# Contents

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

## 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

Last revised on July 15, 2019 at 10:45:08. See the history of this page for a list of all contributions to it.