# nLab Boolean prealgebra object

Contents

category theory

## Applications

#### Limits and colimits

limits and colimits

(0,1)-category

(0,1)-topos

# Contents

## Idea

A version of a Boolean algebra without the preorder being an antisymmetric relation, internal to a finitely complete category.

## Definition

In a finitely complete category $C$, a Boolean prealgebra object is a bicartesian closed preordered object $X$ with a function

$\zeta:((* \to X) \times (* \to X)) \to (* \to R)$

such that for all global elements $a:* \to X$ and $b:* \to X$,

• $s \circ \zeta(a, b) = a \Rightarrow b$
• $t \circ \zeta(a, b) = (a \Rightarrow \bot) \vee b$