nLab bi-pointed set

 Idea

A set with two elements. Equivalently, a bi-pointed object in the category of sets. Equivalently, a pointed set $A$ with a point-preserving function $\mathbb{2} \to A$ from the boolean domain.

A bi-pointed set $S$ is trivial if the two elements $0, 1 \in S$ are equal to each other $(0 = 1)$, and it is non-trivial if they are not equal to each other $(0 \neq 1)$.

Examples of bi-pointed sets include rigs, rings, lattices, frames, pointed abelian groups, absorption monoids, bounded total orders, closed midpoint algebras, scales, and interval coalgebras.

 Properties

The boolean domain is the initial bi-pointed set, and the trivial bi-pointed set is the terminal bi-pointed set.

Related concepts

• boolean domain

• pointed set

• bi-pointed object

• bi-pointed set