nLab bi-pointed set

 Idea

A bipointed set is a set $S$ equipped with the choice of a pair $s_1, s_2 \in S$ of its elements, hence with a function $(s_1, s_2) \colon \mathbb{2} \to S$ from the boolean domain.

Equivalently, this is a bi-pointed object in the category of sets.

Naturally occuring examples of bi-pointed sets include (the underlying sets of rigs, rings, lattices, frames, pointed abelian groups, absorption monoids, bounded total orders, closed midpoint algebras, scales, and interval coalgebras, which are all bipointed by the elements usually denoted “0” and “1”

 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 type