nLab one

Idea

The multiplicative unit in the natural numbers $\mathbb{N}$, integers $\mathbb{Z}$, real numbers $\mathbb{R}$ and complex numbers $\mathbb{C}$ is pronounced one and denoted “$1$”. Originally, in Euclid‘s Elements, Book VII, this element $1 \in \mathbb{N}$ is called a mονάς (cf. here at monad terminology).

More generally, in any group, or even any monoid, the group operation is often called ‘multiplication’ and written as juxtaposition, and then the neutral element $\mathrm{e}$ is sometimes also denoted “$1$”. As a consequence, in any ring, or more generally any rig, the two binary operations are called ‘addition’ and ‘multiplication’, and the identity for multiplication is called one.

Categorifying this idea, in any monoidal category the unit object may be called one, best denoted now by a bolder font, such as $\mathbb{1}$. This is especially true in the case of a cartesian monoidal category, that is a category with (at least finitary) products. In this case the terminal object, which serves as the identity for products, is often denoted “$1$”:

$$x \times 1 \cong x$$

For example, in the category theory-literature the singleton set is often denoted $1$.

Categorifying horizontally instead, any identity morphism may be thought of as an analog of one and is often written $1$.

All these ideas can be, and have been, categorified further.

Related concepts

• numbers

• zero

• two

• unit

• unit object

• monad terminology