# nLab one

The multiplicative identity in the natural numbers, integers, real numbers and complex numbers is called one and written as $1$.

More generally, in any group, or even any monoid, the group operation is often called ‘multiplication’ and written as juxtaposition, and then the identity element is called one and written as $1$. 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 monoidal unit may be called one. 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 called one:

$x \times 1 \cong x$

For example, in the category Set, the singleton set is often written $1$ in the category-theoretic literature.

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.

Created on August 16, 2010 05:52:25 by Toby Bartels (75.117.109.43)