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)