zero

The additive identity in the natural numbers, integers, real numbers and complex numbers is called **zero** and written as $0$.

More generally, in any abelian group, or even any commutative monoid, the group operation is often called ‘addition’ and written as $+$, and then the identity element is called **zero** and written as $0$. As a consequence, in any ring, or more generally any rig, the two binary operations are called ‘multiplication’ and ‘addition’, and the identity for addition is called **zero**.

Categorifying this idea, in any 2-rig the additive identity may be called **zero**. This is especially true in the case of a distributive category, that is a category with (at least finitary) products and coproducts, the former distributing over the latter. In this case the initial object, which serves as the identity for coproducts, is often called **zero**:

$x + 0 \cong x$

For example, in the category Set, the empty set is often written $0$ in the category-theoretic literature.

In an abelian category, the initial object is also terminal, and denoted $0$. More generally, any object with this property is called a zero object.

Categorifying horizontally instead, we get the notion of zero morphism.

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

Revised on August 16, 2010 05:53:00
by Toby Bartels
(75.117.109.43)