# nLab 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)