nLab zero

Definition

The additive neutral element in the natural numbers, integers, real numbers complex numbers and generally in any ring 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 neutral 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.

Categorification

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.

Related concepts

• zero vector

• zero morphism

• trivial group

• one

• two