Zero Object

A zero object in a Category $\mathcal{C}$ is an object $0:\mathcal{C}$ that is both Initial and Terminal.

A category equipped with a zero object is sometimes referred to as a pointed category, though this can be a bit misleading, as a pointed category is not a Pointed Object.

Examples

• The Zero Group is a zero object in the Category of Groups.
• The Unit Type is a zero object in the Category of Pointed Sets.

Properties

• The zero map $0_{X,Y}:X\to 0\to Y$ is a left and right annihilator with respect to composition.