Initial Object

An initial object in a Category $\mathcal{C}$ is an object $\bot:\mathcal{C}$ such that there is a unique morphism $\bot\to X$ to any other object $X$.

More abstractly, an initial object is a Colimit over the Initial Category.

Examples

• The initial object in the Category of Sets is the Empty Type.

• The initial object in the Category of Rings is the Ring of Integers, as Ring Homomorphisms preserve both $0$ and $1$.

• The initial object in the Category of Groups is the Zero Group, which is also the Terminal Object. This makes the zero group a canonical example of a Zero Object.