Zero (Category) is Initial Object

Theorem

Let $\mathbf{Cat}$ be the category of categories.

Let $\mathbf 0$ be the zero category.

Then $\mathbf 0$ is an initial object of $\mathbf{Cat}$.

Proof

Let $\mathbf C$ be an object of $\mathbf{Cat}$, i.e. a small category.

By Empty Mapping is Unique, there are unique mappings:

$F_0: \mathbf 0_0 = \O \to \mathbf C_0$

$F_1: \mathbf 0_1 = \O \to \mathbf C_1$

making $F: \mathbf 0 \to \mathbf C$ a functor by vacuous truth.

That $F_0$ and $F_1$ are actually mappings follows from $\mathbf C$ being a small category.

Hence the result, by definition of initial object.

$\blacksquare$

Sources

• 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 2.2$: Example $2.11$: $2$