---
tag: 93T4M1R5CT
kind: definition
name: Initial and terminal objects
msc: [18A30, 18A05]
defines: EK85VSSRMR
uses: [A2DE66937D, 859F0TFVA3, 0ECVK59Y03]
sources:
  - url: https://en.wikipedia.org/wiki/Initial_and_terminal_objects
---

## Definition

Let $\mathcal{C}$ be a [[category|A2DE66937D]].
An object $I$ of $\mathcal{C}$ is *initial* if for every object $X$ of $\mathcal{C}$ there is exactly one [[morphism|859F0TFVA3]] $I \to X$.
An object $T$ of $\mathcal{C}$ is *terminal* if for every object $X$ of $\mathcal{C}$ there is exactly one morphism $X \to T$.

## Remarks

If $I$ and $I'$ are both initial, the unique morphisms $I \to I'$ and $I' \to I$ compose to the unique endomorphisms, which are the identities, so they are inverse isomorphisms.
So an initial object is unique up to unique isomorphism, and the same holds for terminal objects.
A terminal object of $\mathcal{C}$ is an initial object of $\mathcal{C}^{\mathrm{op}}$.
Common notations are $0$ or $\emptyset$ for an initial object and $1$ or $*$ for a terminal object.

Example: in the category of sets, the [[empty set|0ECVK59Y03]] is initial, since the empty function is the only function $\emptyset \to X$.
Every one-point set $\{*\}$ is terminal, since the constant map is the only function $X \to \{*\}$.
In the category of rings with identity, $\Z$ is initial and the zero ring is terminal.

Non-example: in the category of sets, the two-point set $\{0, 1\}$ is not terminal, since a non-empty set $X$ has $2^{\abs{X}} \ge 2$ functions to it.
The category of fields has no initial object, since a field homomorphism preserves the characteristic, so no field maps to both $\mathbb{F}_2$ and $\Q$.