---
tag: EK85VSSRMR
kind: concept
name: Initial and terminal objects
msc: [18A30, 18A05]
sources:
  - url: https://en.wikipedia.org/wiki/Initial_and_terminal_objects
same_as:
  wikidata: Q529752
  wikipedia: Initial and terminal objects
---

## Summary

An initial object of a [[category|A2DE66937D]] is an object with exactly one morphism to every object.
A terminal object is an object with exactly one morphism from every object.
They are the simplest universal properties: the empty set and the one-point set play these roles among sets.
Each is unique up to a unique isomorphism when it exists, and many constructions are initial or terminal objects of suitable auxiliary categories.

## Explanations

### L1 human

An initial object is a starting point: from it there is one and only one arrow to anything.
A terminal object is an end point: from anything there is one and only one arrow to it.
Among sets, the [[empty set|0ECVK59Y03]] is initial, because there is exactly one function from it to any set, the empty function.
A one-point set is terminal, because every set has exactly one function to it, sending everything to the single point.
Among groups, the trivial group is both initial and terminal.