An initial object has a unique morphism to every other object in the category.

### Strict Initial Object

A **strict** initial object has no morphisms into it (apart from its own identity).

In type theory this could be related to the empty type and way to represent 'false'.

An example of a strict initial object is the empty set.

## Examples in Various Categories

Initial Object | |
---|---|

Notation | 0 |

generalisation | a kind of colimit |

universal cone over diagram | |

examples: set: | Ø = {} empty set |

group (null object) | trivial group (just identity element) |

topological space | empty space |

poset | least element (if exists) |

monoid | trivial monoid |

semigroup | empty semigroup |

Rng | ring of integers Z |

fields | does not have initial object |

Vec | zero object |

Top | empty space |

Grf | the null graph (without vertices and edges) |

Ω-Alg algebra with signature Ω |
initial (term) algebra whose carrier consists of all finite trees. |

Cat | empty category |