# nLab initial object

category theory

## Applications

#### Limits and colimits

limits and colimits

# Contents

## Definition

An initial object in a category $C$ is an object $\emptyset$ such that for any object $x$ of $C$, there is a unique morphism $!:\emptyset\to x$. An initial object, if it exists, is unique up to unique isomorphism, so we speak of the? initial object.

An initial object may also be called coterminal, universal initial, co-universal, or simply universal.

Initial objects are the dual concept to terminal objects: an initial object in $C$ is the same as a terminal object in $C^{op}$. An object that is both initial and terminal is called a zero object.

## Strict initial objects

An initial object $\emptyset$ is called a strict initial object if any morphism $x\to \emptyset$ must be an isomorphism. The initial objects of a poset, of $Set$, $Cat$, $Top$, and of any topos (in fact, any extensive category) are strict. At the other extreme, a zero object is only a strict initial object if the category is trivial (equivalent to the terminal category).

Revised on November 8, 2012 13:50:11 by Urs Schreiber (82.169.65.155)