nLab
initial object

Definition

An initial object in a category $C$ is an object $\varnothing$ such that for any object $x$ of $C$, there is a unique morphism $!:\varnothing \to C$. 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 $C^{\mathrm{op}}$. An object that is both initial and terminal is called a zero object.

Examples

• An initial object in a poset is a bottom element.

• The empty set is an initial object in Set.

• Likewise, the empty category is an initial object in Cat, the empty space is an initial object in Top, and so on.

• The trivial group is the initial object (in fact, the zero object) of Grp and Ab.

• The integers are the initial object of Ring.

Strict initial objects

An initial object $\varnothing$ is called strict if any morphism $x\to \varnothing$ must be an isomorphism. The initial objects of a poset, of $\mathrm{Set}$, $\mathrm{Cat}$, $\mathrm{Top}$, 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).