Initial object

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Definition

Suppose ${\mathcal {C}}$ is a category. An initial object in ${\mathcal {C}}$ is an object $A\in \operatorname {Ob} {\mathcal {C}}$ such that for any object $B\in \operatorname {Ob} {\mathcal {C}}$, there is a unique morphism from $A$ to $B$.

If an initial object exists, then any two initial objects are isomorphic, and there is a unique isomorphism between them. In particular, the automorphism group of any initial object is trivial.

Examples

Examples of initial objects

• The empty set is an initial object in the category of sets.
• A one-point set is an initial object in the category of pointed sets.
• The trivial group is an initial object in the category of groups.
• The ring of integers is an initial object in the category of unital rings.
• The empty topological space is an initial object in the category of topological spaces.

Examples of categories without initial objects

• The category of fields has no initial object. Every field has a unique field embedding of its prime subfield; however, there are multiple isomorphism classes of prime fields.
• The category of nontrivial finite groups has no initial object.

Related notions

• Zero object
• Final object