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.

Related notions

• Zero object
• Final object