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Initial object

In mathematics, an initial object of a category C is an object I in C such that to every object X in C, there exists precisely one morphism I -> X. The dual notion is that of a terminal object: T is terminal, if to every object X in C there exists a single morphism X -> T. Initial objects are also called coterminal and terminal objects are also called final. If an object is both initial and terminal, we call it a zero object.


Not all categories have initial or terminal objects, as will be seen below. Directly from the definition, one can show however that if an initial object exists, then it is unique up to a unique isomorphism. The same is true for terminal objects.


This article is based on PlanetMath's article on examples of initial and terminal objects.