Initial and terminal objects

In category theory, a branch of mathematics, an initial object of a category C is an object I in C such that for every object X in C, there exists precisely one morphism I → X.

The dual notion is that of a terminal object (also called terminal element): T is terminal if for every object X in C there exists exactly one morphism X → T. Initial objects are also called coterminal or universal, and terminal objects are also called final.

If an object is both initial and terminal, it is called a zero object or null object. A pointed category is one with a zero object.

A strict initial object I is one for which every morphism into I is an isomorphism.