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 a single 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.

Initial and terminal objects are not required to exist in a given category. However, if they do exist, they are essentially unique. Specifically, if I₁ and I₂ are two different initial objects, then there is a unique isomorphism between them. Moreover, if I is an initial object then any object isomorphic to I is also an initial object. The same is true for terminal objects.

For complete categories there is an existence theorem for initial objects. Specifically, a (locally small) complete category C has an initial object if and only if there exist a set I (not a proper class) and an I-indexed family (K_i) of objects of C such that for any object X of C there at least one morphism K_i → X for some i ∈ I.

Terminal objects in a category C may also be defined as limits of the unique empty diagram ∅ → C. Since the empty category is vacuously a discrete category, a terminal object can be thought of as an empty product (a product is indeed the limit of the discrete diagram {X_i}, in general). Dually, an initial object is a colimit of the empty diagram ∅ → C and can be thought of as an empty coproduct or categorical sum.

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