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 (strict terminal objects are defined analogously).

Examples

The empty set is the unique initial object in Set, the category of sets. Every one-element set (singleton) is a terminal object in this category; there are no zero objects. Similarly, the empty space is the unique initial object in Top, the category of topological spaces and every one-point space is a terminal object in this category.

In the category Rel of sets and relations, the empty set is the unique initial object, the unique terminal object, and hence the unique zero object.

In the category of pointed sets (whose objects are non-empty sets together with a distinguished element; a morphism from (A, a) to (B, b) being a function f : A → B with f(a) = b), every singleton is a zero object. Similarly, in the category of pointed topological spaces, every singleton is a zero object.