terminal object

A terminal object is a categorical “endpoint” — an object into which there is exactly one way to map from anywhere. It is the most economical sink: every object has precisely one arrow to it, with no freedom. Terminal objects are the dual of initial objects, obtained by reversing every arrow, and they serve as the trivial or “one-point” object in most categories.

Formally, an object T of C is terminal if for every object X there is a unique morphism X -> T. As with initial objects, existence plus uniqueness is the whole content, and any two terminal objects are canonically isomorphic. A terminal object is exactly a limit over the empty diagram, dual to the description of an initial object as an empty colimit.

In familiar categories the terminal object is the one-element structure. In Set it is any one-point set (every function to it is constant, hence unique). In Grp and in the category of vector spaces it is the trivial group / zero space. In Top it is the one-point space, and in the category of pointed sets it coincides with the initial object — a category with an object that is both initial and terminal has a zero object, the situation in additive and abelian categories.