Pointed set - Knowlpedia

Definition

A pointed set is a pair $(X,x_0)$ consisting of a set $X$ and a distinguished element $x_0\in X$ , called the basepoint. A morphism of pointed sets $f:(X,x_0)\to(Y,y_0)$ is a function satisfying $f(x_0)=y_0$ .

Examples

Any group is a pointed set with its identity as basepoint, but a pointed set need not carry a multiplication. Nonabelian cohomology sets are commonly pointed by the class of the trivial cocycle or trivial bundle.

Categorical description

The category of pointed sets is equivalent to the coslice category $\{*\}\!\downarrow\!\mathbf{Set}$ : selecting a map from a singleton to $X$ is the same as selecting one element of $X$ .

References

Saunders Mac Lane, Categories for the Working Mathematician, 2nd ed., Springer, 1998. DOI record. Relevant: pointed objects and comma categories.