Pointed object

of a category $\mathcal{C}$ having a terminal object

A pair $(X,x_0)$ where $X \in \mathrm{Ob}\mathcal{C}$ and $x_0$ is a morphism of the terminal object into $X$. An example is a pointed topological space (see Pointed space). The pointed objects of $\mathcal{C}$ form a category, in which the morphisms are the mappings sending the distinguished point to the distinguished point.

Comments

The category of pointed objects of $\mathcal{C}$ has a zero object (see Null object of a category), namely the terminal object of $\mathcal{C}$ equipped with its unique point. Conversely, if a category $\mathcal{C}$ has a zero object, then it is isomorphic to its own category of pointed objects.