# Pointed object

2020 Mathematics Subject Classification: *Primary:* 18A [MSN][ZBL]

*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 $X$. Examples are pointed sets, and pointed topological spaces (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.

**How to Cite This Entry:**

Pointed object.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Pointed_object&oldid=34807