Difference between revisions of "Pointed object"
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''of a category $\mathcal{C}$ having a terminal 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 $X$. | + | 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 set]]s, 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==== | ====Comments==== | ||
The category of pointed objects of $\mathcal{C}$ has a zero object (see [[Null object of a category|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. | The category of pointed objects of $\mathcal{C}$ has a zero object (see [[Null object of a category|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. | ||
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Latest revision as of 17:36, 22 November 2014
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.
Pointed object. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pointed_object&oldid=34258