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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 $X$. 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.
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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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[[Category:Category theory; homological algebra]]
 

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.

How to Cite This Entry:
Pointed object. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pointed_object&oldid=34261
This article was adapted from an original article by A.F. Kharshiladze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article