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Difference between revisions of "Pointed object"

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''of a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073210/p0732101.png" /> having a terminal object''
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''of a category $\mathcal{C}$ having a terminal object''
  
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A pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073210/p0732102.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073210/p0732103.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073210/p0732104.png" /> is a morphism of the terminal object into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073210/p0732105.png" />. An example is a pointed topological space (see [[Pointed space|Pointed space]]). The pointed objects of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073210/p0732106.png" /> 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 $X$. 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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====Comments====
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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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====Comments====
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{{TEX|done}}
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The category of pointed objects of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073210/p0732107.png" /> has a zero object (see [[Null object of a category|Null object of a category]]), namely the terminal object of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073210/p0732108.png" /> equipped with its unique point. Conversely, if a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073210/p0732109.png" /> has a zero object, then it is isomorphic to its own category of pointed objects.
 

Revision as of 21:55, 2 November 2014

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.

How to Cite This Entry:
Pointed object. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pointed_object&oldid=15650
This article was adapted from an original article by A.F. Kharshiladze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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