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''terminal object, of a category''
 
''terminal object, of a category''
  
A concept formalizing the categorical property of a one-point set. An object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040160/f0401601.png" /> in a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040160/f0401602.png" /> is called final if for every object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040160/f0401603.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040160/f0401604.png" /> the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040160/f0401605.png" /> consists of a single morphism. A final object is also called a right null object of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040160/f0401606.png" />. A left null or initial object of a category is defined in the dual way.
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A concept formalizing the categorical property of a one-point set. An object $T$ in a category $\mathfrak{K}$ is called final if for every object $X$ in $\mathfrak{K}$ the set $H_{\mathfrak{K}}(X,T)$ consists of a single morphism. A final object is also called a right null object of $\mathfrak{K}$. A left null or ''initial object'' of a category is defined in the dual way.
  
In the category of sets the final objects are just the one-point sets. In any category with null objects the final objects are the null objects (cf. [[Null object of a category]]). Other examples of final objects arise in various categories of diagrams, where the concept of a final object is essentially equivalent to that of the limit of a diagram. For example, suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040160/f0401607.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040160/f0401608.png" /> be the category of left equalizers of the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040160/f0401609.png" />; in other words, the objects of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040160/f04016010.png" /> are morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040160/f04016011.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040160/f04016012.png" />, and morphisms in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040160/f04016013.png" /> are morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040160/f04016014.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040160/f04016015.png" />. A final object in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040160/f04016016.png" /> is a kernel of the pair of morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040160/f04016017.png" /> (cf. [[Kernel of a morphism in a category|Kernel of a morphism in a category]]).
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In the category of sets the final objects are just the one-point sets, and the initial object is the empty set.. In any category with null objects the final objects are the null objects (cf. [[Null object of a category]]). Other examples of final objects arise in various categories of diagrams, where the concept of a final object is essentially equivalent to that of the limit of a diagram. For example, suppose that $\alpha,\beta:A \rightarrow B$ and let $\mathrm{Eq}(\alpha,\beta)$ be the category of left equalizers of the pair $(\alpha,\beta)$; in other words, the objects of $\mathrm{Eq}(\alpha,\beta)$ are morphisms $\mu:X \rightarrow A$ for which $\mu\alpha = \mu\beta$, and morphisms in $\mathrm{Eq}(\alpha,\beta)$ are morphisms $\gamma : (X,\mu)\rightarrow (Y,\nu)$ for which $\gamma\nu=\mu$. A final object in $\mathrm{Eq}(\alpha,\beta)$ is a kernel of the pair of morphisms $(\alpha,\beta)$ (cf. [[Kernel of a morphism in a category]]).
  
  
  
 
====Comments====
 
====Comments====
The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040160/f04016018.png" /> is, by definition, the set of morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040160/f04016019.png" />. Note that any two final objects of a category are (canonically) isomorphic, and so are two initial objects.
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The set $H_{\mathfrak{K}}(X,T)$ is, by definition, the set of morphisms in $\mathfrak{K}$ from $X$ to $T$. Note that any two final objects of a category are (canonically) isomorphic, and so are two initial objects.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Adámek,  "Theory of mathematical structures" , Reidel  (1983)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B. Mitchell,  "Theory of categories" , Acad. Press  (1965)  pp. 4</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Adámek,  "Theory of mathematical structures" , Reidel  (1983)</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  B. Mitchell,  "Theory of categories" , Acad. Press  (1965)  pp. 4</TD></TR>
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</table>
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Revision as of 21:16, 21 December 2017

terminal object, of a category

A concept formalizing the categorical property of a one-point set. An object $T$ in a category $\mathfrak{K}$ is called final if for every object $X$ in $\mathfrak{K}$ the set $H_{\mathfrak{K}}(X,T)$ consists of a single morphism. A final object is also called a right null object of $\mathfrak{K}$. A left null or initial object of a category is defined in the dual way.

In the category of sets the final objects are just the one-point sets, and the initial object is the empty set.. In any category with null objects the final objects are the null objects (cf. Null object of a category). Other examples of final objects arise in various categories of diagrams, where the concept of a final object is essentially equivalent to that of the limit of a diagram. For example, suppose that $\alpha,\beta:A \rightarrow B$ and let $\mathrm{Eq}(\alpha,\beta)$ be the category of left equalizers of the pair $(\alpha,\beta)$; in other words, the objects of $\mathrm{Eq}(\alpha,\beta)$ are morphisms $\mu:X \rightarrow A$ for which $\mu\alpha = \mu\beta$, and morphisms in $\mathrm{Eq}(\alpha,\beta)$ are morphisms $\gamma : (X,\mu)\rightarrow (Y,\nu)$ for which $\gamma\nu=\mu$. A final object in $\mathrm{Eq}(\alpha,\beta)$ is a kernel of the pair of morphisms $(\alpha,\beta)$ (cf. Kernel of a morphism in a category).


Comments

The set $H_{\mathfrak{K}}(X,T)$ is, by definition, the set of morphisms in $\mathfrak{K}$ from $X$ to $T$. Note that any two final objects of a category are (canonically) isomorphic, and so are two initial objects.

References

[a1] J. Adámek, "Theory of mathematical structures" , Reidel (1983)
[a2] B. Mitchell, "Theory of categories" , Acad. Press (1965) pp. 4
How to Cite This Entry:
Final object. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Final_object&oldid=42578
This article was adapted from an original article by M.Sh. Tsalenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article