Difference between revisions of "Final object"
From Encyclopedia of Mathematics
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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. | 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. | ||
| − | 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. 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]]). | + | 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]]). |
Revision as of 20:54, 21 December 2017
terminal object, of a category
A concept formalizing the categorical property of a one-point set. An object 
in a category 
is called final if for every object 
in 
the set 
consists of a single morphism. A final object is also called a right null object of 
. 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 
and let 
be the category of left equalizers of the pair 
; in other words, the objects of 
are morphisms 
for which 
, and morphisms in 
are morphisms 
for which 
. A final object in 
is a kernel of the pair of morphisms 
(cf. Kernel of a morphism in a category).
Comments
The set 
is, by definition, the set of morphisms 
. 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=13638
Final object. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Final_object&oldid=13638
This article was adapted from an original article by M.Sh. Tsalenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article