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Difference between revisions of "Null object of a category"

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''zero (object) of a category''
 
''zero (object) of a category''
  
An object (usually denoted by 0) such that for every object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067890/n0678901.png" /> of the category the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067890/n0678902.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067890/n0678903.png" /> are singletons. The null object, if it exists in a given category, is uniquely determined up to isomorphism. In the category of sets with a distinguished point the null object is a singleton, in the category of groups it is the trivial group, in the category of modules it is the zero module, etc. Not every category contains a null object, but a null object can always be formally adjoined to any given category. Every category with a null object has null morphisms.
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An object (usually denoted by $\mathbf{0}$) such that for every object $X$ of the category the sets $H(X,\mathbf{0})$ and $H(\mathbf{0},X)$ are singletons. The null object, if it exists in a given category, is uniquely determined up to isomorphism. In the category of [[pointed set]]s (sets with a distinguished point) a [[singleton]] is a null object, in the category of groups it is the trivial group, in the category of modules it is the zero module, etc. Not every category contains a null object, but a null object can always be formally adjoined to any given category.  
  
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====References====
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* S. MacLane, "Categories for the working mathematician" Graduate Texts in Mathematics '''5''', Springer (1971) {{ISBN|0-387-98403-8}}
  
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====Comments====
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An object $I$ of a category is called initial if there is just one morphism $I\to X$ for any $X$, and terminal (or final) if there is just one morphism $X\to I$ for any $X$. Thus a null object is one which is both an [[initial object]] and a [[terminal object]]. If an initial object exists in a given category, it is unique up to isomorphism, and similarly for terminal objects; but the initial and terminal objects of a category need not be isomorphic. For example, in the category of sets, the empty set is an initial object and any singleton is terminal. A terminal object of a category may be regarded as a limit for the empty diagram in that category (cf. the editorial comments to [[Limit|Limit]] for the concept of a limit of a diagram in a category). Conversely, a limit of an arbitrary diagram may be defined as a terminal object in an appropriate category of cones.
  
====Comments====
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If $A$, $B$ are objects of a category with a null object $\mathbf{0}$, then there is a unique composite map $A \rightarrow \mathbf{0} \rightarrow B$, the ''zero'' or ''null morphism'' from $A$ to $B$.
An object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067890/n0678904.png" /> of a category is called initial if there is just one morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067890/n0678905.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067890/n0678906.png" />, and terminal (or final) if there is just one morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067890/n0678907.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067890/n0678908.png" />. Thus a null object is one which is both initial and terminal. If an initial object exists in a given category, it is unique up to isomorphism, and similarly for terminal objects; but a category may have non-isomorphic initial and terminal objects. For example, in the category of sets, the empty set is an initial object and any singleton is terminal. A terminal object of a category may be regarded as a limit for the empty diagram in that category (cf. the editorial comments to [[Limit|Limit]] for the concept of a limit of a diagram in a category). Conversely, a limit of an arbitrary diagram may be defined as a terminal object in an appropriate category of cones.
 

Latest revision as of 18:38, 13 November 2023

2020 Mathematics Subject Classification: Primary: 18B [MSN][ZBL]

zero (object) of a category

An object (usually denoted by $\mathbf{0}$) such that for every object $X$ of the category the sets $H(X,\mathbf{0})$ and $H(\mathbf{0},X)$ are singletons. The null object, if it exists in a given category, is uniquely determined up to isomorphism. In the category of pointed sets (sets with a distinguished point) a singleton is a null object, in the category of groups it is the trivial group, in the category of modules it is the zero module, etc. Not every category contains a null object, but a null object can always be formally adjoined to any given category.

References

  • S. MacLane, "Categories for the working mathematician" Graduate Texts in Mathematics 5, Springer (1971) ISBN 0-387-98403-8

Comments

An object $I$ of a category is called initial if there is just one morphism $I\to X$ for any $X$, and terminal (or final) if there is just one morphism $X\to I$ for any $X$. Thus a null object is one which is both an initial object and a terminal object. If an initial object exists in a given category, it is unique up to isomorphism, and similarly for terminal objects; but the initial and terminal objects of a category need not be isomorphic. For example, in the category of sets, the empty set is an initial object and any singleton is terminal. A terminal object of a category may be regarded as a limit for the empty diagram in that category (cf. the editorial comments to Limit for the concept of a limit of a diagram in a category). Conversely, a limit of an arbitrary diagram may be defined as a terminal object in an appropriate category of cones.

If $A$, $B$ are objects of a category with a null object $\mathbf{0}$, then there is a unique composite map $A \rightarrow \mathbf{0} \rightarrow B$, the zero or null morphism from $A$ to $B$.

How to Cite This Entry:
Null object of a category. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Null_object_of_a_category&oldid=13145
This article was adapted from an original article by M.Sh. Tsalenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article