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A concept characterizing the notion of a Cartesian product in the language of morphisms. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075030/p0750301.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075030/p0750302.png" />, be an indexed family of objects in the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075030/p0750303.png" />. An object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075030/p0750304.png" /> (together with morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075030/p0750305.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075030/p0750306.png" />) is called a product of the family of objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075030/p0750307.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075030/p0750308.png" />, if for every family of morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075030/p0750309.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075030/p07503010.png" />, there is a unique morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075030/p07503011.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075030/p07503012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075030/p07503013.png" />. The morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075030/p07503014.png" /> are called product projections; the product is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075030/p07503015.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075030/p07503016.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075030/p07503017.png" /> in the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075030/p07503018.png" />. The morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075030/p07503019.png" /> that occurs in the definition of the product is sometimes denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075030/p07503020.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075030/p07503021.png" />. The product of a family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075030/p07503022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075030/p07503023.png" />, is determined uniquely up to isomorphism; it is associative and commutative. The concept of the product of a family of objects is dual to that of a [[Coproduct|coproduct]] of a family of objects.
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A product of the empty family of objects is a right zero (a terminal object) of the category. In most categories of structured sets (categories of sets, groups, topological spaces, etc.) the concept of the product of a family of objects coincides with the concept of the Cartesian (direct) product of these objects. Nevertheless, this coincidence is not necessary: In the category of torsion Abelian groups the product of a family of groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075030/p07503024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075030/p07503025.png" />, is the torsion part of their Cartesian product, which in general is different from the Cartesian product itself.
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In categories with zero morphisms, for any product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075030/p07503026.png" /> there exist uniquely defined morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075030/p07503027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075030/p07503028.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075030/p07503029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075030/p07503030.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075030/p07503031.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075030/p07503032.png" /> is finite and the category is additive, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075030/p07503033.png" /> and the product of the family of objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075030/p07503034.png" /> is also their coproduct.
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A concept characterizing the notion of a Cartesian product in the language of morphisms. Let  $  A _ {i} $,
 +
$  i \in I $,
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be an indexed family of objects in the category  $  \mathfrak K $.
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An object  $  P \in  \mathop{\rm Ob}  \mathfrak K $(
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together with morphisms  $  \pi _ {i} :  P \rightarrow A _ {i} $,
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$  i \in I $)
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is called a product of the family of objects  $  A _ {i} $,
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$  i \in I $,
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if for every family of morphisms  $  \alpha _ {i} :  X \rightarrow A _ {i} $,
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$  i \in I $,
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there is a unique morphism  $  \alpha :  X \rightarrow P $
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such that  $  \alpha \pi _ {i} = \alpha _ {i} $,
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$  i \in I $.
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The morphisms  $  \pi _ {i} $
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are called product projections; the product is denoted by  $  \prod _ {i \in I }  ^  \times  A _ {i} ( \pi _ {i} ) $
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or  $  \prod _ {i \in I }  A _ {i} $,
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or  $  A _ {1} \times \dots \times A _ {n} $
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in the case  $  I = \{ 1 \dots n \} $.
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The morphism  $  \alpha $
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that occurs in the definition of the product is sometimes denoted by  $  \prod _ {i \in I }  \alpha _ {i} $
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or  $  (\times) _ {i \in I }  \alpha _ {i} $.
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The product of a family  $  A _ {i} $,
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$  i \in I $,
 +
is determined uniquely up to isomorphism; it is associative and commutative. The concept of the product of a family of objects is dual to that of a [[Coproduct|coproduct]] of a family of objects.
 +
 
 +
A product of the empty family of objects is a right zero (a [[terminal object]]) of the category. In most categories of structured sets (categories of sets, groups, topological spaces, etc.) the concept of the product of a family of objects coincides with the concept of the Cartesian (direct) product of these objects. Nevertheless, this coincidence is not necessary: In the category of torsion Abelian groups the product of a family of groups  $  G _ {i} $,
 +
$  i \in I $,
 +
is the torsion part of their Cartesian product, which in general is different from the Cartesian product itself.
 +
 
 +
In categories with zero morphisms, for any product $  P = \prod _ {i \in I }  ^  \times  A _ {i} ( \pi _ {i} ) $
 +
there exist uniquely defined morphisms $  \sigma _ {i} : A _ {i} \rightarrow P $,  
 +
$  i \in I $,  
 +
such that $  \sigma _ {i} \pi _ {i} = 1 _ {A _ {i}  } $,  
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$  \sigma _ {i} \pi _ {j} = 0 $
 +
for $  i \neq j $.  
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If $  I $
 +
is finite and the category is additive, then $  \pi _ {1} \sigma _ {1} + \dots + \pi _ {n} \sigma _ {n} = 1 $
 +
and the product of the family of objects $  A _ {1} \dots A _ {n} $
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is also their coproduct.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.Sh. Tsalenko,  E.G. Shul'geifer,  "Fundamentals of category theory" , Moscow  (1974)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.Sh. Tsalenko,  E.G. Shul'geifer,  "Fundamentals of category theory" , Moscow  (1974)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 08:07, 6 June 2020


A concept characterizing the notion of a Cartesian product in the language of morphisms. Let $ A _ {i} $, $ i \in I $, be an indexed family of objects in the category $ \mathfrak K $. An object $ P \in \mathop{\rm Ob} \mathfrak K $( together with morphisms $ \pi _ {i} : P \rightarrow A _ {i} $, $ i \in I $) is called a product of the family of objects $ A _ {i} $, $ i \in I $, if for every family of morphisms $ \alpha _ {i} : X \rightarrow A _ {i} $, $ i \in I $, there is a unique morphism $ \alpha : X \rightarrow P $ such that $ \alpha \pi _ {i} = \alpha _ {i} $, $ i \in I $. The morphisms $ \pi _ {i} $ are called product projections; the product is denoted by $ \prod _ {i \in I } ^ \times A _ {i} ( \pi _ {i} ) $ or $ \prod _ {i \in I } A _ {i} $, or $ A _ {1} \times \dots \times A _ {n} $ in the case $ I = \{ 1 \dots n \} $. The morphism $ \alpha $ that occurs in the definition of the product is sometimes denoted by $ \prod _ {i \in I } \alpha _ {i} $ or $ (\times) _ {i \in I } \alpha _ {i} $. The product of a family $ A _ {i} $, $ i \in I $, is determined uniquely up to isomorphism; it is associative and commutative. The concept of the product of a family of objects is dual to that of a coproduct of a family of objects.

A product of the empty family of objects is a right zero (a terminal object) of the category. In most categories of structured sets (categories of sets, groups, topological spaces, etc.) the concept of the product of a family of objects coincides with the concept of the Cartesian (direct) product of these objects. Nevertheless, this coincidence is not necessary: In the category of torsion Abelian groups the product of a family of groups $ G _ {i} $, $ i \in I $, is the torsion part of their Cartesian product, which in general is different from the Cartesian product itself.

In categories with zero morphisms, for any product $ P = \prod _ {i \in I } ^ \times A _ {i} ( \pi _ {i} ) $ there exist uniquely defined morphisms $ \sigma _ {i} : A _ {i} \rightarrow P $, $ i \in I $, such that $ \sigma _ {i} \pi _ {i} = 1 _ {A _ {i} } $, $ \sigma _ {i} \pi _ {j} = 0 $ for $ i \neq j $. If $ I $ is finite and the category is additive, then $ \pi _ {1} \sigma _ {1} + \dots + \pi _ {n} \sigma _ {n} = 1 $ and the product of the family of objects $ A _ {1} \dots A _ {n} $ is also their coproduct.

References

[1] M.Sh. Tsalenko, E.G. Shul'geifer, "Fundamentals of category theory" , Moscow (1974) (In Russian)

Comments

S. MacLane [a1] is generally credited with being the first to observe that Cartesian products could be described in purely categorical terms, as above.

References

[a1] S. MacLane, "Duality for groups" Bull. Amer. Math. Soc. , 56 (1950) pp. 485–516
[a2] S. MacLane, "Categories for the working mathematician" , Springer (1971) pp. Chapt. IV, Sect. 6; Chapt. VII, Sect. 7
How to Cite This Entry:
Product of a family of objects in a category. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Product_of_a_family_of_objects_in_a_category&oldid=15014
This article was adapted from an original article by M.Sh. Tsalenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article