Difference between revisions of "Product of a family of objects in a category"
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− | In categories with zero morphisms, for any product | + | 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|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==== | ====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> | ||
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====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 |
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=42572