Difference between revisions of "Product of a family of objects in a category"
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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. | 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. | ||
− | 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. | + | 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. |
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. | 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. |
Revision as of 21:00, 21 December 2017
A concept characterizing the notion of a Cartesian product in the language of morphisms. Let ,
, be an indexed family of objects in the category
. An object
(together with morphisms
,
) is called a product of the family of objects
,
, if for every family of morphisms
,
, there is a unique morphism
such that
,
. The morphisms
are called product projections; the product is denoted by
or
, or
in the case
. The morphism
that occurs in the definition of the product is sometimes denoted by
or
. The product of a family
,
, 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 ,
, 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 there exist uniquely defined morphisms
,
, such that
,
for
. If
is finite and the category is additive, then
and the product of the family of objects
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