Category Product
المؤلف:
Joshi, K. D
المصدر:
"Products and Coproducts." Ch. 8 in Introduction to General Topology. New Delhi, India: Wiley
الجزء والصفحة:
...
11-1-2022
1566
Category Product
The product of a family ![<span style=]()
{X_i}_(i in I)" src="https://mathworld.wolfram.com/images/equations/CategoryProduct/Inline1.gif" style="height:18px; width:35px" /> of objects of a category is an object 
, together with a family of morphisms ![<span style=]()
{p_i:P->X_i}_(i in I)" src="https://mathworld.wolfram.com/images/equations/CategoryProduct/Inline3.gif" style="height:18px; width:84px" /> such that for every object 
and every family of morphisms ![<span style=]()
{q_i:Q->X_i}" src="https://mathworld.wolfram.com/images/equations/CategoryProduct/Inline5.gif" style="height:16px; width:69px" /> there is a unique morphism 
such that
for all 
. The product is unique up to isomorphisms.
In the category of sets, the product is the Cartesian product, and in the category of groups it is the group direct product. In both cases, 
, and 
is the projection onto the 
th factor.
REFERENCES:
Joshi, K. D. "Products and Coproducts." Ch. 8 in Introduction to General Topology. New Delhi, India: Wiley, pp. 189-216, 1983.
Kasch, F. "Construction of Products and Coproducts." §4.80 in Modules and Rings. New York: Academic Press, pp. 80-84, 1982.
Rowen, L. "Products and Coproducts." In Ring Theory, Vol. 1. San Diego, CA: Academic Press, pp. 73-76, 1988.
Strooker, J. R. "Products and Sums." §1.5 in Introduction to Categories, Homological Algebra and Sheaf Cohomology Cambridge, England: Cambridge University Press, pp. 14-21, 1978.
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