Product of a family of objects in a category

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