# 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 |

**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=48306