# 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.