Editing Direct product (section)

== Categorical product ==
{{Main|Product (category theory)}}
The direct product can be abstracted to an arbitrary [[category theory|category]]. In a category, given a collection of objects <math>(A_i)_{i \in I}</math> indexed by a set <math>I</math>, a '''product''' of those objects is an object <math>A</math> together with [[morphisms]] <math>p_i \colon A \to A_i</math> for all <math>i \in I</math>, such that if <math>B</math> is any other object with morphisms <math>f_i \colon B \to A_i</math> for all <math>i \in I</math>, there is a unique morphism <math>B \to A</math> whose composition with <math>p_i</math> equals <math>f_i</math> for every <math>i</math>.  
<!-- This is easier to visualize as a [[commutative diagram]]; eventually, somebody should insert a relevant diagram for the categorical product here! -->
Such <math>A</math> and <math>(p_i)_{i \in I}</math> do not always exist. If they exist, then <math>(A,(p_i)_{i \in I})</math> is unique up to isomorphism, and <math>A</math> is denoted <math>\prod_{i \in I} A_i</math>.

In the special case of the category of groups, a product always exists. The underlying set of <math>\prod_{i \in I} A_i</math> is the Cartesian product of the underlying sets of the <math>A_i</math>, the group operation is componentwise multiplication, and the (homo)morphism <math>p_i \colon A \to A_i</math> is the projection sending each tuple to its <math>i</math>th coordinate.