Categorical Product
The direct product can be abstracted to an arbitrary category. In a general category, given a collection of objects Ai and a collection of morphisms pi from A to Ai with i ranging in some index set I, an object A is said to be a categorical product in the category if, for any object B and any collection of morphisms fi from B to Ai, there exists a unique morphism f from B to A such that fi = pi f and this object A is unique. This not only works for two factors, but arbitrarily (even infinitely) many.
For groups we similarly define the direct product of a more general, arbitrary collection of groups Gi for i in I, I an index set. Denoting the cartesian product of the groups by G we define multiplication on G with the operation of componentwise multiplication; and corresponding to the pi in the definition above are the projection maps
- ,
the functions that take to its ith component gi.
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