direct product in nLab

Direct products

Direct products

Definition

In universal algebra, a direct product is simply a product in a concrete category that is created by the forgetful functor.

Compare the direct sum, a more complicated concept.

Examples

Trivially, a cartesian product of sets is a direct product in Set.

One of the requirements of a topological category is that any family of objects must have a direct product, although the term ‘direct product’ is not used in topology.

Many algebraic categories, such as Grp, Ab, Ring, etc, also have all direct products; this is where the term ‘direct product’ originated.

The category of (SetSet-valued) models of any Lawvere theory has all direct products; this includes the examples from algebra above.

• direct sum