Editing Empty product (section)

== Nullary categorical product ==
In any [[category (category theory)|category]], the [[product (category theory)|product]] of an empty family is a [[terminal object]] of that category. This can be demonstrated by using the [[limit (category theory)|limit]] definition of the product. An ''n''-fold categorical product can be defined as the limit with respect to a [[diagram (category theory)|diagram]] given by the [[discrete category]] with ''n'' objects. An empty product is then given by the limit with respect to the empty category, which is the terminal object of the category if it exists. This definition specializes to give results as above. For example, in the [[category of sets]] the categorical product is the usual Cartesian product, and the terminal object is a singleton set. In the [[category of groups]] the categorical product is the Cartesian product of groups, and the terminal object is a [[trivial group]] with one element. To obtain the usual arithmetic definition of the empty product we must take the [[decategorification]] of the empty product in the category of finite sets.

[[Dual (category theory)|Dually]], the [[coproduct]] of an empty family is an [[initial object]].
Nullary categorical products or coproducts may not exist in a given category; e.g. in the [[category of fields]], neither exists.