Distributive category


In mathematics, a category is distributive if it has finite products and finite coproducts such that for every choice of objects, the canonical map
is an isomorphism, and for all objects, the canonical map is an isomorphism. Equivalently, if for every object the endofunctor defined by preserves coproducts up to isomorphisms. It follows that and aforementioned canonical maps are equal for each choice of objects.
In particular, if the functor has a right adjoint, it necessarily preserves all colimits, and thus any cartesian closed category with finite coproducts is distributive.

Example

The category of sets is distributive. Let,, and be sets. Then
where denotes the coproduct in Set, namely the disjoint union, and denotes a bijection. In the case where,, and are finite sets, this result reflects the distributive property: the above sets each have cardinality.
The categories Grp and Ab are not distributive, even though they have both products and coproducts.
An even simpler category that has both products and coproducts but is not distributive is the category of pointed sets.