In mathematics, a category is distributive if it has finite productss and finite coproducts and such that for every choice of objects, the canonical map

is an isomorphism is an isomorphism (where 0 denotes the initial object the endofunctor. and aforementioned canonical maps are equal for each choice of objects.

In particular, if the functor has a right adjoint (i.e., if the category is cartesian closed), it necessarily preserves all colimits, and thus any cartesian closed category with finite coproducts (i.e., any bicartesian closed category) is distributive.

Example

The category of sets is distributive. Let , , and be sets. Then

denotes the coproduct in Set, namely the disjoint union 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.

References

Further reading