In mathematics, a category is distributive if it has finite products and finite coproducts and such that for every choice of objects
, the canonical map is an isomorphism, and for all objects
is an isomorphism (where 0 denotes the initial object).
preserves coproducts up to isomorphisms
and aforementioned canonical maps are equal for each choice of objects.
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.
The category of sets is distributive.
denotes the coproduct in Set, namely the disjoint union, and
denotes a bijection.
In the case where A, B, and C 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.
This category theory-related article is a stub.