Distributive category

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In mathematics, a category is distributive if it has finite products and finite coproducts and such that for every choice of objects ${\displaystyle A,B,C}$, the canonical map

${\displaystyle [{\mathit {id}}_{A}\times \iota _{1},{\mathit {id}}_{A}\times \iota _{2}]:A\!\times \!B\,+A\!\times \!C\to A\!\times \!(B+C)}$

is an isomorphism, and for all objects ${\displaystyle A}$, the canonical map ${\displaystyle 0\to A\times 0}$ is an isomorphism (where 0 denotes the initial object). Equivalently, if for every object ${\displaystyle A}$ the endofunctor ${\displaystyle A\times -}$ defined by ${\displaystyle B\mapsto A\times B}$ preserves coproducts up to isomorphisms ${\displaystyle f}$.^[1] It follows that ${\displaystyle f}$ and aforementioned canonical maps are equal for each choice of objects.

In particular, if the functor ${\displaystyle A\times -}$ 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.