Distributive Category

(Category Theory)

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What is 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 (where 0 denotes the initial object). 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 (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.

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