Distributive category

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

$[1\times\iota_1,1\times\iota_2] : A\times B + A\times C\to A\times(B+C)$

is an isomorphism, and for all objects A, the canonical map $0 \to A\times 0$ is an isomorphism. Equivalently. if for every object A the functor $A\times -$ preserves coproducts up to isomorphisms f ^[1]. It follows that f and aforementioned canonical maps are equal for each choice of objects.

For example, Set is distributive, while Grp is not.

1. ^ Taylor, Paul (1999). Practical Foundations of Mathematics. Cambridge University Press. p. 275.