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

$[\mathit{id}_A \times\iota_1, \mathit{id}_A \times\iota_2] : A\times B + A\times C\to A\times(B+C)$

is an isomorphism, and for all objects $A$, 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.

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

For example, Set is distributive, while Grp is not, even though it has both products and coproducts.

An even simpler category that has both products and coproducts but is not distributive is the category of pointed sets.^[2]

References

1. Taylor, Paul (1999). Practical Foundations of Mathematics. Cambridge University Press. p. 275.
2. F. W. Lawvere; Stephen Hoel Schanuel (2009). Conceptual Mathematics: A First Introduction to Categories (2nd ed.). Cambridge University Press. pp. 296–298. ISBN 978-0-521-89485-2.

Further reading

• Cockett, J. R. B. (1993). "Introduction to distributive categories". Mathematical Structures in Computer Science 3 (3): 277. doi:10.1017/S0960129500000232.
• Carboni, Aurelio (1993). "Introduction to extensive and distributive categories". Journal of Pure and Applied Algebra 84 (2): 145–158. doi:10.1016/0022-4049(93)90035-R.