nLab
distributive category

Definition

Definition

A category $C$ with finite products $(-)\times (-)$ and coproducts $(-)\mathrm{plus}(-)$ is called (finitary) distributive if for any $X,Y,Z\in C$ the canonical distributivity morphism

$$X\times Y+X\times Z\to X\times (Y+Z)$$X\times Y + X\times Z \to X\times (Y+Z)

is an isomorphism. The canonical morphism is the unique morphism such that $X\times Y\to X\times (Y+Z)$ is $X\times i$, where $i:Y\to Y+Z$ is the coproduct injection, and dually for $X\times Z\to X\times (Y+Z)$.

Remark

A linearly distributive category is not distributive in this sense.

This axiom on binary coproducts easily implies the analogous $n$-ary result for $n>2$. In fact it also implies the analogous 0-ary statement that the projection

$$X\times 0\to 0$$X\times 0 \to 0

is an isomorphism for any $X$. Moreover, for a category with finite products and coproducts to be distributive, it actually suffices for there to be any natural family of isomorphisms $X\times Y+X\times Z\cong X\times (Y+Z)$, not necessarily the canonical ones; see the paper of Lack referenced below.

A category $C$ with finite products and all small coproducts is infinitary distributive if the statement applies to all small coproducts. One can also consider $\kappa$-distributivity for a cardinal number $\kappa$, meaning the statement applies to coproducts of cardinality $<\kappa$.

Any extensive category is distributive, but the converse is not true.

Related concepts

• distributive law

• bipermutative category

References

• Aurelio Carboni, and Stephen Lack, and Robert F. C. Walters, Introduction to extensive and distributive categories, Journal of Pure and Applied Algebra 84 (1993) 145-158,

• Stephen Lack, Non-canonical isomorphisms. arXiv:0912.2126.