# nLab distributive category

### Context

#### Monoidal categories

monoidal categories

category theory

# Contents

## Definition

###### Definition

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

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

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 $i\colon Y\to Y +Z$ is the coproduct injection, and dually for $X\times Z \to X\times (Y+Z)$.

###### Remark

This notion is part of a hierarchy of distributivity for monoidal structures, and generalizes to distributive monoidal categories and rig categories. A linearly distributive category is not distributive in this sense.

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

$X\times 0 \to 0$

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 $\lt\kappa$.

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

## References

Revised on October 31, 2013 06:04:19 by Danel? (94.175.93.67)