nLab
distributive category

Context

Monoidal categories

Category theory

Contents

Definition

Definition

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

X×Y+X×ZX×(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×YX×(Y+Z)X\times Y \to X\times (Y+Z) is X×iX\times i, where i:YY+Zi\colon Y\to Y +Z is the coproduct injection, and dually for X×ZX×(Y+Z)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 nn-ary result for n>2n\gt 2. In fact it also implies the analogous 0-ary statement that the projection

X×00 X\times 0 \to 0

is an isomorphism for any XX. 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×Y+X×ZX×(Y+Z)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 CC with finite products and all small coproducts is infinitary distributive if the statement applies to all small coproducts. One can also consider κ\kappa-distributivity for a cardinal number κ\kappa, meaning the statement applies to coproducts of cardinality <κ\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)