# nLab distributive category

Contents

### Context

#### Monoidal categories

monoidal categories

category theory

# Contents

## Definition

###### Definition

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

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

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 $X$ (see Proposition below). 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 (Lack).

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.

## Properties

###### Proposition

In a category with products and coproducts, if products distribute over binary coproducts, then coproduct coprojections are monic.

###### Proof

Let $i_B: B \to B + C$ be a coproduct coprojection, and suppose given maps $f, g: A \to B$ such that $i_B f = i_B g$. We observe that the coprojection

$i: A \times B \to A \times B + A \times C$

is monic because it has a retraction $(1_{A \times B}, \phi): A \times B + A \times C \to A \times B$. (All we need here is the existence of a map $\phi: A \times C \to A \times B$, for example the composite $A \times C \stackrel{\pi_A}{\to} A \stackrel{\langle 1_A, f \rangle}{\to} A \times B$.)

The composite of the coprojection $i$ with the canonical isomorphism $A \times B + A \times C \cong A \times (B + C)$, namely $1_A \times i_B: A \times B \to A \times (B + C)$, is therefore also monic. Given that $\langle 1_A, i_B f \rangle = \langle 1_A, i_B g \rangle: A \to A \times (B + C)$, we conclude

$(1_A \times i_B)\langle 1_A, f \rangle = \langle 1_A, i_B f \rangle = \langle 1_A, i_B g \rangle = (1_A \times i_B)\langle 1, g \rangle,$

whence $\langle 1_A, f\rangle = \langle 1_A, g\rangle: A \to A \times B$ since $1_A \times i_B$ is monic. It follows that $f = g$, as was to be shown.

###### Proposition

If products distribute over binary coproducts, then products distribute over nullary coproducts (i.e., the projection $X \times 0 \to 0$ is an isomorphism for all objects $X$).

###### Proof

We show that $X \times 0$ is initial. Clearly $\hom(X \times 0, Y)$ is inhabited by $X \times 0 \to 0 \to Y$ for any object $Y$. On the other hand, since the two coprojections $0 \to 0 + 0$ coincide, the same holds for the two coprojections $X \times 0 \to (X \times 0) + (X \times 0)$, by applying the distributivity isomorphism $X \times (0 + 0) \cong (X \times 0) + (X \times 0)$. This is enough to show that any two maps $X \times 0 \to Y$ coincide, since given maps $f, g : X \times 0 \to Y$, we have $f = [f, g] \circ i_1 = [f, g] \circ i_2 = g$.

###### Proposition

In a distributive category, the initial object is strict.

###### Proof

Given an arrow $f: A \to 0$, we have that $\pi_A: A \times 0 \to A$ is a retraction of $\langle 1, f \rangle: A \to A \times 0$, so that $A$ is a retract of $A \times 0 \cong 0$. But retracts of initial objects are initial.

## Examples

For example:

are distributive categories (hence distributive monoidal categories, hence rig categories).

These categories have in common that they are extensive. An example1 of a distributive category that is not extensive is given by

• the category of rectangular bands which has objects semigroups $X$ satisfying $x x = x$ for all $x\in X$ and $x y x=x$ for all $x,y\in X$.

Since for posets viewed as categories, finite products and coproducts are given by meets and joins we find:

• A poset viewed as a category is distributive iff it is a distributive lattice. Accordingly, non distributive lattices provide instances of categories with finite products and coproducts that do not distribute over each other.

A further non-example:

• The category $Pfn$ of sets and partial functions has finite coproducts and products. The former given by disjoint unions as in Set and the latter by $(X\times Y)+X+Y$ (where $\times,+$ indicate the usual product and coproduct in $Set$) with projections $\pi_1,\pi_2$ given on the domains of definition $(X\times Y)+X$ (resp. $(X\times Y)+Y$) by $\pi_1((x,y))=x$ and $\pi_1(x)=x$ (resp. $\pi_2((x,y))=y$ and $\pi_2(y)=y$). Since the initial object $\empty$ is not strict, $Pfn$ is not distributive.

## References

1. Pointed out by Peter Freyd in this discussion.

Last revised on March 16, 2021 at 00:20:13. See the history of this page for a list of all contributions to it.