nLab
disjoint coproduct

Idea

The notion of disjoint coproduct is a generalization to arbitrary categories of that of disjoint union of sets.

One says that a coproduct $X+Y$ of two objects $X,Y$ in a category $𝒞$ is disjoint if the intersection of $X$ with $Y$ in $X\coprod Y$ is empty. In this case one writes $X\coprod Y≔X+Y$ for the coproduct and speaks of the disjoint union of $X$ with $Y$.

Definition

In a category

A coproduct $a+b$ in a category is disjoint if

1. the coprojections $a\to a+b$ and $b\to a+b$ are monic, and

2. their intersection is an initial object.

Equivalently, this means we have pullback squares

$$\begin{array}{ccccccccccccc}a& \to & a& & & b& \to & b& & & 0& \to & b\\ \downarrow & & \downarrow & & & \downarrow & & \downarrow & & & \downarrow & & \downarrow \\ a& \to & a+b& & & b& \to & a+b& & & a& \to & a+b\end{array}$$\array{ a & \to & a &&& b & \to & b &&& 0 & \to & b\\ \downarrow && \downarrow &&& \downarrow && \downarrow &&& \downarrow && \downarrow \\ a & \to & a+b &&& b & \to & a+b &&& a & \to & a+b}

An arbitrary coproduct ${\coprod }_i{a}_i$ is disjoint if each coprojection $a_i\to {\coprod }_i{a}_i$ is monic and the intersection of any two is initial. Note that every 0-ary coproduct (that, is initial object) is disjoint.

In a bicategory

Properties

Characterization of extensivity and of sheaf toposes

A category having all finitary disjoint coproducts is half of the condition for a category to be extensive.

Having all small disjoint coproducts is one of the conditions in Giraud's theorem characterizing sheaf toposes.

In coherent categories

Proposition

Let $𝒞$ be a coherent category. If $X,Y\hookrightarrow Z$ are two subobjects of some object $Z\in 𝒞$ and are disjoint, in that their intersection in $Z$ is empty, $X\cap Y\simeq \varnothing$, then their union $X\cup Y$ is their (disjoint) coproduct.

This apears as (Johnstone, cor. A1.4.4).

Definition

A coherent category in which all coproducts are disjoint is also called a positive coherent category.

(Johnstone, p. 34)

Example

Every extensive category is in particular positive, by definition.

In a positive coherent category, every morphism into a coproduct factors through the coproduct coprojections:

Proposition

Let $𝒞$ be a postive coherent category, def. 1, and let $f:A\to X\coprod Y$ be a morphism. Then the two subobjects $f^{*}(X)\hookrightarrow A$ and $f^{*}(Y)\hookrightarrow Y$ of $A$, being the pullbacks in

$$\begin{array}{ccc}{f}^{*}(X)& \to & X\\ \downarrow & & {\downarrow }^{i_X}\\ A& \stackrel{f}{\to }& X\coprod Y\end{array}\begin{array}{ccc}{f}^{*}(Y)& \to & Y\\ \downarrow & & {\downarrow }^{i_Y}\\ A& \stackrel{f}{\to }& X\coprod Y\end{array}$$\array{ f^* (X) &\to& X \\ \downarrow && \downarrow^{\mathrlap{i_X}} \\ A &\stackrel{f}{\to}& X \coprod Y } \;\;\;\; \array{ f^* (Y) &\to& Y \\ \downarrow && \downarrow^{\mathrlap{i_Y}} \\ A &\stackrel{f}{\to}& X \coprod Y }

are disjoint in $A$ and $A$ is their disjoint coproduct

$$A\simeq f^{*}(X)\coprod f^{*}(Y).$$A \simeq f^*(X) \coprod f^*(Y) \,.

This appears in (Johnstone, p. 34).

Remark

This means that if $A\in 𝒞$ itself is indecomposable in that it is not a coproduct of two objects in a non-trivial way, for instance if $𝒞$ is an extensive category and $A\in 𝒞$ is a connected object, then every morphism $A\to X\coprod Y$ into a disjoint coproduct factors through one of the two canonical inclusions.

References

For instance page 34 in section A1.4.4 in

• Peter Johnstone, Sketches of an Elephant