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 $\mathcal{C}$ is disjoint if the intersection of $X$ with $Y$ in $X \coprod Y$ is empty. In this case one writes $X \coprod Y \coloneqq X + Y$ for the coproduct and speaks of the disjoint union of $X$ with $Y$.
A coproduct $a+b$ in a category is disjoint if
the coprojections $a\to a+b$ and $b\to a+b$ are monic, and
their intersection is an initial object.
Equivalently, this means we have pullback squares
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.
Having all small disjoint coproducts is one of the conditions in Giraud's theorem characterizing sheaf toposes.
Let $\mathcal{C}$ be a coherent category. If $X, Y \hookrightarrow Z$ are two subobjects of some object $Z \in \mathcal{C}$ and are disjoint, in that their intersection in $Z$ is empty, $X \cap Y \simeq\emptyset$, then their union $X \cup Y$ is their (disjoint) coproduct.
This apears as (Johnstone, cor. A1.4.4).
A coherent category in which all coproducts are disjoint is also called a positive coherent category.
Every extensive category is in particular positive, by definition.
In a positive coherent category, every morphism into a coproduct factors through the coproduct coprojections:
Let $\mathcal{C}$ be a postive coherent category, def. , and let $f \colon A \to X \coprod Y$ be a morphism. Then the two subobjects $f^*(X) \hookrightarrow A$ and $f^*(Y) \hookrightarrow Y$ of $A$, being the pullbacks in
are disjoint in $A$ and $A$ is their disjoint coproduct
This appears in (Johnstone, p. 34).
This means that if $A \in \mathcal{C}$ itself is indecomposable in that it is not a coproduct of two objects in a non-trivial way, for instance if $\mathcal{C}$ is an extensive category and $A \in \mathcal{C}$ is a connected object, then every morphism $A \to X \coprod Y$ into a disjoint coproduct factors through one of the two canonical inclusions.
For instance page 34 in section A1.4.4 in
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