The notion of disjoint coproduct is a generalization to arbitrary categories of that of disjoint union of sets.
One says that a coproduct of two objects in a category is disjoint if the intersection of with in is empty. In this case one writes for the coproduct and speaks of the disjoint union of with .
A coproduct in a category is disjoint if
the coprojections and are monic, and
their intersection is an initial object.
Equivalently, this means we have pullback squares
An arbitrary coproduct is disjoint if each coprojection is monic and the intersection of any two is initial. Note that every 0-ary coproduct (that, is initial object) is disjoint.
In the category of sets and functions, the coproduct is given by disjoint union and is, unsurprisingly, disjoint. In the category of sets and partial functions the coproduct is equally given by disjoint union and total injections and is disjoint as well.
Since having all finitary disjoint coproducts is half of the condition for a category to be extensive, extensive categories provide examples for categories with disjoint finite coproducts. In the preceding discussion instantiates this case whereas does not: since the initial object in is not strict, the latter category is not extensive.
In the category of (real) vector spaces coproducts are given by direct sum and are disjoint but not stable under pullback: pulling back the colimit diagram along the diagonal morphism yields which is not a colimit diagram. Whence is not extensive.
Non-example: the interval category has coproducts but is they are not all disjoint: . There are plenty more examples of posets that have non-disjoint coproducts besides this one. In a Boolean algebra two elements and are disjoint in the sense that if and only if is their disjoint coproduct.
Having all small disjoint coproducts is one of the conditions in Giraud's theorem characterizing sheaf toposes.
Let be a coherent category. If are two subobjects of some object and are disjoint, in that their intersection in is empty, , then their union 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 be a postive coherent category, def. , and let be a morphism. Then the two subobjects and of , being the pullbacks in
are disjoint in and is their disjoint coproduct
This appears in (Johnstone, p. 34).
This means that if itself is indecomposable in that it is not a coproduct of two objects in a non-trivial way, for instance if is an extensive category and is a connected object, then every morphism 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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