Let $C$ be a category, $X$ an object of $C$, and $i\colon A\to X$ and $j\colon B\to X$ monomorphisms (we say that $A$ and $B$ are subobjects of $X$, by abuse of language). Suppose these subobjects are disjoint. Does it follow that the union of $A$ and $B$ as subobjects is abstractly isomorphic to the coproduct $A+B$? The union of $A$ and $B$ is a new monomorphism $i\cup j\colon A\cup B\to X$. And the question is whether the domain of this morphism, $A\cup B$, is isomorphic to the coproduct $A+B$.
Are disjoint unions coproducts
category-theory
Best Answer
Counterexamples are easily found in categories of algebraic objects (as long as you avoid abelian categories). For instance, in the category of groups, take $X=S_3$ and let $A$ and $B$ be the subgroups generated by a 2-cycle and a 3-cycle, respectively. These subgroups are disjoint (their meet in the lattice of subgroups is the least element, the trivial subgroup) and their join in the lattice of subgroups is the entire group $X$. However, $X$ is not a coproduct of $A$ and $B$.