I’m looking for examples of categories whose coproducts are not automatically disjoint.
A coproduct $X\rightarrow X\coprod Y\leftarrow Y$ is disjoint iff the pullback of the above diagram is an initial object.
I’ve considered Set, Group and Top, but all seem to have naturally disjoint coproducts.
In Group I expected non-disjoint coproducts since the underlying set of the coproduct of groups isn’t a disjoint union, but the pullbacks are singletons just like the initial group.
Any examples are appreciated.
Best Answer
Perhaps this is a cheap example, but given a set $X$, its power set $2^X$ is in particular a poset by the inclusion relation, and is thus a category. In this case, the coproduct of $U,V\subseteq X$ is given by their union, which is disjoint only if $U$ and $V$ were already disjoint as sets since $\require{AMScd}$ \begin{CD} U\cap V @>>> U \\ @VVV @VVV \\ V @>>> U\cup V \end{CD} is a pullback square (and a pushout square).