Are the injections of a coproduct a cover in the canonical pretopology?

Question

Assume we're in a category $C$ with all pullbacks and finite coproducts.

Recall that the canonical coverage of $C$ is the finest Grothendieck (pre) topology for which all representables are sheaves. A family is covering in the canonical coverage iff every representable is a sheaf for every pullback of the family.

Is it the case that $\{i_1, i_2\}$ is in the canonical coverage of $A+B$ in $C$, where these are the injections of the coproduct?

I was able to prove it when $C$ is LCCC because then pullback and coproduct commute, but I couldn't figure it out in general.

Answer 1

This will not be the case in general. A family is a cover in the canonical topology if all its pullbacks are jointly regular epimorphism.

So this will for example be the case if coproducts are universal (for example if the category is LCC) but this has no reason to be true in general.

For example, in the category of abelian group, we have the diagonal map $\Delta: G \to G \oplus G$ and the two pullbacks along the coproduct inclusion $G \to G \oplus G$ are $0$, so the two coproduct inclusion don't form a canonical cover (unless $G = 0$).