In an additive category, What are sufficient conditions for the canonical morphism from the coproduct to the product of arbitary collection of objects to be monic (when they both exist)? the conditions can be either on the category or on the objects.


(**Helpful related question:** proving this statement in abelian category without using the Freyd-Mitchell embedding theorem.) -irrelevant in view of Qiaochu's answer



Is the morphism coproduct -> product in additive category monic?
Since you mentioned locally presentable categories, I'll give one sufficient condition involving that.


Let $\mathcal{A}$ be a locally *finitely* presentable additive category. Then the canonical morphism $\sum\_{i \in I} A\_i \to \prod\_{i \in I} A\_i$ is a monomorphism. This is true when $I$ is finite because $\mathcal{A}$ is additive. Suppose $I$ is (possibly) infinite. Then $\sum\_{i \in I} A\_i$ is a filtered colimit of all its finite "sub-coproducts". But for any finite subset $I' \subseteq I$, the canonical morphism $\sum\_{i \in I'} A\_i \cong \prod\_{i \in I'} A\_i \to \prod\_{i \in I} A\_i$ is a (split!) monomorphism, and filtered colimits preserve monomorphisms, so the canonical morphism $\sum\_{i \in I} A\_i \to \prod\_{i \in I} A\_i$ is indeed a monomorphism.


In fact, all we really needed was Grothendieck's axiom AB5 (in addition to AB3 and AB3\*, of course), i.e. that filtered colimits in $\mathcal{A}$ are exact.


