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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

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2 Answers 2

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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.

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  • $\begingroup$ You seem to use that a finitely presentable category has the property that filtered colimits of monics are monic. Where can I find this? $\endgroup$ Oct 2, 2013 at 20:08
  • $\begingroup$ In a locally finitely presentable category, filtered colimits preserve finite limits. See my comment here. $\endgroup$
    – Zhen Lin
    Oct 2, 2013 at 20:25
  • $\begingroup$ I marked the question as answered, since this was the only answer available, although I wonder if there is another conditions, or maybe being locally finitely presented is also necessary. $\endgroup$
    – Yitzhak Z
    Oct 8, 2013 at 22:52
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    $\begingroup$ It is not necessary: AB5 is enough, and there are abelian categories that satisfy AB5 but are not locally finitely presentable (such as abelian sheaves). $\endgroup$
    – Zhen Lin
    Oct 8, 2013 at 22:58
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This statement isn't true in all abelian categories. The morphism $\bigoplus \mathbb{Z} \to \prod \mathbb{Z}$ from an infinite coproduct to an infinite product of copies of $\mathbb{Z}$ is monic but not epic in $\text{Ab}$, so the corresponding morphism in $\text{Ab}^{op}$ is epic but not monic. Freyd-Mitchell doesn't help here because the embedding it gives is only guaranteed to be exact; it isn't guaranteed to preserve infinite products or coproducts.

(Note that by Pontrjagin duality $\text{Ab}^{op}$ is the category of compact Hausdorff abelian groups, with $\mathbb{Z}$ being sent to $S^1$, so the claim is that the canonical morphism from an infinite coproduct to an infinite product of copies of $S^1$ is epic but not monic. This might seem strange, but the infinite coproduct in compact Hausdorff abelian groups is not the infinite direct sum: it's the Bohr compactification of the infinite direct sum. Also note that $\text{Ab}$ is locally finitely presentable and $\text{Ab}^{op}$ isn't, so maybe that's a hint as to what the correct conditions to impose are?)

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  • $\begingroup$ Thanks for the parenthetical remark, it was quite helpful. $\endgroup$ Oct 2, 2013 at 18:25
  • $\begingroup$ thanks for the clarification! however, the main question remains... (maybe I'll look into the topic of presentable categories, which I didn't learn yet) $\endgroup$
    – Yitzhak Z
    Oct 2, 2013 at 19:28
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    $\begingroup$ You don't have to, AB5 suffices. $\endgroup$ Oct 2, 2013 at 20:11

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