Consider $A_i, \; i \in I$ a collection of objects in an Abelian category with arbitrary products and coproducts $\mathcal{C}$. Is there always a functorial monomorphism $\coprod_{i}A_i \hookrightarrow \prod_{i}A_i$?

Maybe a useful set of definitions is the following: Call a direct sum $\oplus_{i}A_i$ the image of $r:\coprod_i A_i\rightarrow \prod_i A_i$ given by

$$r_{ij}:A_i \rightarrow A_j, \;\; r_{ij}=1_{A_i}, i=j, \;\; r_{ij}=0, i\neq j.$$

(This is a definition taken from nLab.)

Then what I am actually interested in is the question

In a complete cocomplete Abelian category, is it true that $r$ is mono, i.e. $\coprod \simeq \bigoplus$?

I think I've read it on several places (e.g. planetmath, first sentence in "infinite products and coproducts"), but I am still suspicious, mainly because I don't see a reason why this should be true e.g. in categories of quasi-coherent sheaves. If this is not true, are there any hands-on examples, where this fails?

Thanks in advance for any help.

  • 4
    $\begingroup$ Well, no: if it were true then the dual result would force it to be an isomorphism, which is absurd. $\endgroup$ – Zhen Lin May 21 '15 at 14:15
  • $\begingroup$ @Zhen lin: thanks, it was silly of me to ask this. $\endgroup$ – Pavel Čoupek May 21 '15 at 16:05
  • $\begingroup$ @Pavel: I don't know how the first sentence of that section on planetmath should be read, mostly because I don't know what "direct sum" means in an arbitrary abelian category unless it means "coproduct." $\endgroup$ – Qiaochu Yuan May 21 '15 at 16:54
  • $\begingroup$ (Well, you propose a definition in the question, but it's not clear to me that it's what planetmath intended. It also doesn't do the obvious thing in $\text{Ab}^{op}$.) $\endgroup$ – Qiaochu Yuan May 21 '15 at 17:10
  • $\begingroup$ @Qiaochu Yuan: Yes, that was the main reason for my doubts. I could not make sense from the sentence otherwise, so I assumed it was this. Turns out that it is probably just talking about modules. $\endgroup$ – Pavel Čoupek May 21 '15 at 17:58

An explicit counterexample is $\text{Ab}^{op}$ (an excellent counterexample in general), which by Pontryagin duality is equivalent to the category of compact (Hausdorff) abelian groups. The product is what you think it is but the coproduct is more complicated: it is not the direct sum but the Bohr compactification of the direct sum, and because this compactification adds a lot of extra elements the natural map $\coprod A_i \to \prod A_i$ is an epimorphism but usually not a monomorphism.


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