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I have a monad $T: \mathcal{C} \to \mathcal{C}$ on a (Grothendieck) abelian category which preserves filtered colimits and direct sums (but is not exact). There is a finite collection $G$ of compact, projective objects which generate $\mathcal{C}$ in the sense that every object $X$ is presented as a cokernel $R_1 \to R_0\to X\to 0$ of objects $R_1$ and $R_0$ which are (possibly infinite) direct sums of objects in $G$.

I know that $T$ is idempotent on $G$: Can I conclude that $T$ is idempotent on $\mathcal{C}$?

If the answer is 'no': what sorts of extra conditions do I need to check? In my case, I have access to the category of algebras for this monad as well, and what I really want to know is that the forgetful functor is fully faithful. Any ideas would be appreciated.

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    $\begingroup$ Two very quick reactions; I haven't thought about your question properly yet. (1) If $T$ preserves finite coproducts and filtered colimits, then it preserves infinite coproducts (perhaps you meant infinite coproducts when you said direct sums?). (2) A monad $T$ is idempotent iff the forgetful functor from the category of algebras is fully faithful, so your "really want to know" is already equivalent to the original question. $\endgroup$
    – Todd Trimble ♦
    Mar 29, 2017 at 18:03
  • $\begingroup$ Thank you- I was aware of both of these. The reason I mentioned the "really want to know" piece was in case there was a criterion to check that involved knowing something about the category of algebras. Equivalent statements are not always equivalently easy to check :) $\endgroup$
    – Dylan Wilson
    Mar 29, 2017 at 18:06
  • $\begingroup$ (Tone doesn't parse well on the internet. Sorry if that came across brusque!) $\endgroup$
    – Dylan Wilson
    Mar 29, 2017 at 18:22
  • $\begingroup$ Nope! You were totally fine. :-) $\endgroup$
    – Todd Trimble ♦
    Mar 29, 2017 at 19:08
  • $\begingroup$ It would help if you add the details of the specific situation where this question arises from. $\endgroup$
    – HeinrichD
    Mar 31, 2017 at 7:07

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