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Let $T$ be a triangulated category closed with respect to (small) coproducts, and $t$ be (an arbitrary!) a $t$-structure on $T$. I have noted that the heart $\underline{Ht}$ of $t$ is closed with with respect to coproducts as well (that it, it is an AB3 abelian category).

Indeed, the aisle $T^{t\le 0}$ is closed with with respect to coproducts. Hence for $H_i\in \underline{Ht}$ the zeroth $t$-homology of the $T$-coproduct of $M_i$ gives the $\underline{Ht}$-coproduct of these objects (by the corresponding adjunction).

Is this correct? Did anybody formulate this (rather counter-intuitive) statement earlier?

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    $\begingroup$ I think one has to be careful about using the word 'closed with respect to coproducts' here. The heart obviously has coproducts, but in the generality you're working, I don't think they are preserved by the inclusion into your triangulated category T. $\endgroup$
    – Clark Barwick
    Commented Jul 25, 2019 at 9:04
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    $\begingroup$ Dear Clark, my problem was that I believed that this (obvious) statement is wrong.:) Besides, in my question I have described the relation between coproducts in the heart and in $T$ precisely; so I hope that this will not cause any confusion. $\endgroup$
    – Mikhail Bondarko
    Commented Jul 25, 2019 at 13:24
  • $\begingroup$ Agreed. I was initially confused by the phrase 'closed under', so I wanted to underscore the distinction, in case that was contributing to your sense of surprise! (Incidentally, do you have an easy example of such a T for which the coproduct (in T) of objects in the heart do not remain in the heart?) $\endgroup$
    – Clark Barwick
    Commented Jul 25, 2019 at 16:39
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    $\begingroup$ @ClarkBarwick, for such an example consider the dual situation: consider a Grothendieck category $A$ where products are not exact (this happens for several categories of q.coh. sheaves). Then $A$ is the heart of the canonical $t$-structure in its derived category $D(A)$, but it is not closed under products taken in $D(A)$. Now, for your example, pass to the opposite category. $\endgroup$
    – Simone Virili
    Commented Jul 25, 2019 at 18:12
  • $\begingroup$ Of course. Thanks! $\endgroup$
    – Clark Barwick
    Commented Jul 25, 2019 at 20:34

2 Answers 2

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This is Proposition 3.1.2 in the thesis of Parra, "Hearts of $t $-structures which are Grothendieck or module categories".

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That is true and the same is true for products. More than this, if your category has an enhancement that allows you to talk about homotopy co/limits, e.g., $T$ is the homotopy category of a stable model category, or of a bicomplete stable $(\infty,1)$-category or it is a base of a stable derivator, then not only the heart of any $t$-structure in $T$ is a bicomplete Abelian category, but any co/limit is computed as the $0$-th $t$-cohomology of the corresponding homotopy co/limit. For this, see Lemma 7.3 in this paper: https://arxiv.org/pdf/1708.07540.pdf

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