For any abelian category $\mathcal{A}$, we can consider its derived category $\mathcal{D(A)}$, which is naturally triangulated. If $\mathcal{A}$ is endowed with a monoidal structure (bilinear with respect to biproducts) does this imply some extra structure on $\mathcal{D(A)}$? If so, how does this extra structure interact with the triangular structure of $\mathcal{D(A)}$?
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1$\begingroup$ In principle you need "enough flat objects" in $\mathcal{A} $. Using the acyclicity properties to derived the monoidal product you may proceed as on sheaves of modules over a ringed space where there are not enough projectives. $\endgroup$– Leo AlonsoCommented Jan 24, 2021 at 20:30
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4$\begingroup$ Presumably one would hope that at least under suitable hypotheses on $\mathcal{A}$, the category $\mathcal{D}(A)$ would itself be monoidal, compatibly with the triangulated structure as in the papers cited at ncatlab.org/nlab/show/tensor+triangulated+category. $\endgroup$– Mike ShulmanCommented Jan 24, 2021 at 20:54
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