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Let us call a triangulated category algebraic if it admits a differential graded enhancement (i.e., an enrichment in complexes of abelian groups). Certainly, there is a notion of a tensor product on differential graded categories. My question is: does there exist an algebraic tensor triangulated category such that this tensor product is not compatible with any its differential enhancement (as well as with enhancements of equivalent categories)? If the answer is not known, do there exist any reasonable candidates for examples of this sort?

In particular, what can one say about the tensor product on the category of modules over the Morava's $K(n)$ (for $n>0$ and a prime $p$); cf. Do there exist "topologically significant" (and not "algebraic") triangulated categories killed by the multiplication by $p$?

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  • $\begingroup$ Maybe the tensor product over the sphere spectrum $\mathbb{S}$ on $H\mathbb{Z}$-modules? I don't know what this looks like though. $\endgroup$
    – Qiaochu Yuan
    Commented Dec 4, 2015 at 21:43
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    $\begingroup$ @QiaochuYuan Is that even well-defined? What of the two $H\mathbb{Z}$-module structures do you put on the product? $\endgroup$
    – Denis Nardin
    Commented Dec 5, 2015 at 1:48
  • $\begingroup$ Ah right, I suppose you land in a different category (namely $H\mathbb{Z}$-bimodules). $\endgroup$
    – Qiaochu Yuan
    Commented Dec 5, 2015 at 1:53

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