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A DG category can be considered as an infinity category, say by taking Dold-Kan of the coconnective part of Hom spaces, thus obtaining a simplicial category.

My question is, are the following equivalent: this underlying infinity category having finite colimits and limits, and our DG category being pre-triangulated.

Thank you, sasha

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    $\begingroup$ I thought pre-triangulated dg-categories were the same as stable oo-categories: ie a square is a pushout if and only if it is a pullback. (modulo ignoring linearity over the base field) $\endgroup$
    – bananastack
    Commented Mar 20, 2014 at 6:33

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This paper worked out in gory detail the equivalence between pre-triangulated dg-categories and certain stable categories.

http://arxiv.org/abs/1308.2587

The definition of stable category can be found in Lurie's higher algebra: Defn 1.1.1.9.

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  • $\begingroup$ Thank you. But I asked a bit different question, I think: not whether the totality of pre-tri. dgcats model the same homotopy theory as stable cats, but whether a dgcat which, as an inf-cat, admits small colimits, is then automatically pre-tri. $\endgroup$
    – Sasha
    Commented Mar 21, 2014 at 5:28
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    $\begingroup$ I see. The answer should be no, as I don't think there's any reason to expect that in a dg-category a square is cartesian iff it is cocartesian. I'll see if it makes sense and if I can find a reference. $\endgroup$
    – bananastack
    Commented Mar 21, 2014 at 9:12
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At least one direction of the question is answered by the paper

  • Giovanni Faonte, Simplicial nerve of an A-infinity category, arXiv:1312.2127.

See section 4.2, where he proves that for a pre-triangulated dg-category, its dg-nerve is a stable $\infty$-category. Here dg-nerve means either the construction you described (applying Dold-Kan and then the simplicial nerve) or a construction from Lurie's Higher algebra that associates directly an $\infty$-category to a dg-category; the above paper shows that both constructions are equivalent. The author does not address explicitly the converse statement, that if the dg-nerve is stable then the dg-category is pre-triangulated, but I would expect it to be true as well.

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