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Let $I$ be a filtered category and $\{k_i\}_{i\in I}$ be a system of commutative rings over $I$. Toen proved that there is an equivalence of categories $$ \text{Colim}-\otimes^{\mathbb{L}}_{k_i} k:\text{Colim}_{i\in I}\text{Ho}(\text{smooth proper dg-alg}/k_i) \simeq \text{Ho}(\text{smooth proper dg-alg}/k). $$ (see https://www.semanticscholar.org/paper/Anneaux-de-d%C3%A9finition-des-dg%E2%80%90alg%C3%A8bres-propres-et-To%C3%ABn/72decd287a7bb55e4ee1a49e23022200073a6f35)

Does this theorem still hold if we replace dg-algebras with dg-categories? That is, is there an equivalence of categories $$ \text{Colim}-\otimes^{\mathbb{L}}_{k_i} k:\text{Colim}_{i\in I}\text{Ho}(\text{smooth proper dg-cat}/k_i) \simeq \text{Ho}(\text{smooth proper dg-cat}/k)? $$ 　

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  • $\begingroup$ I havent read the paper carefully but from what it seems, most of the list of facts Toën gives (numbered F 1-7) are true for dg-categories when looking at the reference in To-Ve, however fact F7 is used strongly and its proof takes about half of this paper. The sublemma 3 uses the fact that you're dealing with dg-algebras by checking equivalences through passing to the total cohomology of the algebra. If T is a dg-cat not homotopically eq to a dg-algebra then I dont know how to use that lemma, or prove F7 directly. $\endgroup$
    – AT0
    Commented Nov 1, 2023 at 10:25

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