Given a k-linear dg category Cdg, I can produce an (A∞-quasi-equivalent) k-linear A∞ category CA∞ by taking the homotopy category H0(Cdg) with zero differential and the given composition, and then defining all the higher A∞ structure morphisms to be the Massey products. Possibly up to some adjectives which I’ve omitted, this is an equivalence between the ∞-category of k-linear dg categories and the ∞-category of k-linear A∞ categories, and it has the benefit that if the category Cdg with which I began was reasonably simple, then the results of this process might not too difficult to compute.

My question is: is there an equally nice description of the inverse functor? I mean: if I start with an A∞ category CA∞, is there a short but explicit description of the algorithm which produces a dg category A∞-quasi-isomorphic to CA∞? One obvious guess is might involve some inductive procedure of adding higher-degree morphisms “by hand” in order to produce all the Massey products you want, but that’s a bit ugly and also tricky to keep track of, and I hoped there might be something better.

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**Attribution***Source : Link , Question Author : Ben G , Answer Author : Community*