Given a $k$-linear dg category $\mathcal{C}_{dg},$ I can produce an ($A_\infty$-quasi-equivalent) $k$-linear $A_\infty$ category $\mathcal{C}_{A_\infty}$ by taking the homotopy category $H^0(\mathcal{C_{dg}})$ with zero differential and the given composition, and then defining all the higher $A_\infty$ structure morphisms to be the Massey products. Possibly up to some adjectives which I've omitted, this is an equivalence between the $\infty$-category of $k$-linear dg categories and the $\infty$-category of $k$-linear $A_\infty$ categories, and it has the benefit that if the category $\mathcal{C}_{dg}$ 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 $A_\infty$ category $\mathcal{C}_{A_\infty},$ is there a short but explicit description of the algorithm which produces a dg category $A_\infty$-quasi-isomorphic to $\mathcal{C}_{A_\infty}$? 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.

  • 5
    $\begingroup$ cobar bar is a functorial rectification of an A_\infty thing into a dg thing. But I think your question is backwards. Without extra information, how can you make the requisite choices to define the Massey products on the homotopy category functorially? $\endgroup$ Sep 8, 2016 at 1:16


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy