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Let $k$ be a field.

A folklore theorem states that dg-categories (over $k$), $A_{\infty}$-categories (over $k$) and stable ($k$-linear) $(\infty, 1)$-categories are "the same" (see for example Stable infinity categories vs dg-categories).

I would like to understand in what sense (and why) dg- and $A_{\infty}$-categories are "the same".

There is a canonical inclusion functor from the category of dg categories to the category of $A_{\infty}$ categories. However, this functor is not full, so it does not define an equivalence of categories.

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  • $\begingroup$ Any $A_\infty$-algebra is quasiequivalent to a dg-algebra through the cobar-bar construction. $\endgroup$
    – Aaron Bergman
    Commented Jan 24, 2020 at 1:38
  • $\begingroup$ Ah, so perhaps the equivalence of categories I'm looking for sends an $A_{\infty}$ algebra $A$ to the dg algebra $\Omega B A$. I need to think a bit more, but I think this answers the question at the level of algebras. $\endgroup$
    – user142700
    Commented Jan 24, 2020 at 5:45

2 Answers 2

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By Corollary 9.2.1 in the paper https://arxiv.org/abs/1410.5675 the model category of small A_∞-categories is Quillen equivalent to the model category of small categories (with a fixed set of objects for simplicity, but see also Proposition 9.2.3 for the general case), where both types of categories are enriched over a fixed monoidal tractable model category, such as chain complexes over a commutative ring, simplicial sets, simplicial modules over a commutative ring, symmetric spectra, etc.

The proof there also gives explicit formulas for the rectification operation.

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  • $\begingroup$ Could you please clarify how Corollary 9.2.1. relates to dg-categories? (I apologize if this is obvious -- I have very little background in this area). $\endgroup$
    – user142700
    Commented Jan 24, 2020 at 5:38
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    $\begingroup$ @user142700: If C=V=Ch, then algebras over the operad Cat^O_W are A_∞-dg-categories respectively dg-categories if O=A_∞ respectively As (the associative operad) with W as a set of objects. $\endgroup$
    – Dmitri Pavlov
    Commented Jan 24, 2020 at 5:40
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    $\begingroup$ But A_infinity categories don't have a model structure! $\endgroup$
    – Francesco Genovese
    Commented Jun 14, 2021 at 15:50
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    $\begingroup$ @FrancescoGenovese they do. $\endgroup$
    – Fernando Muro
    Commented Jun 14, 2021 at 22:01
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    $\begingroup$ @FrancescoGenovese: The paper you referenced is talking about a very different category, the category of A_∞-categories and A_∞-functors (not ordinary enriched functors) between them. As far as I am aware, we don't know how to define A_∞-functors for arbitrary A_∞-operads in arbitrary monoidal model categories, which is the context of my answer. $\endgroup$
    – Dmitri Pavlov
    Commented Jun 22, 2021 at 15:51
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I think that the best answer so far comes from this paper: https://arxiv.org/abs/1811.07830 , where it is proved that homotopy categories of dg-categories and various flavours of A-infinity categories are equivalent.

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