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.

  • $\begingroup$ Any $A_\infty$-algebra is quasiequivalent to a dg-algebra through the cobar-bar construction. $\endgroup$ – Aaron Bergman Jan 24 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 Jan 24 at 5:45

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.

  • $\begingroup$ 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 Jan 24 at 5:38
  • 1
    $\begingroup$ @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 Jan 24 at 5:40

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