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A functor $N\colon\mathrm{Cat}_{A_\infty}\longrightarrow\mathrm{Cat}_\infty$ is constructed in a paper [1] by Faonte. This gives a way to get an $\infty$-category by starting with an $A_\infty$-category.

Going the other way, is it possible to define linear $A_\infty$-categories as special $\infty$-categories?

References

[1] Simplicial nerve of an A-infinity category (Giovanni Faonte, arXiv:1312.2127), suggested by DamienC in an answer to MO152370.

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  • $\begingroup$ If Faonte's functor is fully faithful, then yes. $\endgroup$
    – David Roberts ♦
    Commented Jan 13, 2019 at 20:03
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    $\begingroup$ I'm pretty sure that an $A_∞$-category is just going to be simply a $k$-linear $∞$-category, but I don't know if anyone has written a proof yet. I'm pretty sure also that it is well known that an $A_∞$-algebra (whatever definition you're using) is just an $E_1$-algebra in the derived category of your base ring, although I'll let someone more familiar than me with algebraic operads hunt down the references. $\endgroup$
    – Denis Nardin
    Commented Jan 13, 2019 at 20:55
  • $\begingroup$ @DenisNardin Thanks for the pointer! Do you know good sources for learning about $k$-linear $\infty$-categories? In particular, do they appear also on Lurie's Higher Algebra or only on SAG? (The relevant nLab page points only to Section 6 of DAG-VII for linear $\infty$-categories.) $\endgroup$
    – Emily
    Commented Jan 14, 2019 at 1:43
  • $\begingroup$ @Untitled I don't know a particular reference, I'd define them as $D(k)$-enriched $∞$-categories, but I'm not aware of whether someone has actually tried to use them for something. $\endgroup$
    – Denis Nardin
    Commented Jan 14, 2019 at 11:31
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    $\begingroup$ @DenisNardin, I think the term $k$-linear $\infty$-categories usually refers to stable $\infty$-categories which are modules over $Perf(k)$. These are automatically $D(k)$-enriched, but not the other way around. In either case the embedding in $\infty$-categories is not fully-faithful, and in the case of $D(k)$-enriched categories (which is what $A_\infty$-categories are probably a model of) it is not even conservative. In particular, the answer to the OP's question is no. $\endgroup$
    – Yonatan Harpaz
    Commented Jan 14, 2019 at 21:13

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An affirmative to the "conjecture" above is fully recorded in this work: https://arxiv.org/abs/2003.05806

In Remark 1.2, for example, we comment how Gepner-Haugseng's results imply that k-linear A-infinity categories are precisely k-chain-complex-enriched infinity-categories.

This passes through the infinity-categorical equivalence between k-linear dg-categories and k-linear A-infinity categories. The proof is completely in line with Rune's comments--in fact, it was based on a discussion I had with Rune back before COVID.

By the way, if you define a k-linear infinity-category to be an infinity-category enriched over k-chain-complexes, you're fine. (Re: Denis's comment.) But some people define k-linear infinity-categories as those with an action of the infinity-category of k-linear chain complexes; then you're not fine. Not all A-infinity-categories have (even finite) colimits, for example. I think this is what Yonatan points out in the comments.

So, to address OP's original question: No, k-linear A-infinity-categories are not a full subcategory of infinity-categories.

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  • $\begingroup$ Hi Hiro, thank you so much for your answer (and sorry for taking quite a while to reply)! $\endgroup$
    – Emily
    Commented Feb 22, 2023 at 18:33

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