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One can define certain higher categorical truncations. For example, a discussion from the quasi-categories point of view can be found in HTT 5.5.6.

On the other hand, as a model of linear $\infty$-category, $A_\infty$-categories show up more often in symplectic geometry. So, does anyone discuss similar truncation from $A_\infty$-categories point of view?

In particular, I am more interested in the following specific question: For two different $A_\infty$-categories, can one define any kind of $A_n$-truncated functor between them? For example, I could probably define an $A_n$-truncted functor by only requiring the first $n$-"components" that define an $A_\infty$-functor satisfying the first $n$-functor equations.

Does any kind of discussion show up somewhere? Also, feel free to just think of the target category as a DG category, which is the situation I am really considering.

Btw: I would probably not be thinking about $(A_\infty, n)$-theory, I feel I am just thinking about a kind of "$(A_n,1)$-theory".

I realized that in principle I can write down everything using results from quasi-categories side via $A_\infty$-nerve construction. So, now, it is more like a reference request question: Is there anything written down explicitly anywhere?

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  • $\begingroup$ maybe vaguely relevant (I think) mathoverflow.net/questions/448092/… $\endgroup$
    – Tim
    Commented Dec 9 at 15:06
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    $\begingroup$ @Tim Thx! It is relevant. It is basically what in my mind the store on the quasi-category side. But here, my point is does anyone write down anything using the language of $A_\infty$-category without passing to the $A_\infty$-nerve. $\endgroup$
    – Bingyu Zhang
    Commented Dec 9 at 17:06
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    $\begingroup$ In the case of quasicategories, the "truncation" is really about truncating the homotopy types of the mapping spaces. You seem to be asking about the truncating the operad itself however, in your third paragraph (i.e. restricting to $A_n$-operads). Do you mean these to be different notions? $\endgroup$
    – Jonathan Beardsley
    Commented Dec 9 at 22:40
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    $\begingroup$ If you're trying to mimic the idea of truncation in a quasicategory then you probably want to do something like require your mapping complexes to have trivial homology below some degree. But this seems qualitatively different that saying you only want coherence of your composition up to $A_n$. I could be totally wrong though! $\endgroup$
    – Jonathan Beardsley
    Commented Dec 9 at 22:41

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$A_n$-spaces are already discussed in the original paper of Stasheff, Homotopy associativity of H-spaces, I and II.

In the linear setting, $A_n$-algebras are discussed e.g. in A∞-algebras, spectral sequences and exact couples, but I'm quite sure this is not the first place they appear.

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  • $\begingroup$ Excellent! This is exactly the kind of thing I am looking! It is compatible with my imagination that those things should have existed quite early. $\endgroup$
    – Bingyu Zhang
    Commented Dec 9 at 19:21

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