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Is the infinity category of $A_{\infty}$-categories complete? By complete I mean do there exist arbitrary homotopy limits in the infinity category of $A_{\infty}$-categories?

I felt like this result should be somewhere inside Lurie's books, Higher topos theory and Higher algebra, and I did find a statement saying that the completeness of an algebra over some infinity operad can be deduced from completeness of the underlying infinity category, but it seems to be a $1$-categorical result (i.e. we can regard $A_{\infty}$-category as an algebra over $\mathrm{Assoc}$), but what if we consider infinity category of $A_{\infty}$-categories?

There should also be a Proposition saying that the infinity category of dg categories is complete, which I could not find in Lurie's book either. Although it's not known whether the infinity category of $A_{\infty}$-categories and the infinity category of dg-categories are equivalent, it would also be very helpful.

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    $\begingroup$ Note that the notion of $A_\infty$ only makes sense 1-categorically. From the $\infty$-category theoretic point of view, the $A_\infty$ operads is equivalent to the Assoc operads. Given the way you write your question it feels like you have a very precise concrete definition of $A_\infty$-categories in mind. In this case you probably have a model structure for those, and the $\infty$-category associated to a model category is always complete and co-complete. $\endgroup$
    – Simon Henry
    Commented Oct 13, 2023 at 20:19
  • $\begingroup$ @SimonHenry The 1-category of $A_{\infty}$-categories does not have a model structure, and in fact there're simple counterexamples showing that the category of $A_{\infty}$-algebras does not have arbitrary limits and colimits (with reference: Localizations of the category of $A_{\infty}$-categories and internal Homs). However, the 1-categorical description is a "fake category", which means we should really look at the infinity category, which might have a model structure but I cannot find in the literature. $\endgroup$
    – TheWildCat
    Commented Oct 15, 2023 at 7:22
  • $\begingroup$ Ok. The mathematical content of the paper is not wrong, but I think there is a big misunderstanding of how model structures are supposed to work: (strictly unital or non-unital) $A_\infty$ categories are an essentially algebraic notion so the category of $A_\infty$-categories has all limits and colimits. The problem is that the category the paper considers use a different notion of morphism, the $A_\infty$ functors, corresponding to "weak functors". These are not meant to be the morphism in the model category. They corresponds to arrows $CX \to Y$ where C is (...) $\endgroup$
    – Simon Henry
    Commented Oct 15, 2023 at 13:41
  • $\begingroup$ $C$ is a cofibrant replacement comonad (probably the bar resolution). So the thing the paper looks at is a CoKleisli category, so it is not surprising it doesn't have all limits. Once you have the model structure the correct notion of morphism you use to compute the homotopy category are these "weak" or $A_\infty$ functor but the model structure itself is meant to be on the category of strict algebraic morphisms. The situation is similar to the Bergner model structure on simplicial categories. The model structure is on the category of strict functors, which are obviously not (...) $\endgroup$
    – Simon Henry
    Commented Oct 15, 2023 at 13:46
  • $\begingroup$ ... the right notion homotopically speaking. but the correct notion of "weak functor" appears when you look at morphism out of a cofibrant replacement. I have never really looked at the theory of $A_\infty$-categories itself. But for $A_\infty$-categories with fixed set of objects, it is well known there is a model structure (they are algebras for a cofibrant operad) and I would be extremely surprised if varying the set of objects would create any problems. $\endgroup$
    – Simon Henry
    Commented Oct 15, 2023 at 13:50

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