Consider first the following $2-$groupoid of Algebras over $\mathbb{C}$. Objects are Algebras, $1-$morphisms are isomorphisms, and a $2-$morphism between the isoms $f$ and $g$ from $A$ to $B$ is an element of $B^{\times}$ such that $f(a) b = b g(a)$ for all $a \in A$.

This is a certain sub$2$groupoid of the $2-$category of categories where the objects are the linear categories $A-$mod, the $1-$morphisms are (certain) functors (I think preserving the tensor structure), and the $2-$morphisms are natural transformations. Alternatively, take linear categories with a point as object and morphisms $A$ and consider functors and natural tranformations of those.

Is there a natural analogue of this in the case of $A_{\infty}$ algebras? I was hoping to understand some systematic way of writing down all the higher morphisms in an $\infty-$groupoid of $A_{\infty}$ algebras and hoping that they are all non-trivial. However, I don't even understand the analogue of the elements $b$ from above.

Is this done in the literature somewhere where one can extract explicit formulae? One guess would be to consider $A_{\infty}-$algebras as $A_{\infty}-$categories with one object, but this does not help unless one can explicitly write down some explicit $\infty-$groupoid structure on $A_{\infty}-$categories which I guess would require one to first convert these $A_{\infty}-$categories to $\infty-$categories



2 Answers 2


Notice that you are looking at the 2-category of algebras, bimodules and bimodule homomorphism.

Because of this: if you regard a morphism $f : A \to B$ of algebras as an $A$-$B$ bimodule $B_f$ ($B$ equipped with the obvious right $B$-action and with left $A$-action induced by $f$) then the 2-morphisms that you are looking at are bimodule homomorphism $B_g \to B_f$ given on $B$ by left multiplication with $b \in B$ (this trivially respects the right $B$-action and the equation $f(a)b = b g(a)$ is precisely the condition that it also respects the left $A$-action.)

So you are looking for the $A_\infty$-version of (the maximal higher groupoid inside) the 2-category of algebras, bimodules and bimodule homomorphisms.

Now, in

Berger, Moerdijk, Resolution of coloured operads and rectification of homotopy algebras http://arxiv.org/PS_cache/math/pdf/0512/0512576v2.pdf

there is described in section 6 a model-category theoretic construction of a simplicial category whose objects are $A_\infty$-algebras, morphisms are bimodules of $A_\infty$-algebras, 2-morphisms are bimodule homomorphisms, and so on.

This simplicial category you may think of as presenting an $\infty$-category of $A_\infty$-algebras.

(Notice that this applies to $A_\infty$-algebras over any suitable enriching category, say tor $A_\infty$-spaces You are probably thinking of the standard dg-case, enriched over chain complexes, to which it applies in particular.)

Some paragraphs on this you can also find here:


  • $\begingroup$ Excellent, I am especially eager to read about the "so on" part, thanks! Yes, I was thinking of the dg enriched case (although unfortunately when one considers curved $A_{\infty}$ algebras and curved morphisms its not quite that case anymore) Although this could be related to the Seidel construction via reinterpreting $A_{\infty}-$cats as $\infty-$cats, that sounds like a difficult path, whereas Berger, Moerdijk sounds more directly related to what I need. $\endgroup$ Nov 17, 2010 at 20:25
  • $\begingroup$ It seems that the first new feature that is present for $A_{\infty}$ algebras as opposed to regular algebras is that the bimodule homomorphisms themselves have non-identity morphisms between them, even when the bimodules come from algebra isomorphisms. Do the bimodules between two $A_\infty$ algebras form a simplicial category in some standard way? What are the morphisms between two isomorphisms $f_{1}$ and $f_{2}$ between the $A_{1}-A_{2}$ bimodules $B_{1}$ and $B_{2}$? $\endgroup$ Nov 17, 2010 at 23:32
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    $\begingroup$ You mean for a fixed pair of $A_\infty$-algebras, what's the $\infty$-category of bimodules betweem them? Recall from Berger-Moerdijk the strategy to get that: there is an operad whose algebas are bimodules. Pass to the correspnding model category of its homotopy algebras as described by Berger-Moerdijk. That presents the $\infty$-category of all $A_\infty$-bimodules. It comes with two functors to that of just $A_\infty$-algebras, so take the fiber over your chosen ones. ... $\endgroup$ Nov 17, 2010 at 23:46
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    $\begingroup$ ... This is actually the precise way to get $\infty$-categories by presenting them by model categories. Regarding that simplicial category that we talked about as an $\infty$-category really requires a bit more discussion. $\endgroup$ Nov 17, 2010 at 23:47

Seidel's book Fukaya categories and Picard-Lefschetz theory, sections (1a-e), describes in explicit terms the $A_\infty$-category of non-unital functors between two fixed (small) $A_\infty$-categories, as well as the functors between functor-categories obtained by composing with a fixed functor on the left or right. This is not quite the whole story concerning composition.

Unsurprisingly, you can formulate everything as sums over trees, but the signs are nasty.

  • $\begingroup$ No need to climb higher: an $A_\infty$-category is already a model for a (stable) $\infty$-category. $\endgroup$ Nov 17, 2010 at 19:25
  • $\begingroup$ Urs: thanks! I'll remove that sentence. $\endgroup$
    – Tim Perutz
    Nov 17, 2010 at 19:43
  • $\begingroup$ Thanks for this reference! I also see that this issue was discussed over here: golem.ph.utexas.edu/category/2006/11/…. There they discussed how to cook up a quasicategory of $L_{\infty}$ algebras and gave some references. It would be great to even have some explicit understanding of what the invertible $2$ and $3-$morphisms look like, even in a case where $m_{4}$ and higher vanish for all the algebras involved. $\endgroup$ Nov 17, 2010 at 20:05
  • $\begingroup$ Notice the difference between the oo-categories of homotopy algebras whose 1-morphisms are just plain morphisms and those where the 1-morphisms are more generally bimodules. For the first version a comprehensive construction and discussion of oo-categories of homotopy algebras over any suitable operad is at ncatlab.org/nlab/show/… . In the Examples-section is discussed how to generalize this to $\infty$-categories whose 1-morphisms are allowed to be bimodules. $\endgroup$ Nov 17, 2010 at 20:23

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