At Hausdorff series I have written some general facts about **Magnus algebras** and **Magnus group**. Many related entries need improvement, like Mal’cev completion. The abstract Hausdorff series is about the exponential map in certain completion of the free associative algebra which contains free Lie algebra on the same symbols: the exponentials send Lie elements to invertible elements and all invertibles (elements of the Magnus group) are in the image. If we were dealing with locally nilpotent Lie algebras, instead of free then we do not need to go into completion and do everything within the enveloping algebra which is a Hopf algebra and the exponential there sends primitive elements to group like and all group likes are in the image.

Now what is the infinite-categorical analogue of that picture ? Baranovsky has a version

- Vladimir Baranovsky,
*A universal enveloping for L-infinity algebras*, Math. Res. Lett. 15 (2008), no. 6, 1073–1089, arxiv/0706.1396, MR2011a:18014, journal pdf

(arxiv) of the enveloping algebra of an $L_\infty$-algebra $A$ which is of right size (there is a homotopically correct (up to quasi-isomorphism in fact) answer which is too big, namely the cobar construction $\Omega CE(A)$ of the Cartan-Chevalley-Eilenberg coalgebra of $A$, and Baranovsky’s paper aimed at correcting this size issue by producing a smaller version of the right size in the sense of PBW like counting).

On the other hand, Getzler has a hierarchy what he calls “generalized Hausdorff formulas” (or higher Hausdorff formulas) in his Lie integration paper in Annals:

- Ezra Getzler,
*Lie theory for nilpotent $L_\infty$-algebras*, math.AT/0404003, Ann. of Math. (2) 170 (2009), no. 1, 271–301, MR2010g:17026, doi

Getzler works with nilpotent case only. I would like to find the group like part and the primitive part both within the infinity version of enveloping and see the role of exponential map. The group like part of enveloping infinity algebra should span a Kan complex. In Baranovsky, the universal enveloping is an A-infinity algebra with coalgebra structure which is strict and the two structures are strictly compatible in my memory. But I do not know how to extract a group-like part from it. What one does for such infinity-bialgebra to extract a Kan complex ?

Next, for non-nilpotent version, what can one say about a completion. Is there an infinity analogue of Magnus algebra and “inside” constructible a Magnus Kan complex ?

For me these questions are very central to the formal infinity-Lie theory.

]]>The page internal ∞-groupoid claimed that the case of “internal ∞-groupoids in an (∞,1)-category” was discussed in detail at groupoid object in an (∞,1)-category. That doesn’t seem right to me—I think the groupoid objects on the latter page are really only internal 1-groupoids, not internal ∞-groupoids. They’re “∞” in that their composition is associative and unital only up to higher homotopies, but those are homotopies in the *ambient* (∞,1)-category; they themselves contain no “higher cells” as additional data. In particular, if the ambient (∞,1)-category is a 1-category, then an internal groupoid in the sense of groupoid object in an (∞,1)-category is just an ordinary internal groupoid, no ∞-ness about it. Does that seem right?