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Did anybody attend the Richard Garner’s talk The Campbell–Baker–Hausdorff adjunction at CT2015 ? Having a writeup or partial notes ? This is the abstract:
The Lie algebra associated to a Lie group G encodes the first-order infinitesimal structure of G near the identity; on the other hand, the formal group law associated to G is a collection of formal power series which encode all finite-order infinitesimal behaviour. One obtains the formal group law from the Lie group by Taylor expanding the multiplication with respect to some chart around the identity; alternatively, one may obtain the formal group law from the Lie algebra by applying the Campbell– Baker–Hausdorff (CBH) formula, which expresses the group multiplication power series near the identity purely in terms of iterated Lie brackets.
There are a number of ways of deriving the CBH formula, some geometric and some algebraic in nature. The aim of this talk is to describe a categorical approach drawing on synthetic differential geometry. We consider a category E of microlinear spaces wherein formal group laws may be construed as genuine internal groups; we then construct an adjunction, the Campbell–Baker–Hausdorff adjunction of the title, between internal groups and internal Lie algebras in E. Applying the left adjoint to a finite dimensional Lie algebra yields its associated formal group law; applying it to the free Lie algebra on two generators yields the free group on two non-commuting tangent vectors, whose multiplication may be seen as a pure combinatorial manifestation of the CBH formula.
I would be interested also if there were any signs of trying to do higher categorical version; because Getzler had some analogues in higher categorical dimension in his integration paper…
I attended the talk; unfortunately the notes are in my office and I am away for a few weeks. I also can’t remember many of the details, but I don’t think there were any higher-categorical aspects.
Thanks, let me know once you have the hold of the notes again.
I uploaded the scans. I’m not sure I wrote down everything.
Thanks a lot!
P.S. some elements of his approach quite remind me of Durov’s treatment in Ch. 7 of our paper
Garner’s $\mathcal{E}$ has the role like and similarity to Durov’s $\mathcal{E}$ which is the category of covariant presheaves on $\mathbf{k}\backslash\mathcal{P}$, for $\mathbf{k}=\mathbb{R}$ (the role of nilpotents, of BCH formula etc.). But I want to think more of it…
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