The Campbell-Baker-Hausdorff adjunction

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 extracts the formal group law from the Lie group by Taylor expanding the multiplication with respect to some chart around the identity; alternatively, one may build 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.