I am aware of the following: in the context of synthetic differential geometry (SDG) one obtains a Lie algebra by exponentiating a microlinear group by a standard infinitesimal object and taking the infinitesimal commutator, and that the functor expressed by this operation factors through formal group laws (FGLs) in the usual way. This reveals that Lie groups are FGLs with respect to first-order infinitesimals.
Now I would like to consider a lined topos equipped with higher-order infinitesimals, and develop in this context a modified notion of microlinearity. I have not yet developed the details of this. But does modifying microlinearity in this way, to yield R-modules by exponentiating FGLs with higher-order infinitesimals, sound reasonable? It is worth saying that in general we want certain polynomial identities to hold in the resulting R-modules, e.g. the Jacobian identity.
While FGLs have been thought of in this way (e.g. Didry in [1], an attempt to extend Lie theory to include Leibniz algebras), I have not found sources discussing modifications of microlinearity to subsume FGLs in the language of SDG. Some suggestive remarks can be found in Nishimura's work, such as in the introduction of the paper [2], where the author discusses prolongations of spaces with respect to polynomial algebras as generalizations of Weil algebras.
Any sources or ideas that sound relevant to this speculation would be great. Maybe it's already been done! Feel free to let me know.
