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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 polynomials algebras as generalizations of Weil algebras. What do you think, nForum?
[1] Didry, M. Construction of Groups Associated to Lie- and to Leibniz- Algebras
[2] Nishimura, H. Axiomatic Differential Geometry II-2, Chapter 2: Differential Forms
Thanks for your message here on the $n$Forum.
Right now I am not exactly sure what your question is. You say you are after a good definition generalizing microlinearity to the non-linear realm of higher order jets. (What’s might be a good term, for this?) What if you just follow your nose with this? It feels like this should be pretty straightforward; but I haven’t really thought about it. Do you encounter any obstacles or encounter ambiguities of choosing the right definition?
Hello Urs, thanks for having the nForum and for your role in the nLab!
Ok, so when we think about microlinearity we have a concept that certain objects “look microscopically like the line R”, which satisfies the KL axiom scheme etc. More specifically, since the functor $R^{(-)}$ takes cones of Weil algebras (in particular those corresponding to infinitesimal objects) to genuine limits, what we are searching for is something that takes a larger class of R-algebras to genuine limits, via exponentiation. Moreover, we require this class of R-algebras to correspond to higher-order infinitesimals, so that the notion of differentiation has been genuinely extended. Hence we still want to work in a lined topos, but this topos may or may not be smooth. Rather, objects which infinitesimally exponentiate to R-modules will simply be “microalgebraic” in this expanded sense. My uncertainty is in which class to choose, and how to control the polynomial identities in these exponential objects. And I’ll admit, Nishimura’s remarks on the universality of the Lie bracket and Jacobi identity in [1] have my head spinning a bit about the role these might play in the local structure of “microalgebraic” groups. I guess it’s time to experiment.
Your mention of jets is prompting me think of it explicitly from a vector bundle perspective, so I’ll put some more thought into that.
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