Higher Lie Theory
Final Report Abstract
Lie theory concerns itself with the relationship of global and infinitesimal symmetries of geometric objects. Mathematically, this relationship is given by an adjunction between Lie groups (e.g. the space of rotations in space) and Lie algebras (e.g. the space of skew-symmetric matrices). The procedure for getting from global to infinitesimal is called differentiation and the converse direction is called integration. Higher Lie theory vastly generalizes this approach by allowing the study of symmetries, which carry symmetries themselves. The global objects in this context are Lie-infinity groups (simplicial manifolds satisfying certain Kan conditions) and the infinitesimal ones are Q-manifolds. In this project we have studied the differentiation and integration in the context of Higher Lie theory. On the differentiation side, our main result is the construction of a Van-Est homomorphism for general simplicial manifolds, allowing to relate cohomological properties of a simplicial manifold to its associated Q-manifold (its tangent complex) including mathematical tools for the calculation of the homomorphism in concrete example classes (e.g. Lie groupoids or strict Lie-2 groups). The methods developed for this concrete instance of differentiation will also help us understand more general differentiations, especially since it gives us hints about how vector fields on lower levels of the simplicial manifold should be extendible to higher levels. With regards to integration, the most significant advancement is the construction of an parasimplicial manifold out of any acyclic Q-manifold. Many Q-manifolds admit an ’acyclic shadow’ - a version of them stripped of all the isotropy data. Such an acyclic Q-manifold is fully described by a singular foliation and the integration is constructed using methods from the area of singular foliations. Finally, we started working on Tepui fibrations, a diffeological approach to the differentiation and integration of singular Lie groupoids and Lie algebroids. This approach should capture ’first-layer-data’ of higher Lie theory, and can be applied even when a given singular foliation does not come from an acyclic Q-manifold, or even from no Q-manifold whatsoever. This provides a conceptual explanation, why e.g. holonomy Lie groupoids of singular foliations are well-behaved, even when they are not extensible to a simplicial manifold.
Publications
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An invitation to singular foliations
Camille Laurent-Gengoux, Ruben Louis & Leonid Ryvkin
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The holonomy Lie 8-groupoid of a singular foliation I
Ruben Louis & Camille Laurent-Gengoux
