Homotopical algebra of foliations
par
Salle Katherine Johnson, bâtiment 1R3
Institut de Mathématiques de Toulouse
Partition Lie algebroids are fully homotopy-coherent analogues of Lie–Rinehart algebras, which classify formal moduli stacks over general bases. Meanwhile, infinitesimal derived foliations are equivariant stacks whose structure sheaves are closely related to the completed Hodge filtration
of Grothendieck’s infinitesimal cohomology. This thesis is dedicated to proving that these two notions are equivalent under certain finiteness conditions. This equivalence is expected to lay a solid homotopical algebraic foundation for the future study of pathological foliations in general
characteristics. The crucial tool is a filtered Chevalley–Eilenberg complex constructed via an ∞-categorical Koszul duality. Moreover, this filtration also provides a feasible tool for studying the homotopy operations of concrete partition Lie algebroids arising from geometry. Additionally, I have included a comparison of partition Lie algebroids with existing literature at the end.