Lie algebroids in derived differential topology

Abstract

A classical principle in deformation theory asserts that any formal deformation problem is controlled by a differential graded Lie algebra. This thesis studies a generalization of this principle to Lie algebroids, and uses this to examine the interactions between the theory of Lie algebroids and the derived geometry of moduli spaces. The first half of the thesis develops the homotopy theory of differential graded Lie algebroids over a fixed affine derived manifold. We prove that any deformation problem over such a derived manifold is controlled by a Lie algebroid, by constructing an equivalence between the homotopy theory of Lie algebroids and the homotopy theory of formal moduli problems. This equivalence furthermore extends to an equivalence between representations of Lie algebroids and quasi-coherent sheaves over formal moduli problems. In the second half of the thesis, we develop the theory of derived differential topology and apply it to study Lie algebroids arising from derived differentiable stacks. Using the results of the first half, we show that the relative tangent bundle of a derived manifold over a derived stack has a Lie algebroid structure. We then provide a criterion for maps between Lie algebroids to integrate to maps between stacks, generalizing classical theorems of Lie and Van Est. This result is applied to show that finite-dimensional L-infinity algebras can be integrated to higher Lie groups.