The Lie group-Lie algebra correspondence in tangent categories

Source abstract

Classic Lie theory establishes a correspondence between Lie groups, which are internal group objects in the category of smooth manifolds, and Lie algebras. An analogous correspondence also exists for group objects in affine schemes. Both smooth manifolds and affine schemes form tangent categories, which provide a categorical context for differential geometry. Therefore, it is natural to ask whether the Lie correspondence can be constructed entirely from the tangent structure. Building on work of Cockett and Schwarz, we develop an internal Lie group-Lie algebra correspondence in tangent categories. We introduce Lie group objects as group objects in a tangent category which admit a tangent space at the unit. For any group object, we construct a Lie functor and show it is tangentially representable exactly when the group object is a Lie group. Representability then yields an internal Lie algebra whose underlying object is the tangent space at the unit. We prove this Lie algebra is a differential Lie algebra, with a bilinear Lie bracket induced by the adjoint representation, and we extend the construction to a functor from Lie groups to differential Lie algebras. Finally, we recover the usual Lie correspondences in both differential and algebraic geometry.