The functorial semantics of Lie theory

TL;DR

This paper develops a tangent-categorical framework to unify Lie algebroids and Lie functor using involution algebroids, bridging two major research directions in Lie theory.

Contribution

It introduces involution algebroids within tangent categories, providing a functorial semantics perspective on Lie algebroids and the Lie functor.

Findings

Category of Lie algebroids equals involution algebroids in smooth manifolds.

Weil algebras form the classifying category of involution algebroids.

Lie functor is represented as a tangent-categorical functor.

Abstract

Ehresmann's introduction of differentiable groupoids in the 1950s may be seen as a starting point for two diverging lines of research, many-object Lie theory (the study of Lie algebroids and Lie groupoids) and sketch theory. This thesis uses tangent categories to build a bridge between these two lines of research, providing a structural account of Lie algebroids and the Lie functor. To accomplish this, we develop the theory of involution algebroids, which are a tangent-categorical sketch of Lie algebroids. We show that the category of Lie algebroids is precisely the category of involution algebroids in smooth manifolds, and that the category of Weil algebras is precisely the classifying category of an involution algebroid. This exhibits the category of Lie algebroids as a tangent-categorical functor category, and the Lie functor via precomposition with a functor $\partial: \mathsf{Weil}_1 \to \mathcal{T}_{\mathsf{Gpd}},$ bringing Lie algebroids and the Lie functor into the realm of functorial semantics.