A Synthetic Version of Lie's Second Theorem

TL;DR

This paper generalizes Lie's second theorem to categories beyond smooth manifolds, including synthetic differential geometry, by formulating an axiomatic system and extending the theorem to homomorphisms between formal group laws and Lie groups.

Contribution

It introduces a synthetic framework that broadens Lie theory to non-smooth categories and formalizes the structures needed for such generalizations.

Findings

Generalization of Lie's second theorem to categorical settings

Development of an axiomatic system for abstract structures

Introduction of enriched mono-coreflective subcategories

Abstract

We formulate and prove a twofold generalisation of Lie's second theorem that integrates homomorphisms between formal group laws to homomorphisms between Lie groups. Firstly we generalise classical Lie theory by replacing groups with categories. Secondly we include categories whose underlying spaces are not smooth manifolds. The main intended application is when we replace the category of smooth manifolds with a well-adapted model of synthetic differential geometry. In addition we provide an axiomatic system that specifies the abstract structures that are required to prove Lie's second theorem. As a part of this abstract structure we define the notion of enriched mono-coreflective subcategory which makes precise the notion of a subcategory of local models.