A Synthetic Version of Lie's Second Theorem

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.