Synthetic Lie theory

Traditionally an infinitesimal neighbourhood of the identity element of a Lie group is studied indirectly by using an appropriately chosen algebraic structure such as a Lie algebra to represent it. In this thesis we use the theory of synthetic differential geometry to work directly with this infinitesimal neighbourhood and reformulate Lie theory in terms of infinitesimals. We show how to carry out this reformulation for the established generalisation of Lie theory involving Lie groupoids and Lie algebroids and make a further generalisation by replacing groupoids with categories. Our main result is a proof of Lie's second theorem in this context. Finally we show how our new constructions and definitions relate to the classical ones.