Macquarie University
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Synthetic Lie theory

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posted on 2022-03-28, 21:36 authored by Matthew Burke
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.


Table of Contents

Introduction -- 1. Synthetic differential geometry -- 2. Factorisation systems -- 3. Paths in categories -- 4. Synthetic lie theory -- 5. Relationship to classical lie theory -- Conclusion.


Bibliography: pages 159-161 Empirical thesis.

Awarding Institution

Macquarie University

Degree Type

Thesis PhD


PhD, Macquarie University, Faculty of Science and Engineering, Department of Mathematics

Department, Centre or School

Department of Mathematics

Year of Award


Principal Supervisor

Richard Garner

Additional Supervisor 1

Dominic Verity


Copyright Matthew Burke 2015. Copyright disclaimer:




1 online resource (161 pages)

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