Applied Categorical Structures

, Volume 26, Issue 4, pp 767–798 | Cite as

A Synthetic Version of Lie’s Second Theorem

  • Matthew Burke


We formulate and prove a 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 groupoids. Secondly we include groupoids 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 provides all the abstract structure that is required to prove Lie’s second theorem.


Lie theory Lie groupoid Lie algebroid Category theory Factorisation system Synthetic differential geometry Intuitionistic logic 


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The author is very grateful for the constructive comments offered by and the corrections indicated by the reviewer. The author would like to acknowledge the assistance of Richard Garner, my Ph.D. supervisor at Macquarie University Sydney, who provided valuable comments and insightful discussions in the genesis of this work. In addition the author is grateful for the support of an International Macquarie University Research Excellence Scholarship and a Postdoctoral Scholarship from the Department of Mathematics and Statistics at the University of Calgary.


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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of CalgaryAlbertaCanada

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