Mathematical applications of category theory in synthetic calculus of variations

Vafa Khalighi – 23 May 2007

1. Introduction: In various lectures on categorical dynamics given in 1967, Lawvere laid down the foundations of a new discipline which in recent years has undergone a rapid development and has come to be known as "synthetic differential geometry". As the title of Lawvere's lectures suggests the motivation came from the search for a suitable complete category theory, small Greek delta, in which to develop dynamics. Such a category, small Greek delta, unlike the category m of smooth paracompact manifolds and smooth mappings was to possess all finite limits and exponentials. 2. Some simple questions in the calculus of variations: We look at some simple questions in the calculus of variations like: to find the shortest curve between two points on a surface or to find a closed curve of given length and maximal enclosed area, and we analyse how category theory can answer these simple questions.