Differential Geometry in Cartesian Closed Categories of Smooth Spaces

• Main Author: Laubinger, Martin
• Other Authors: David Kirshner
• Format: Others
• Language: en
• Published: LSU 2008
• Subjects: Mathematics
• Online Access: http://etd.lsu.edu/docs/available/etd-02212008-165645/

The main categories of study in this thesis are the categories of diffeological and Fr\"olicher spaces. They form concrete cartesian closed categories. In Chapter 1 we provide relevant background from category theory and differentiation theory in locally convex spaces. In Chapter 2 we define a class of categories whose objects are sets with a structure determined by functions into the set. Fr\"olicher's $M$-spaces, Chen's differentiable spaces and Souriau's diffeological spaces fall into this class of categories. We prove cartesian closedness of the two main categories, and show that they have all limits and colimits. We exhibit an adjunction between the categories of Fr\"olicher and diffeological spaces. In Chapter 3 we define a tangent functor for the two main categories. We define a condition under which the tangent spaces to a Fr\"olicher space are vector spaces. Fr\"olicher groups satisfy this condition, and under a technical assumption on the tangent space at identity, we can define a Lie bracket for Fr\"olicher groups.