Type of Document 
Dissertation 
Author 
Laubinger, Martin

URN 
etd02212008165645 
Title 
Differential Geometry in Cartesian Closed Categories of Smooth Spaces 
Degree 
Doctor of Philosophy (Ph.D.) 
Department 
Mathematics 
Advisory Committee 
Advisor Name 
Title 
Jimmie D. Lawson 
Committee Chair 
Ambar Sengupta 
Committee Member 
Gestur Olafsson 
Committee Member 
Patrick Gilmer 
Committee Member 
Pramod Achar 
Committee Member 
David Kirshner 
Dean's Representative 

Keywords 
 Groups of Mappings
 Lie Theory
 Category Theory

Date of Defense 
20080219 
Availability 
unrestricted 
Abstract
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.

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