Comparative Smootheology, IV
Posted by John Baez
For some time now we’ve been comparing different approaches to ‘smooth spaces’ — generalizations of manifolds that have proved handy in math and physics. Here’s a thesis on the subject:
- Martin Laubinger, Differential Geometry in Cartesian Closed Categories of Smooth Spaces, Ph.D. thesis, Louisiana State University, February 2008.
I thank Eugene Lerman for pointing this out. Alex Hoffnung has contacted Laubinger and let him know that there’s a community of people out here studying this subject.
Here’s the abstract of Laubinger’s thesis:
The main categories of study in this thesis are the categories of diffeological and Frölicher 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. Frölicher’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 Frölicher 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 Frölicher space are vector spaces. Frölicher groups satisfy this condition, and under a technical assumption on the tangent space at identity, we can define a Lie bracket for Frölicher groups.
Some of these results overlap with results we’ve already discussed here… but some were obtained first by Laubinger! Alex and I will rewrite our paper to cite his work.
The construction of Lie algebras for Frölicher groups is something we haven’t seen here.
Re: Comparative Smootheology, IV
Half way through your post you change his name from Laubinger to Lautinger; Laubinger was correct.