## April 23, 2009

### 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:

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.

Posted at April 23, 2009 12:43 AM UTC

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### Re: Comparative Smootheology, IV

Half way through your post you change his name from Laubinger to Lautinger; Laubinger was correct.

Posted by: Scott Morrison on April 23, 2009 3:34 AM | Permalink | Reply to this
Read the post Smooth Structures in Ottawa II
Weblog: The n-Category Café
Excerpt: A summary of some talks at the Fields Workshop on Smooth Structures in Logic, Category Theory and Physics.
Tracked: May 9, 2009 9:05 PM

### Re: Comparative Smootheology, IV

Any interest here in Beyond the Regnant Philosophy of Manifolds?

Posted by: David Corfield on December 7, 2009 11:31 AM | Permalink | Reply to this

### Re: Comparative Smootheology, IV

I tagged it as a “read soon”, but it’s the exam period here so “soon” might be a relative term …

Posted by: Andrew Stacey on December 7, 2009 12:16 PM | Permalink | Reply to this

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