Books on Lie Groups via nonstandard analysis? Is there any book or online note that covers the basics of lie groups using nonstandard analysis? Another thing I would like is to see these things in category theory (along the lines of Algebra: Chapter 0 except for differential geometry?)
 A: I'm not aware of any Lie theory being done with Robinson's nonstandard analysis, but a different approach to infinitesimals via Kock and Lawvere's smooth infinitesimal analysis has certainly been used to develop some Lie theory. This might be more fruitful to look into since it's an axiomatization of some of Grothendieck's methods in algebraic geometry (formal schemes and such), which are in mainstream use. There's some stuff on Lie groups and Lie algebras in Kock's Synthetic Differential Geometry and Kock's Synthetic Geometry of Manifolds which are both available for free. Lavendhomme's Basic Concepts of Synthetic Differential Geometry also covers some Lie theory.
Smooth infinitesimal analysis is very nice and intuitive, more so than classical differential geometry in many ways (tangent vectors as actual curves and vector fields as infinitesimal transformations). It is certainly categorical in style, as you can probably guess since it was pioneered by Kock and Lawvere.
A: I never looked at that part of the book in any detail at all, but Abraham Robinson's Nonstandard Analysis includes some material on Lie groups via nonstandard analysis.
A little bit of googling found a paper on the arXiv that might be relevant.
A: You can find references to such literature in Isaac Goldbring's statement of research. In particular see the work of Robinson [13], Bate [1], Singer [14] and Hirschfeld [6] referenced there.
