Some History of Geometry
I am almost finished with a book, an advanced
introduction to differential manifolds, bundles, and groups, that began as
the lecture notes for my 2011--13 seminars. There is also an appendix that
may be integrated into the book, Manifolds in Fluid
Dynamics by J. Ryan. In addition to the exercises in my book,
there are also some good ones in Bröcker &
Jänich and in Warner, even though some of
his notation is archaic, to put it kindly.
The category theory needed is almost covered by 713 as I have taught it. One additional result is used exactly twice: that right adjoints preserve products, found in MacLane's Categories for the Working Mathematician (CWM) on p.114. It lies just beyond 713; a highly selective reading of CWM will suffice to understand the proof of the result. A more elementary reference is Dodson's little book. General topology at the 525+ (old 725) level is essential. Algebra and analysis prerequisites are about at the level of 513/713 (M & B) and 640 (Bartle, part II); the most essential things are groups, group actions, and differential analysis in real n-space. Group actions are covered thoroughly in the book.
Remarks and comments, however, may assume much more: homotopy, (co)homology, nuclear HLCTVSs, topological function spaces, etc. I mention nuclear spaces because I'd like to add a chapter about analysis on section spaces someday.
Indeed, I am looking for someone to assist me in developing analysis on complete, nuclear k-spaces, using that on Fréchet spaces in particular and on Hausdorff locally convex TVSs in general as a model and/or guide. It should make a good MS thesis at least. If you might be interested, let me know.