Hofer’s theory of polyfolds may be viewed a certain kind of generalized smooth spaces. The bulk of a theory developed by Helmut Hofer is a generalized approach to Fredholm operator analysis in geometrical situations; it can be also considered as an “analytic framework designed to resolve the issue of transversality systematically”. It gives a unified framework for symplectic field theory, Floer homology, some constructions in contact geometry etc.
Katrin Werheim’s quick introduction.
Hofer’s lectures at Stanford on polyfolds (a more complete version is here but for the time being (?) it’s password protected).
Helmut Hofer, Polyfolds and a general Fredholm theory, arXiv/0809.3753
Helmut Hofer, Kris Wysocki, Eduard Zehnder, Integration theory for zero sets of polyfold Fredholm sections, arXiv/0711.0781
Helmut Hofer, Kris Wysocki, Eduard Zehnder, A general Fredholm theory I: A splicing-based differential geometry, arXiv/0612604; A general Fredholm theory II: implicit function theorems, arXiv/0705.1310; A general Fredholm theory III: Fredholm functors and polyfolds, arXiv
Helmut H. Hofer, A general Fredholm theory and applications, arXiv/math/0509366
We thank Eugene Lerman for some of the information here.