Generalized smooth spaces

The category of smooth manifolds is badly behaved: it is not cartesian closed and it does not have all equalizers or coequalizers, among other things. The category of smooth manifolds with boundary is even worse: it doesn’t have products! Many authors have attempted to enlarge the standard categories of smooth manifolds into a “convenient category” of generalized smooth spaces .

There is a bewildering variety of proposals for generalized smooth spaces:

• Diffeological space , aka Souriau space
• Chen space
• Differential space, aka differential module or Sikorski space
• Frölicher space
• Smith space
• Smooth set , aka smooth space

Literature comparing these notions:

• Stacey, 2011: Comparative smootheology (pdf, arxiv, nCat Cafe 1 ,2 ,3 ,4 )
  • Compares many definitions of a “category of smooth objects,” namely those of Chen, Frolicher, Sikorski, Smith, and Souriau
• Baez & Hoffnung, 2011: Convenient categories of smooth spaces (doi, arxiv, nCat Cafe )
  • Takes “smooth space” to be diffeological space or Chen space
• Watts, 2012, PhD thesis: Diffeologies, differential spaces, and symplectic geometry (arxiv, pdf)
  • Chapter 2 compares diffeological and differential spaces and shows that their “intersection” is Frolicher spaces