Differential forms on fuzzy manifolds

Solution 1:

I think your intuition is right: you already have the structure in place to talk about fuzzy tangent bundles (Definition 18). So you can talk about (fuzzy) sections of bundles, hence fuzzy vector fields, fuzzy forms and all that. So, if your question is "does the notion of fuzzy $C^1$ manifold you've defined lead to a natural notion of fuzzy differential forms?" then the answer is certainly yes.

However, my question would be this: is the resulting notion sufficiently strong to prove useful results? Specifically, you'd like to have a fuzzy version of de Rham cohomology; it would be nice if fuzzy exact forms are fuzzy-closed, for example, so that the fuzzy exterior derivative is fuzzy exact.

You'd also like to marry this up with fuzzy integral theory, which seems to be well developed. Perhaps this has all been done, but I'm not seeing it.