Most higher-genus curves cannot be smoothly embedded in the plane, but they fit nicely in three-space.

Is anyone other than me in favor of changing the word "non-trivial" to "interesting" in the title? I, for one, would feel much less intimidated.

By the way, do you know a good reference for the reconstruction result?

Do you mean the statement that a set is the union of the preimages of the image elements under a map? I think both statements are categorifications of the existence of addition, but linear categorifications are intrinsically more powerful.

I think your argument is more effective against a slogan like, "all interesting functions are differentiable". In my (limited) experience, differentiation tends to be algorithmic in practice, although it can be unstable in numerical applications. This is in contrast to integrals, which exist much more often and tolerate numerical error well, but are generally very difficult to compute.

There are also profinite Sylow theorems, yielding the existence of a maximal pro-p subgroup. The proofs are relatively straightforward extensions of the finite proofs.

That sounds like a good idea. I'll ask specific questions for details when I understand the generalities a little better.

Sorry, my previous comment was unnecessarily confrontational. I'll delete it.