Did I mention Google? :)

Yes, this is Hartshorne terminology, so it's standard :) My question was more, why you don't simply use affine opens.

Of course, everything is holomorphic!

Why do you use "quasi-affine" instead of "affine"?

Thanks for the edit, Daniel! Now, I'm not confused anymore... :)

I don't understand your example. But in general: how do you check smoothness? This should work in both cases (algebraic and analytic) by taking an affine cover and then apply the Jacobi criterion. This is in both cases identical and so a point on the algebraic variety should be smooth if and only if the point in the analytic space is smooth. But maybe, I'm just missing something obvious...

Moreover, if X is a variety over C and X^{an} is the corresponding (reduced) analytic space, then X^{an} should be smooth if and only if X is smooth. Or am I absolutely wrong? (Sorry, if this is the case!)

The first sentence: seriously? How do you see that? (I may be just too dumb)

Amazing, although I don't really know what to do with this (\infty,1)-stuff...I remember reading your notes/thoughts/questions (andreasholmstrom.org/research/Cohomology1.pdf) some time ago and I was happy that indeed people are thinking about these questions (people always tell me that I'm crazy when I ask such questions!). So, I'm looking forward to whatever you will add to your comment :)

Well, good point. I'm looking of course for a presentation with less elements than the group order. Something that makes it possible (at least for small n and m) to determine/handle these groups with a computer without doing this by brute force.

Well, sometimes Wikipedia is better if you don't know what an (\infty,1)-category is, because the number of entries in nlab you can read without knowing this is close to 0 :) (serious)