I am new to the topic of copulas and my math is limited. I have different questions:
1. Is it correct to say, that $C: [0,1]^d \rightarrow [0,1]$ is a mapping from a multidimensional distribution to its one dimensional margins? Or that this is a mapping from the unit cube to the uniform margins?
2. I am often reading about the Fréchet-Hoeffding bounds, these seem to basically give the co-domain of the copula? Or what do they state and why do I need them? When are they reached? If I understood it correctly, than the lower bound is reached, when d=2 and the random variables are perfectly negatively dependent and the upper bound is reached, when the random variables are perfectly positively dependent?
3. In every book about copulas I read about the Sklar's Theorem. I do not understand the intuition of this mathematical theorem. What does it say and why do I need it in the context of copulas?
4. The definition of a copula is: It is a multivariate distribution with uniform marginal distributions, but I thought, that I can take any distribution for the marginal distributions and not only the uniform distribution?
I know these are beginners questions, but I could not find a good introductory book about copulas. Every book in this topic gives a lot of mathematical definitions and theorems and I am not on this level, unfortunately.