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I understand the Taylor series and I understand what a germ is.
"Any vector $g = (z_0, α_0, α_1, ...)$ is a germ if it represents a power series of an analytic function around $z_0$"
And a sheaf is a set of germs.
But then I read
"Sheaves of analytic functions were once called multi-valued functions"
I am not sure how to make sense of this. Do they just mean that there are multiple vectors with the same $z_0$ but having different values for $α_0, α_1,...$?
For example, the square root function has two branches. On the reals, you may choose the principal branch, but over the complex plane there is no consistent way to do this without excluding a branch cut.
One solution is to treat the square root as a multi-valued function. $\sqrt{r^2e^{i2\theta}}=\{re^{i\theta},re^{i(\theta+\pi)}=-re^{i\theta}\}.$
The more modern approach uses the language of sheaf theory. The sheaf of germs of the square root function defines a two-sheeted cover of the complex plane.
A sheaf is an assignment of a fiber to each point (subject to some conditions), so the answers are equivalent.
And yes, to your last sentence. For the sheaf of a function to be multi-valued, viewing it as the set of germs of that function, means there are multiple Taylor series with the same $z_0$. For example, the principal square root and its negative have different Taylor series (they differ by a sign).
