It seems to me that multivalued functions and/or indefinite integrals can be thought of as sheaves.
For example:
The real square-root function can be viewed as the sheaf $\mathcal{F}$ defined on the open sets of $\mathbb{R}$ by letting $\mathcal{F}(U)$ denote the set of all continuous function $f : U \rightarrow \mathbb{R}$ satisfying $\forall x \in U(f(x)^2 = x).$
The complex square-root function can be viewed as the sheaf $\mathcal{F}$ defined on the open sets of $\mathbb{C}$ by letting $\mathcal{F}(U)$ denote the set of all continuous function $f : U \rightarrow \mathbb{C}$ satisfying $\forall z \in U(f(z)^2 = z).$
The complex logarithm function can be viewed as the sheaf $\mathcal{F}$ defined on the open sets of $\mathbb{C}$ by letting $\mathcal{F}(U)$ denote the set of all continuous function $f : U \rightarrow \mathbb{C}$ satisfying $\forall z \in U(e^{f(z)} = z).$
If $f$ is a continuous real-valued function, we can define $\int f(x) dx$ to be the sheaf $\mathcal{F}$ such that $\mathcal{F}(U)$ is the set of all differentiable functions $F$ on $U$ satifying $F'=f$. More generally, if $\mathfrak{f}$ is a sheaf of continuous functions, we can define $\int \mathfrak{f}(x)dx$ to be the sheaf $\mathcal{F}$ such that $\mathcal{F}(U)$ is the set of all differentiable function $F : U \rightarrow \mathbb{R}$ satisfying $F' \in \mathfrak{f}(U).$
Similar statements to the above probably hold for the complex case, allowing us to prove claims like $$\log(z) \subseteq \int \frac{1}{z}dz,$$ etc. where $\log$ is viewed as the sheaf-theoretic inverse of $z \mapsto e^z$ as described above, and the $\subseteq$ in this context really means something like: for all open $U \subseteq \mathbb{C}$, we have $$(z \mapsto \log(z))(U) \subseteq \left(z \mapsto \int \frac{1}{z}dz\right)(U).$$
Question. Do any published books or articles take this point of view?
