Let $(X\_n)$ be a tower of finite covering maps of compact smooth manifolds, with $f\_{s,t} : X\_t\to X\_s$ the maps, and $\Lambda\_n := f\_{n,0}^{-1}\Lambda$, with $\Lambda$ the constant abelian sheaf on $X\_0$ with value $\Lambda$, an abelian group.


Clearly $\Lambda\_n$ is the constant sheaf on $X\_n$ with value $\Lambda$ too.


There are, then, transition maps $f\_{n+1,n}^{-1}\Lambda\_n\to\Lambda\_{n+1}$.


A base for the topology on $X := \varprojlim\_{n\ge 0}X\_n$ is given by opens of the form $U:= \varprojlim\_{m\in I}U\_m$, $I\subset\mathbf{N}$, $U\_m\subset f^{-1}\_{m,k}(U\_k)$ open, $m,k\in I$, and the assignment:


$$U\mapsto \varinjlim\_{m\in I}\Lambda\_m(U\_m)$$


uniquely determines a sheaf $\mathcal{F}\_{\Lambda}$ on $X$.


One can give $X$ the structure of a solenoid in the sense of Smale (see [here](https://en.wikipedia.org/wiki/Solenoid_(mathematics)) and the book by Moore and Sochet).


Say $\Lambda = \mathbf{C}$ the sheaf of locally constant smooth functions with value in the complex numbers. On $X$ one can talk about the sheaf $\mathcal{F}’$ of continuous complex valued functions that are smooth along the “leaves” and continuous along the transversals (these are copies of the Cantor set). 



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> Are $\mathcal{F}\_{\Lambda}$ and $\mathcal{F}’$ the same sheaf on $X$? Do they at least have the same cohomology? 
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In case this question has a negative answer, then:



> 
> Is $\mathcal{F}\_{\Lambda}$ the constant abelian sheaf on $X$ with value $\Lambda$?
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I expect the answer to this last question is “no”.


**Related:** [this](https://mathoverflow.net/questions/317382/on-limits-of-manifolds) seems a related question, esp the answer by Ian Agol. 



Sheaves on solenoids
I think that the sheaf $\mathcal{F}\_\Lambda$ is the constant sheaf with stalk $\Lambda$.


A basis of open sets in a solenoidal space is given by the sets $f\_k^{-1}(U\_k)$ where $U\_k \subseteq X\_k$ is open and $f\_k \colon X \to X\_k$ is the canonical projection. Now if you take $U\_k \subseteq X\_k$ so small that it is regularly covered, that is
$$ f\_{k,m}^{-1}(U\_k) \cong F\_m \times U\_k $$ for all $m > k$ and finite sets $F\_m$, then
$$f\_k^{-1}(U\_k) \cong F \times U\_k$$ where $F = \varprojlim F\_m$ is a profinite set.
Then the group of locally constant $\Lambda$-valued functions on $f\_k^{-1}(U\_k)$ is the direct limit of the locally constant $\Lambda$-valued functions on $F\_m \times U\_k$ and this is, by your definition, the group of sections of the sheaf $\mathcal{F}\_\Lambda$ over $f\_k^{-1}(U\_k)$.


The sheaf $\mathcal{F}'$ is much larger and moreover, it seems to me that it is a soft sheaf. This implies that $\mathcal{F}'$ is acyclic, i.e. trivial higher cohomology groups. If you want to compute the cohomology of $\mathcal{F}\_\mathbb{C}$ then the canonical map $\mathcal{F}\_\mathbb{C} \to \mathcal{F}'$ is a useful first step in constructing a resolution of $\mathcal{F}\_\mathbb{C}$ by acyclic sheaves.


