In Manin's *A Course in Mathematical Logic for Mathematicians*, he defines (p.201) a structure $(\mathcal{E},R)$ given an enumerable set $E \subset (\mathbb{Z}^+)^n$ by:

- $\mathcal{E}$ is the set of all enumerable subsets of $E$.
- For each $E' \in \mathcal{E}$, $R(E')=\{f|domain(f)=E', f:E'\rightarrow (\mathbb{Z}^+) \text{ is recursive}\}.$

He then demonstrates (cumulating on p. 205-6) that there is an analogy between $(\mathcal{E},R)$ and (his quotes) "a topological space together with a sheaf", and a way to define "recursive Cech cohomology of groups" of some complexes that arise from $(\mathcal{E},R)$. He then claims that "it would be interesting to study such cohomology groups".

My question: is this a well known construction/analogy? Has it been studied further? Are there any relevant references?