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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:

  1. $\mathcal{E}$ is the set of all enumerable subsets of $E$.
  2. 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?

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