References regarding a connection between recursion theory and sheaves

Question

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 $E' \in \mathcal{E}$, $R(E')=\{f|\text {domain}(f)=E', f:E'\rightarrow (\mathbb{Z}^+) \text{ is recursive}\}.$

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?

Answer 1

Here is a somewhat positive answer in terms of

germs and equivalence modulo finite differences.

There has been a good deal of impressive lattice-theoretical work on the lattice of recursively enumerable sets. (This topic is a subtopic of recursively enumerable sets and degrees (AMS classification 03D25); these are also known as r.e. sets, computably enumerable sets, and c.e. sets.)

This work in particular has dealt with the lattice of c.e. subsets of a given c.e. set $E$, $\mathcal L(E)$, which has different properties depending on properties of $E$ such as hypersimplicity and creativity.

About half of the work concerns the quotient structures modulo finite differences. So in a sense they are studying germs and there is some analogy with what you are asking for. That is, one studies $\mathcal E$, the lattice of r.e. sets, and $\mathcal E^*$, the lattice of equivalence classes of elements of $\mathcal E$, where $A$ and $B$ are equivalent if $$ \{n : A(n)\ne B(n)\} $$ is a finite subset of $\omega=\mathbb N$.

The focus is on the automorphism group, its orbits, and first order definability. See for instance

Cholak, Peter A.; Downey, Rodney; Harrington, Leo A. On the orbits of computably enumerable sets. J. Amer. Math. Soc. 21 (2008), no. 4, 1105–1135.