References regarding a connection between recursion theory and sheaves - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T20:13:57Z http://mathoverflow.net/feeds/question/25287 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/25287/references-regarding-a-connection-between-recursion-theory-and-sheaves References regarding a connection between recursion theory and sheaves Michael Wan 2010-05-19T22:08:28Z 2010-05-19T22:08:28Z <p>In Manin's <em><a href="http://books.google.com/books?id=8NTWRFD5lZ8C&amp;lpg=PR2&amp;dq=manin%2520logic&amp;pg=PR2#v=onepage&amp;q&amp;f=false" rel="nofollow">A Course in Mathematical Logic for Mathematicians</a></em>, he defines (p.201) a structure $(\mathcal{E},R)$ given an enumerable set $E \subset (\mathbb{Z}^+)^n$ by:</p> <ol> <li>$\mathcal{E}$ is the set of all enumerable subsets of $E$.</li> <li>For each $E' \in \mathcal{E}$, $R(E')=\{f|domain(f)=E', f:E'\rightarrow (\mathbb{Z}^+) \text{ is recursive}\}.$</li> </ol> <p>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". </p> <p>My question: is this a well known construction/analogy? Has it been studied further? Are there any relevant references? </p>