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Sheaves on subanalytic sites. (English) Zbl 1171.32002
Let \(k\) be a field and \(X\) be a real analytic manifold. The spaces of functions which are not defined by local properties do not define sheaves on \(X\), they define sheaves on a site associated to \(X\), the subanalytic site \(X_{sa}\).
M. Kashiwara and P. Schapira [Ind-sheaves. Astérisque. 271. Paris: Société Mathématique de France (2001; Zbl 0993.32009)] introduced the notion of ind-sheaves and defined the so-called six Grothendieck operations in this framework. Restriction to \(\mathbb{R}\)-constructible sheaves gives an equivalence between the category of ind \(\mathbb{R}\)-constructible sheaves on \(X\) and the categroy \(\text{Mod}(k_{X_{sa}})\) of sheaves on the subanalytic site associated to \(X\).
The paper gives a direct, self-contained construction of the six Grothendieck operations on \(\text{Mod}(k_{X_{sa}})\) without using the theory of ind-sheaves.

MSC:
32B99 Local analytic geometry
14P05 Real algebraic sets
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
35C05 Solutions to PDEs in closed form
32S99 Complex singularities
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