### Conic sheaves on subanalytic sites and Laplace transform

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Here we prove a Poincaré - Verdier duality theorem for the o-minimal sheaf cohomology with definably compact supports of definably normal, definably locally compact spaces in an arbitrary o-minimal structure.

Given a real analytic manifold Y, denote by ${Y}_{sa}$ the associated subanalytic site. Now consider a product Y = X × S. We construct the endofunctor $\mathcal{F}\mapsto {\mathcal{F}}^{S}$ on the category of sheaves on ${Y}_{sa}$ and study its properties. Roughly speaking, ${\mathcal{F}}^{S}$ is a sheaf on ${X}_{sa}\times S$. As an application, one can now define sheaves of functions on Y which are tempered or Whitney in the relative sense, that is, only with respect to X.

In this paper we work in o-minimal structures with definable Skolem functions, and show that: (i) a Hausdorff definably compact definable space is definably normal; (ii) a continuous definable map between Hausdorff locally definably compact definable spaces is definably proper if and only if it is a proper morphism in the category of definable spaces. We give several other characterizations of definably proper, including one involving the existence of limits of definable types. We also prove the...

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