Rigid analytic spaces with overconvergent structure sheaf

Abstract: We introduce a category of 'rigid spaces with overconvergent structure sheaf' which we call dagger spaces --- this is the correct category in which de Rham cohomology in rigid analysis should be studied. We compare it with the (usual) category of rigid spaces, give Serre and Poincar\'e duality theorems and explain the relation with Berthelot's rigid cohomology.