On the geometry of metric sites
A metric site is a pair \(({\mathcal C}, \tau)\) where \({\mathcal C}\) is a category and \(\tau \) a covariant functor into a category of topological spaces supplying \({\mathcal C}\) with a geometric structure in terms of a Grothendieck topology; the covers of the latter are universal liftings of open covers of the ``underlying'' spaces. The author uses this notion to describe a broad class of categories (including those of topological spaces, differential manifolds, (affine) schemes, algebraic spaces, etc.) arising in modern geometry. There is a notion of complete metric site; any so-called strict metric site has a fibre product-preserving completion as certain sheaves on \({\mathcal C}\) called ``Dedekind cuts''. Example: the dual of the category of commutative rings is a strict metric site of which the category of schemes is a completion.
scientific article; zbMATH DE number 802376