Birational spaces

In this paper we construct the category of birational spaces as the category in which the relative Riemann–Zariski spaces of [9] are naturally included. Furthermore we develop an analogue of Raynaud's theory. We prove that the category of quasi-compact and quasi-separated birational spaces is naturally equivalent to the localization of the category of pairs of quasi-compact and quasi-separated schemes with an affine schematically dominant morphism between them localized with respect to simple relative blow ups and relative normalizations.