Zariski Toposes

Abstract The category of schemes on a Zariski category can be fully and left-exactly embedded in a large topos - its large Zariski topos -while the category of locally finitely presentable schemes can be fully and left-exactly embedded in a topos-its Zariski topos. Modelled toposes and locally modelled toposes can be defined relatively to a Zariski category. They efficiently play the role of ringed toposes and locally ringed toposes. The Zariski topos is the classifying topos for locally modelled toposes, and to any modelled topos is associated a universal locally modelled topos- its prime spectrum. In order to describe the relations existing between the different kinds of schemes, modelled spaces, modelled toposes, and Zariski toposes associated to different Zariski categories, it is time to introduce the notion of morphisms of Zariski categories. There are, in fact three notions: morphisms, geometrical morphisms, and cogeometrical morphisms.