Limits, Colimits, and Spectra of Modelled Spaces

It is well-known that the construction of Zariski spectra (commutative) rings yields a dual adjunction between category and locally ringed spaces. There are many constructions algebras in various contexts giving such adjunctions. Michel Coste unified them language categorical logic by showing that, for an appropriate triple $ (T_0,T,\Lambda) (which we call spatial context), each T_0 $-model can be associated with T $-modelled space this $-models spaces "admissible" morphisms. However, most his proofs remain unpublished. In paper, introduce alternative give another proof adjunction. Moreover, also extend to relative prove existence limits colimits involved categories modelled We deduce, instance, whose stalks fields complete cocomplete.