  We give the site-theoretic account of the spectral construction as first
introduced by Coste. We provide a detailed examination of the geometric
properties of the spectrum, in particular what classes of topoi it produces
when applied to the different classes of objects and maps in a geometry. We
also give a new proof of the spectral adjunction for set-valued models and the
classifying property of the structure sheaf. We also discuss the opposition
between ``gros and petit" spectra and relation with some canonical classifying
sites. We then describe the general case of a modelled topos. We also prove
that the spectrum of a locally modelled topos is local over its base and deduce
a new proof of the spectral adjunction for the general case. We finally give
several examples from the world of propositional algebras, and in particular
recover classifying topoi of first order theories in various fragment of logic
as spectra of some suitably modelled topoi.
