  We provide a synthesis of different topos-theoretical approaches of the
general construction of spectra, exploring jointly its purely categorical,
model theoretic and geometric aspects. We detail also the exact relation with
Diers notion of spectrum. Then we give a more detailed account of Cole's
original construction from bicategorical universal properties. We also provide
new formulas to compute the spectrum of arbitrary etale maps as pseudolimits,
as well as generalizations of results on fibered topos we then apply to compute
spectrum of a modelled topos.
