On Diers theory of Spectrum I : Stable functors and right multi-adjoints

Diers developed a general theory of right multi-adjoint functors leading to a purely categorical, point-set construction of spectra. Situations of multiversal properties return sets of canonical solutions rather than a unique one. In the case of a right multi-adjoint, each object deploys a canonical cone of local units jointly assuming the role of the unit of an adjunction. This first part revolves around the theory of multi-adjoint and recalls or precises results that will be used later on for geometric purpose. We also study the weaker notion of local adjoint, proving Beck-Chevalley conditions relating local adjunctions and the equivalence with the notion of stable functor. We also recall the link with the free-product completion, and describe factorization aspects involved in a situation of multi-adjunction. The relation between accessible right multi-adjoints and locally finitely multipresentable categories is also revisited.