Diers developed a general theory of right multiadjoint 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 multiadjoint, 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 multiadjoint 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-Chevaley 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.

We provide bicategorical analogs of several aspects of the notion of geometry in the sense of the theory of spectrum. We first introduce a notion of local right biadjoint, and prove it to be equivalent to a notion of bistable pseudofunctor, categorifying an analog 1-categorical result. We also describe further laxness conditions, giving some properties of the already known lax familial pseudofunctors. We also describe 2-dimensional analogs of orthogonality and factorization systems, and use them to construct examples of bistable pseudofunctors through inclusion of left objects and left maps. We apply the latter construction to several examples of factorization systems for geometric morphisms to produce geometry-like situations for Grothendieck topoi, recovering in particular a local geometry involved in the general construction of spectra.

This second part comes to the construction of the spectrum associated to a situation of multi-adjunction. Exploiting a geometric understanding of its multi-versal property, the spectrum of an object is obtained as the spaces of local unit equipped with a topology provided by orthogonality aspects. After recalling Diers original construction, this paper introduces new material. First we explain how the situation of multi-adjunction can be corrected in a situation of adjunction between categories of modeled spaces as in the topos-theoretic approach. Then we come to the 2-functorial aspects of the process relatively to a 2-category of Diers contexts. We propose an axiomatization of the notion of spectral duality through morphisms between fibrations over a category of spatial objects, and show how such situations get back right multi-adjoint functors.

We categorify cocompleteness results of monad theory, in the context of pseudomonads. We first prove a general result establishing that, in any 2-category, weighted bicolimits can be constructed from oplax bicolimits and bicoequalizers of codescent objects. After prerequisites on pseudomonads and their pseudo-algebras, we give a 2-dimensional Linton theorem reducing bicocompleteness of 2-categories of pseudo-algebras to existence of bicoequalizers of codescent objects. Finally we prove this condition to be fulfilled in the case of a bifinitary pseudomonad, ensuring bicocompleteness. MSC2020. 18N15; 18N10; 18C15;

With a model of a geometric theory in an arbitrary topos, we associate a site obtained by endowing a category of generalized elements of the model with a Grothendieck topology, which we call the antecedent topology. Then we show that the associated sheaf topos, which we call the over-topos at the given model, admits a canonical totally connected morphism to the given base topos and satisfies a universal property generalizing that of the colocalization of a topos at a point. We first treat the case of the base topos of sets, where global elements are sufficient to describe our site of definition; in this context, we also introduce a geometric theory classified by the over-topos, whose models can be identified with the model homomorphisms towards the (internalizations of the) model. Then we formulate and prove the general statement over an arbitrary topos, which involves the stack of generalized elements of the model. Lastly, we investigate the geometric and 2-categorical aspects of the over-topos construction, exhibiting it as a bilimit in the bicategory of Grothendieck toposes. NotationThe notation employed in the paper will be standard; in particular, − We shall denote by Set the category of sets (within a fixed model of set theory).− Given a geometric theory T, we shall denote by (C T , J T ) its geometric syntactic site and by Set[T] its classifying topos, which, as is well-known, can be represented as Sh(C T , J T ) (for background on classifying toposes, the reader may refer to [2]).− For any geometric theory T, we denote by T[E] the category of T-models in a geometric category E and model homomorphisms between them. For any E, we have an equivalence T[E] ≃ Cart J T (C T , E) between T[E] and the category of cartesian (that is, finite-limitpreserving) functors C T → E which are J T -continuous (that is, which send J T -covering families to covering families in E). The functor C T → E corresponding to a T-model M in E will be denoted by F M ; it sends any geometric formula { x.φ} over the signature of T to its interpretation [[ x.φ]] M in M , and acts accordingly on arrows (for more details, see, for instance, Chapter 1 of [2]).