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.

§ INTRODUCTION

Many prominent dualities in mathematics are instances of a common construction centered on the notion of spectral functor. Roughly stated, one starts with a finite limit theory, equiped with a geometric extension whose models are tough as local objects encoding point-like data, together with a factorization system (Etale maps, Local maps), where the etale maps behave as duals of distinguished continuous maps. Several manners of axiomatizing the correct relation between those ingredients have been proposed, either through topos theoretic methods by “localizing" local objects with a Grothendieck topology generated by etale maps, or in an alternative (though tightly related) way based on the notion of stable functor. Moreover those data make still sense in categories of models of the ambient finite limit theory in arbitrary toposes. Then the spectrum of a given object is constructed as a topos classifying etale maps under this given object toward local objects, equiped with a structural sheaf playing the role of the “free local object" under it. This defines a spectral functor, left adjoint of a inclusion of a category of locally modelled toposes into a category of modelled spaces.

The history of this construction goes back to <cit.>, where the classifying topos of the theory of local rings was introduced as well as the notion of “(locally) ringed topos" and a first generalization of the notion of locally ringed spaces in the context of algebraic geometry. This work inspired then <cit.>, where a first systematic construction was provided by means of 2-dimensional universal properties and the ingredients of the general situation were isolated, while a first axiomatization of the notion was introduced in <cit.>. <cit.> then gave a first syntactic approach of the process from the point of view of categorical model theory. On his side, <cit.> had provided an alternative approach which does not make use of toposes and processes in a more point-set and concrete way. Later, <cit.> provided an $\infty$-categorical version of the construction from a purely geometric point of view, while <cit.> provided a topological interpretation together with a suited version of the small object argument.

This construction subsumes a large number of examples, beside the classical examples from algebraic geometry, as for instance <cit.>, <cit.>, <cit.>.

In this paper, we sum up and compare different approaches, especially <cit.>, <cit.>, <cit.> and <cit.>. Beside the expository aspects of this work, we provide comparative results expressing how those different processes and the situations they start from are related.

In the first section we give prerequisites on the three ingredients of this constructions:

* the notion of finite limit theory whose models are the “ambient objects", describing the different equivalent manners to see those theories and their categories of models;

* the notion of geometric extension coding for the local objects, and again how to switch from functorial, geometric or model theoretic point of views;

* and the notion of left generated factorization system.

In each case we emphasize the multiplicity of presentations of the same notions, for the rest of the paper will require to switch from one to another with fluidity.

The second section revolves around the notion of geometry and admissibility. The latter is a condition entangling the data above into a factorization condition, which can also be captured by the notion of stable functor, while the former is a very light requirement, whose topological content, once unravelled, produces the condition encoded in admissibility. Moreover, it is remarkable that in those situations, set-valued data happen to be sufficient to induce the same structure on arbitrary toposes, the condition of admissibility being inherited in categories of models in arbitrary toposes. We also devote a subsection to the notion of stable functor, not only how geometries produce stable functors, but also how one can produce an “axiomatization" of a stable functor into a geometry.

The third section is about the notion of (locally) modelled toposes, and the construction of the spectrum from a geometry by means of abstract 2-dimensional properties. Though the process was suggested in <cit.>, we try to be as explicit as possible and give some new insight and precision on the matter.

In the fourth section, we recall <cit.> construction of the corresponding spectral site, either for a set valued object or a modelled topos, and a pseudolimit formula <ref> generalizing the correspondence between etale maps and etale geometric morphisms; after introducing a new form of a classical result from <cit.> about fibered toposes, we prove that the spectrum of a modelled topos is a topos of continuous section of a canonical fibered topos, giving also new fibrational aspects of the construction.

We end with two examples, spectral accounts of Stone duality and Jipsen and Moshier.

§ LEFT GENERATED FACTORIZATION SYSTEMS AND GEOMETRIC EXTENSIONS

In this first part we introduce the three ingredients involved in the construction of a spectrum, the ambient data, the local data and the factorization data, and how they are entangled to constitute a geometry that will be deployed into the spectrum.

§.§ The yoga of Gabriel-Ulmer

This first section just contains very standard facts about finite limit theories and locally finitely presentable categories. Its purpose is to recall all the equivalent ways to describe a finite limit theory, and how one can toggle between the logical, functorial and toposic way of seeing models. We emphasize the way the data of a finite limit theory, its syntactic category, its classifying topos and its class of set valued models are mutually determined by given as much as possible of the involved equivalence of categories, for we will need to have in mind those translations to make more intuitive the rest of the paper.

While all the well known examples of the spectral construction are defined relatively to algebraic categories, which are the object of interest of universal algebra and are axiomatized by Lawvere theories, those latter are not the most suited notion to proceed for our purpose. Indeed, when working with toposes as we are going to do, it is more natural to consider locally finitely presentable categories and their syntactic counterparts, the finite limit theories. The reason is that each Grothendieck topos admits a standard presentation, exhibiting it as a category of sheaves over a small site with finite limits, and that the extensions theorem as Diaconescu are formulated relatively to finite limits. In the following, we call functors preserving finite limits lex functors and denote as $\Lex$ the 2-category of small categories with finite limits, lex functors between them, and natural transformations between lex functors; for concision we also call small categories with finite limits lex categories. For a lex category $\mathcal{C}$ and a Grothendieck topos $\mathcal{E}$ we denote as $\Lex[\mathcal{C}, \mathcal{E}]$ the category of lex functors and natural transformations between them. We also denote as $\mathcal{S}$ the topos of sets.

In the following we fix a first order language $\mathcal{L}$. For a theory $\mathbb{T}$ in $\mathcal{L}$, a sequent is said to be $ \mathbb{T}$-provable if it can be deduced from the axioms of $\mathbb{T}$ and the structural rules of first order logic, and we denote as $ \vdash_\mathbb{T}$ the relation of $\mathbb{T}$-provability. The relation of $\mathbb{T}$-equivalence is the equivalence relation $\dashv \vdash_\mathbb{T}$ on formulas in $\mathcal{L}$ defined as

\[ \phi(\overline{x}) \dashv \vdash_\mathbb{T} \psi(\overline{x}) \textrm{ iff } \phi(\overline{x}) \vdash_\mathbb{T} \psi(\overline{x}), \; \psi(\overline{x}) \vdash_\mathbb{T} \phi(\overline{x}) \]

Its classes are denoted as $ [ \phi(\overline{x})]_\mathbb{T}$.

We recall that a finite limit theory in $\mathcal{L}$ is a first order theory in $\mathcal{L}$ whose sequents consist of cartesian formulas, that are formulas built from atomic formulas with finite conjunctions and existentials with proof of uniqueness.
A $\mathbb{T}$-provably functional formula in $\mathcal{L}$ is a formula $ \theta[\overline{x}, \overline{y}]$ such that the following sequents are $\mathbb{T}$-provable:

* $ \theta(\overline{x},\overline{y}) \vdash_{\mathbb{T}} \psi( \overline{y}) $

* $ \phi(\overline{x}) \vdash_{\mathbb{T}} \exists \overline{y} \theta(\overline{x},\overline{y})$

* $ \theta(\overline{x},\overline{y}) \wedge \theta(\overline{x},\overline{y}') \vdash_{\mathbb{T}} \overline{y} = \overline{y}' $

A provably functional formula $\theta[\overline{x}, \overline{y}]$ should be thought as a statement of equality exhibiting $ \overline{y} $ as a function of $\overline{x}$. The first condition says that this image of this function should satisfy the codomain formula; the second says that this function is defined over witnesses of the domain formula, and the third is the condition of functionality, ensuring that an object has a unique image through this function.

To any first order theory $\mathbb{T}$ we can associate a category $\mathcal{C}_\mathbb{T}$, the syntactic category of $\mathbb{T}$ , whose

* objects are formulas in context $\{\overline{x}, \phi\}$ in the language of $ \mathbb{T}$

* morphisms $\{\overline{x},  \phi \} \rightarrow \{\overline{y}  ,\psi \}$ are equivalence classes $ [\theta(\overline{x},\overline{y})]_\mathbb{T}$ for the relation of $\mathbb{T}$-provability of functional formulas.

If $ \mathbb{T}$ is cartesian then $ \mathcal{C}_\mathbb{T}$ is a small category with finite limits. Conversely, any lex category $ \mathcal{C}$ is the $ \mathcal{C}_\mathbb{T} $ of some cartesian theory $\mathbb{T}$. However in the following we shall not need to explicitly give a syntactic presentation of the finite limit theory associated to a lex category, so we do not recall the process here.

A set-valued model of $\mathbb{T}$ is a lex functor $  F : \mathcal{C}_\mathbb{T} \rightarrow \mathcal{S}et$. The intuition behind this definition is that such a functor associates to any formula in context its set of witnesses in the corresponding model

\[ F(\{ \overline{x}, \phi \}) = \{ \overline{a} \in F \mid \; F \models \phi(\overline{a}) \}  \]

and to any morphism of the syntactic category, the application
\[      F(\{ \overline{x}, \phi \})  \stackrel{ F([\theta(\overline{x}, \overline{y})]_\mathbb{T})}{\longrightarrow}  F(\{ \overline{y}, \psi \})
\]

associating to each $  \overline{a}$ in $F(\{ \overline{x}, \phi \}) $  the unique  $\overline{b}$ in $ F(\{ \overline{y}, \psi \}) $ such that  $ F \models \theta[\overline{a},\overline{b}]$.

A morphism of $\mathcal{L}$-structures between $ \mathbb{T}$-models $ F, \; G$ is a natural transformation $F \Rightarrow G$ in $\Lex[\mathcal{C}_\mathbb{T}, \mathcal{S}]$. Set-valued $\mathbb{T}$-models and morphisms of $ \mathcal{L}$-structures between them form a category $ \Lex[\mathcal{C}_\mathbb{T}, \mathcal{S}]$ which we denote as $\mathbb{T}[\mathcal{S}]$.

Now we turn to the general properties of the categories of set-valued models of finite limits theories. A locally finitely presentable category $\mathcal{B}$ is a category with: 0em

* filtered colimits

* small limits

* a small generator $ \mathcal{B}_{\fp} $ of finitely presented objects such that any object $ B $ is the filtered colimit of $ \mathcal{B}_{\fp}\downarrow B$

In particular any locally finitely presentable category also has small colimits. Moreover, the full subcategory $ \mathcal{B}_{\fp}$ of finitely presented objects is closed in $\mathcal{B}$ under finite limits, and the inclusion $ \mathcal{B}_{\fp} \hookrightarrow \mathcal{B}$ exhibits $ \mathcal{B}$ as the inductive completion
\[\mathcal{B} \simeq \Ind(\mathcal{B}_{\fp})\]

where $ \Ind(\mathcal{B}_{\fp})$ is the full subcategory of $ [\mathcal{B}_{\fp}^{\op}, \mathcal{S} ]$ consisting of functors that are filtered limits of representable.

A morphism of locally finitely presentable category is a functor $ F :  \mathcal{B}_1 \rightarrow \mathcal{B}_2$ preserving filtered colimits and small limits. It can be shown by variation of Freyd adjoint functor theorem that any such functor has a left adjoint $F^*$ sending finitely presented objects of $\mathcal{B}_2$ to finitely presented objects of $\mathcal{B}_1$. We denote as $ \mathcal{LFP}$ the 2-category of locally finitely presentable categories, with natural modifications between locally finitely presentable categories as 2-cells.

Locally finitely presentable categories are exactly the categories of models of Cartesian theories; this is the content of Gabriel-Ulmer duality:

\[ \begin{array}{rcl}
\Lex^{op} & \simeq & \mathcal{LFP}   \\
\mathcal{C} = \mathcal{C}_\mathbb{T} & \mapsto & \Lex[\mathcal{C}, \mathcal{S}] \\
\mathcal{B}^{op}_{\fp} & \mapsfrom & \mathcal{B}
\end{array} \]

where the left to right process associates to any locally finitely presentable category the lex category made of the opposite category of the small subcategory of finitely presented objects. Objects of the small generator of finitely presented objects in $ \Lex[\mathcal{C}_\mathbb{T}, \mathcal{S}]$ are the corepresentable functors, which we denote as $ \hirayo^*_{\{ \overline{x}, \phi \}} = \mathcal{C}_\mathbb{T}[\{ \overline{x}, \phi \}, - ] $.  Moreover, for any object $ B$ in $\mathcal{B}$, the restriction to $\mathcal{B}_\fp$ of the corresponding representable functor $ \hirayo_B = \mathcal{B}[-,B]$ defines a lex functor
\[ \mathcal{B}^{\op}_{\fp} \stackrel{\hirayo_B}{\longrightarrow} \mathcal{S} \]

This defines an equivalence of category $ \mathcal{B} \simeq \Lex[\mathcal{B}^{\op}_{\fp}, \mathcal{S}]$.

For a language $ \mathcal{L}$ and any string of variables $ \overline{x}$, we can construct the free $\mathcal{L}$-structure $ \langle \overline{x} \rangle_\mathcal{L}$ consisting of all terms in $\mathcal{L}$ over the variables $\overline{x}$. For a finite limit theory $\mathbb{T}$, this can be completed into the free model of $\mathbb{T}$ by forcing the equality of terms that are $\mathbb{T}$-provable in $\mathbb{T}$, and we denote it as $ \langle \overline{x} \rangle_\mathbb{T}$.

For a formula in context $ \{ \overline{x} , \phi \}$ in $\mathcal{C}_\mathbb{T}$, the corresponding finitely presented set-valued models of $\mathbb{T}$ can be seen as the quotient $ \langle \overline{x} \rangle_\mathbb{T}/\phi(\overline{x})$. We denote this finitely presented model as $K_\phi$, and seen as an object of $\Lex[\mathcal{C}_\mathbb{T}, \mathcal{S}]$, $ K_\phi $ coincides with the corepresentable functor $ \hirayo^*_{ \{ \overline{x}, \phi \} }$.

Because any lex category is the syntactic category of a finite limit theory, Gabriel-Ulmer duality says that any locally finitely presentable $\mathcal{B}$ category is the category of models of a finite limit theory $ \mathbb{T}$ such that  $ \mathbb{T}[\mathcal{S}] \simeq \Lex[\mathcal{C}_\mathbb{T}, \mathcal{S}]$. Then it can be shown that any finitely presented model $K$ in $\mathcal{B}$ is uniquely determined by a presentation formula $ \{ \overline{x}, \phi \} $ with $\phi$ a formula in the language of $\mathbb{T}$, exhibiting $K$ as $\langle \overline{x} \rangle_\mathbb{T}/\phi(\overline{x})$. Similarly, any map of finite presentation $ f : K \rightarrow K'$ such that $ K = \langle \overline{x} \rangle_\mathbb{T}/\phi(\overline{x}) $ and $K' =\langle \overline{y} \rangle_\mathbb{T}/\psi(\overline{y})$ can be presented by a formula $ \theta_f(\overline{y}, \overline{x})$ encoding the expression of the image by $f$ of the generators of $K$ in terms of the generators of $K'$: that is, if $\overline{x} = x_1, ..., x_n$ one has $f(x_i) = \tau_i [ \overline{y} ]$ for $ i =1,...,n$, then $\theta_f$ can be chosen as the formula

\[ \theta_f(\overline{y},\overline{x}) \Leftrightarrow  \bigwedge_{i =1,...,n}x_i = \tau_i[\overline{y}]  \]

This correspondence is part of an equivalence of category

\[ \mathcal{C}_\mathbb{T} \simeq \mathcal{B}^{\op}_{\fp} \]

In the following we shall denote $ K_\phi=\langle \overline{x}\rangle / \phi$ the finitely presented model presented by the formula $ \phi$ and $ f_\theta$ the finitely presented arrow presented by $\theta$.

Because $ K_\phi$ corresponds to the representable functor $ \hirayo_{ \{ \overline{x}, \phi \}}^* $ in $\mathcal{B} \simeq \Lex[\mathcal{C}_\mathbb, \mathcal{S}]$, Yoneda lemma tells us that any arrow $  K_\phi \rightarrow B $ in $\mathcal{B}$ is the name of some witness in $\mathcal{B}$ of the formula $ \phi(\overline{x})$:

\[\mathcal{B}[K_\phi, B] \simeq B( \{ \overline{x}, \phi \}) = \{ \overline{a} \in B \mid \; B \models \phi(\overline{a}) \}\]

In the following, if $ \overline{a}$ is a witness in $B$ of some $ \phi(\overline{x})$, then we denote as $ \ulcorner a \urcorner :  K_\phi \rightarrow B$ the corresponding arrow in $\mathcal{B}$.

As well as we considered set-valued models of a finite limit theory $\mathbb{T}$, we can also consider models in arbitrary Grothendieck toposes: those are the Lex functors

\[ \mathbb{T}[\mathcal{E}] \simeq \Lex[\mathcal{C}_\mathbb{T}, \mathcal{E} ]\]

with natural transformations between them. In the case of finite limit theories, a $\mathbb{T}$-model in a sheaf topos $\mathcal{E}\simeq \Sh(\mathcal{C},J)$ is the same as a sheaf of $\mathbb{T}$-models over $ (\mathcal{C},J)$. Indeed, if $ F : \mathcal{C}_\mathbb{T} \rightarrow \mathcal{E}$ is a $\mathbb{T}$-model, then not only for each $ \{ \overline{x}, \phi \}$ we have a sheaf $F(\{ \overline{x}, \phi \})$ over $ (\mathcal{C},J)$, sending an object $c$ to a set $ F(\{ \overline{x}, \phi \})(c)$, but for any $c$ in $\mathcal{C}$, we have functor
\[ \mathcal{C}_\mathbb{T} \stackrel{ F(-)(c)}{\longrightarrow} \mathcal{S} \]

sending any $ \{ \overline{x}, \phi \}$ to $ F(\{ \overline{x}, \phi \})(c)$, and this functor is lex: indeed evaluation at an object preserves (finite) limits, because limits are computed pointwisely in categories of functors. Hence $ F(-)(c)$ is a set-valued model of $\mathbb{T}$.

Now we turn to the classifying topos for $\mathbb{T}$. For a category $\mathcal{C}$, we denote as $ \widehat{\mathcal{C}}$ the presheaf topos $ [\mathcal{C}^{\op}, \mathcal{S}]$. For any Grothendieck topos $\mathcal{E}$, we have an equivalence of category

\[ \Geom[\mathcal{E}, \widehat{\mathcal{C}} ] \simeq \Lex[ \mathcal{C}, \mathcal{E}] \]

sending any lex functor $ F : \mathcal{C} \rightarrow \mathcal{E}$ to its Yoneda extension, that is, the left Kan extension along the Yoneda embedding

\[\begin{tikzcd}
{\mathcal{C}} && {\mathcal{E}} \\
\arrow["{F}", from=1-1, to=1-3]
\arrow["{\lan_\hirayo F}"{name=0, swap}, from=2-1, to=1-3]
\arrow["{\hirayo}"', from=1-1, to=2-1, hook]
\arrow[Rightarrow, "{\simeq}" description, from=1-1, to=0, shorten <=4pt, shorten >=4pt, phantom, no head]
\end{tikzcd}\]

where the canonical transformation $ F \Rightarrow \lan_\hirayo F \hirayo$ is invertible for $\hirayo$ being full and faithful. Moreover the functor $ \lan_\hirayo F$ is left adjoint to the functor $ F_* :\mathcal{E} \rightarrow \widehat{\mathcal{C}}$. Moreover $ \lan_\hirayo F$ is still lex, hence it defines a geometric morphism $ \lan_\hirayo F \dashv F_*$. In the following we will denote $ F^*=\lan_\hirayo F $. The functoriality of this process is ensured by the universal property of the left Kan extension. Conversely any geometric morphism $ F^*\dashv F_*$ induces a lex functor $ F^*\hirayo : \mathcal{C} \rightarrow \mathcal{E}$.

Hence in particular for any finite limit theory $ \mathbb{T}$, we have an equivalence of category $ \mathbb{T}[\mathcal{E}] \simeq \Geom[\mathcal{E}, \widehat{\mathcal{C}_\mathbb{T}}]$. For this reason, we call the presheaf topos $ \widehat{\mathcal{C}_\mathbb{T}}$ the classifying topos of $\mathbb{T}$ and denote it $ \mathcal{S}[\mathbb{T}]$. In particular the set-valued models of $\mathbb{T}$ are the points of the classifying topos $ \mathcal{S}[\mathbb{T}]$, that is

\[ \mathbb{T}[\mathcal{S}] \simeq \Ind(\mathcal{C}_\mathbb{T}^{\op}) \simeq  \Lex[\mathcal{C}_\mathbb{T}, \mathcal{S}] \simeq \Geom[\mathcal{S}, \widehat{\mathcal{C}_\mathbb{T}}]  \]

If one starts with a locally finitely presentable category $\mathcal{B}$ this gives the following equivalence of categories
\[        \mathcal{B} \simeq \Ind(\mathcal{B}_{fp})\simeq \Lex[\mathcal{B}_{fp}^{op}, \mathcal{S}et]\simeq \Geom[\mathcal{S}et, \widehat{\mathcal{B}_{fp}^{op} }]
\]

We also have the following fact for finite limit theories: if a Grothendieck topos $ \mathcal{E}$ is equivalent to the sheaf topos $ \Sh(\mathcal{C},J)$, then $\mathbb{T}$-models in $\mathcal{E}$ are exactly sheaves of set-valued $\mathbb{T}$-models over $(\mathcal{C},J)$, that are functors $F :\mathcal{C}^{\op} \rightarrow \mathbb{T}[\mathcal{S}]$ such that for any covering family $ (s_i : c_i \rightarrow c)_{i \in I}$ in $J$ the restrictions maps of $F$ at the $s_i$ exhibits $F(c)$ as a limit in $\mathbb{T}[\mathcal{S}]$

\[ F(c) \simeq \lim \bigg( \prod_{i \in I} F(c_i) \rightrightarrows \prod_{i,j \in I} F(c_i \times_c c_j) \bigg) \]

This comes from the fact that evaluation at an object preserves limits, hence in particular finite limits.

To sum up, the following notions are equivalent:

* a finite limit theory

* a lex category

* a topos of presheaves over a lex category

* a locally finitely presentable category

In the following we will fix once for all a finite limit theory $ \mathbb{T}$, and denote as $ \mathcal{B}$ the associated category of set-valued models, and $ \mathcal{S}[\mathbb{T}]$ its classifying topos.

\[\begin{array}{cccc}
\mathbb{T} & \mathcal{C}_\mathbb{T} & \widehat{\mathcal{C}_\mathbb{T}} & \mathbb{T}[\mathcal{S}]\\
\mathbb{T}_\mathcal{C} & \mathcal{C} & \widehat{\mathcal{C}} & \Lex[\mathcal{C},\mathcal{S}] \\
\mathbb{T}_\mathcal{E} & \mathcal{E}_{\fp} & \mathcal{E} & \Geom[\mathcal{E}, \mathcal{S}] \\
\mathbb{T}_\mathcal{B} & \mathcal{B}^{\op}_{\fp} & \widehat{\mathcal{B}^{\op}_{\fp}} & \mathcal{B}
\end{array}\]

Observe that, from what was said, a finite limit theory can be recovered (up to Morita equivalence) from its categories of models in $\mathcal{S}$: this is a situation where we have “enough set-valued models", in the sense that the points of the classifying topos are jointly conservative. In some sense, finite limit theories are determined by their semantics in sets.

While the notion of free object is very well known in the context of algebraic categories, locally finitely presentable categories also have a notion of free object. For a finite limit theory $\mathbb{T}$, a finite limit extension $ \mathbb{T}'$ of $\mathbb{T}$ is a finite limit theory in the same language $\mathcal{L}$ and whose axioms contain the axioms of $\mathbb{T}$. Then models of $\mathbb{T}'$ are in particular models of $\mathbb{T}$ and it can be checked that the inclusion functor $ \iota_{\mathbb{T}, \mathbb{T}'}: \mathbb{T}'[\mathcal{S}] \hookrightarrow \mathbb{T}[\mathcal{S}]$ is a morphism of locally finitely presentable category: hence it admits a left adjoints $ \iota_{\mathbb{T}, \mathbb{T}'}$, assigning to each $\mathbb{T}$-model $B$ a $\mathbb{T}'$-model $  \iota_{\mathbb{T}, \mathbb{T}'}^*(B)$ which we can see as the free $\mathbb{T}'$-model under $B$. However, for a arbitrary extension of $\mathbb{T}$ that is not a finite limit theory, it is no longer true that such a free construction exists in $\mathcal{S}$. As we shall see, spectra will provide a way to fix this.

§.§ Geometric extensions of finite limit theories and failure of the free construction

In this section, we introduce what will play the role of the local data as announced in the introduction. The previous section was devoted to “well behaved" theories. In this section we give generalities about the more general class of theories whose models will be used as local objects in the context of spectra.

Recall that a site is a category equiped with a Grothendieck topology; it is qualified of lex site if the underlying category is lex, and standard site if it is lex and the topology on it is subcanonical.

For a Grothendieck topology $ J$ on a category $\mathcal{C}$ and a Grothendieck topos $\mathcal{E}$, a functor $F : \mathcal{C} \rightarrow \mathcal{E}$ is said to be  $J$-continuous if for any $J$-covering family $ (u_i : c_i \rightarrow c)_{i \in I}$, the induced arrow

\[\begin{tikzcd}
{\underset{i \in I}{\coprod}F(c_i)} && {F(c)}
\arrow["{\langle F(u_i) \rangle_{i \in I}}", from=1-1, to=1-3]
\end{tikzcd}\]
is an epimorphism.

As well as the Yoneda extension of a lex functor produced the inverse image part of a geometric morphism in the context of presheaf toposes, lex continuous functors also enjoy an extension property, as stated in Diaconescu theorem: for a lex site $(\mathcal{C},J)$ and a Grothendieck topos $\mathcal{E}$, we have an equivalence of category

\[ \Lex_J[\mathcal{C}, \mathcal{E}] \simeq \Geom[\mathcal{E}, \Sh(\mathcal{C},J)] \]

where $\Lex_J[\mathcal{C}, \mathcal{E}]$ denotes the category of lex $J$-continuous functors. Indeed it can be shown that a geometric morphism $F : \mathcal{E} \rightarrow \widehat{\mathcal{C}}$ factorizes through $ \Sh(\mathcal{C},J)$ if and only if the restriction $ F^* \hirayo : \mathcal{C} \rightarrow \mathcal{E}  $ is $J$-continuous. Hence, even when the induced functor $ a_J \hirayo : \mathcal{C} \rightarrow \Sh(\mathcal{C},J)$ is not full and faithful, any $J$-continuous lex functor coincides with the restriction of its extension along $ a_J \hirayo$. When $ J$ is subcanonical, $a_J$ is full and faithful and this becomes true for any functor. In particular, we have points of $\Sh(\mathcal{C},J)$, which are the geometric functors $ \mathcal{S} \rightarrow \Sh(\mathcal{C},J)$, are obtained as

\[  \Lex_J[\mathcal{C},\mathcal{S}]  \simeq   \Geom[\mathcal{S}, \Sh(\mathcal{C},J)] \]

A Geometric theory is a theory whose sequents involve formulas built from atomic formulas using finite $ \wedge$, arbitrary $ \exists $ and small $\bigvee$. Beware that the existentials are no longer supposed to have a proof of uniqueness, and the disjunction may be infinite as long as they are indexed by a set.

For a geometric theory $\mathbb{T}$, the finite limit part of $\mathbb{T}$ is the finite limit $\mathbb{T}_0$ theory made of all the sequents of $\mathbb{T}$ involving formulas containing only conjunctions and existentials with proof of uniqueness. Conversely, for $\mathbb{T}$ a finite limit theory, a geometric extension of $\mathbb{T}$ is a geometric theory $\mathbb{T}'$ whose finite limit part is $\mathbb{T}$.

Suppose we have a geometric extension $ \mathbb{T} \rightarrow \mathbb{T}'$. Then $ \mathbb{T}'$ can be shown to be built from $\mathbb{T}$ by adding sequents of the form

\[ \phi(\overline{x}) \vdash \bigvee_{i \in I} \exists \overline{y}_i  \big{(} \psi(\overline{y}_i) \wedge \theta_i(\overline{y}_i, x) \big{)} \]

with each $\theta_i(\overline{y}_i, x)$ $\mathbb{T}$-provably functional from $\psi(\overline{y}_i) $ to $\phi(\overline{x})$.

Then on the cartesian site $ \mathcal{C}_\mathbb{T}$, one can define a topology $ J_{\mathbb{T}'}$ whose covering families consist of the corresponding families

\[ \big{(} % https://q.uiver.app/?q=WzAsMixbMCwwLCJcXHsgXFxvdmVybGluZXt5fV9pLCBcXHBzaV9pIFxcfSJdLFsyLDAsIlxceyBcXG92ZXJsaW5le3h9LCBcXHBoaSBcXH0iXSxbMCwxLCJbIFxcdGhldGFfaShcXG92ZXJsaW5le3l9X2ksIFxcb3ZlcmxpbmV7eH0pXV9cXG1hdGhiYntUfSJdXQ==
\begin{tikzcd}
{\{ \overline{y}_i, \psi_i \}} && {\{ \overline{x}, \phi \}}
\arrow["{[ \theta_i(\overline{y}_i, \overline{x})]_\mathbb{T}}", from=1-1, to=1-3]
\end{tikzcd} \big{)}_{i \in I}  \]

Then a model of $\mathbb{T}'$ in a Grothendieck topos $\mathcal{E}$ is a lex functor $ F: \mathcal{C}_\mathbb{T} \rightarrow \mathcal{S}$ which is moreover $\mathcal{J}_{\mathbb{T}'}$-continuous. In particular, this is a lex functor: this means that we have a full inclusion of categories

\[ \mathbb{T}'[\mathcal{S}] \hookrightarrow \mathbb{T}[\mathcal{S}] \]

It will be relevant in the following to characterize models of $\mathbb{T}'$ amongst models of $\mathbb{T}$. The condition of being a model of the geometric extension can be reexpressed by an injectivity condition. Observe that, if $\mathcal{B}$ is the locally finitely presentable category $ \mathbb{T}[\mathcal{S}]$, those families correspond in $\mathcal{B}_{\fp}^{\op}$ to families of cone $ ( f_{\theta_i} : K_\phi \rightarrow K_{\psi_i})_{i \in I}$ under the corresponding finitely presented objects. By abuse of notation, we also call $J$ the data of all the families $ (f_{\theta_i})_{i \in I}$ in $\mathcal{B}$.

For a family $J$ of cones $(f_i : K \rightarrow K_i)_{i \in I}$ in $\mathcal{B}$, a $J$-local object (or also $J$-injective) is an object $A$ such that for any cone $ (f_i : K \rightarrow K_i)_{i \in I}$ in $J$ and any arrow $ f : K \rightarrow A$, $f $ admits a factorization through some of the $f_i$
\[ \begin{tikzcd}
& K \arrow[]{ld}[swap]{k_i} \arrow[]{rd}{k_j} \arrow[]{rr}{f} & & A \\ K_i & ... & K_j \arrow[dashed]{ru}[swap]{\exists}
\end{tikzcd} \]

Then for a geometric extension $\mathbb{T}'$ of a finite limit theory $ \mathbb{T}$, $\mathbb{T}'$-models are exactly the $ J_{\mathbb{T}'}$-local objects, that is we have

\[ \mathcal{L}oc_{J_{\mathbb{T}'}} \simeq \mathbb{T}'[\mathcal{S}]  \]

Indeed, in $ \mathbb{T}[\mathcal{S}] = \Lex[ \mathcal{C}_\mathbb{T}, \mathcal{S}]$, we have $ \mathbb{T}[\mathcal{S}](K_\phi, F) \simeq \Lex[\hirayo^*_{ \{ \overline{x}, \phi \} }, \mathcal{F} ] $ which is $ F(\{ \overline{x}, \phi \} ) $ by Yoneda lemma, hence requiring that $J_{\mathbb{T}'}$-local object exactly amounts to say that for any covering family as above we have a surjection

\[\begin{tikzcd}[sep=large]
{\underset{i \in I}{\coprod} F(\{ \overline{y}_i, \psi_i \} )} && {F(\{ \overline{x}, \phi \} )}
\arrow["{\langle F([\theta_i(\overline{y}_i, \overline{x})]_\mathbb{T}) \rangle_{i \in I}}", from=1-1, to=1-3]
\end{tikzcd}\]

Syntactically, this should be understood as follows: for any covering $ (f_{\theta_i} : K_{\phi} \rightarrow K_{\psi_i})_{i \in I}  $, an arrow $ g :  K_{\phi} \rightarrow B $ just defines some $ \overline{b} \in B$ such that $ B \models \phi(\overline{b})$, so as it is a model of $ \mathbb{T}_\mathcal{V}$, there is some $ i \in I$ and a $ \overline{b}_i \in B$ such that $ B \models \psi_i(\overline{b}_i)$ and $ B \models \theta_{i}(\overline{b}_i,\overline{b}))$. Hence we have the factorization through the name of this witness of $ \psi_i$:

\[ \begin{tikzcd}
K_{\phi} \arrow[]{d}[swap]{f_{\theta_i}} \arrow[]{r}{g = \ulcorner \overline{b} \urcorner} & B \\
K_{\psi_i} \arrow[dashed]{ru}[swap]{\exists \ulcorner \overline{b}_i \urcorner } &
\end{tikzcd} \]

In the other direction, from the equivalence between arrows from finitely presented objects and names of witnesses of their presentation formula in the codomain, it is clear that local objects are models of the geometric extension $ \mathbb{T}_J$.

In particular, we have that set-valued models of $\mathbb{T}'$ are the points of the sheaf topos $\Sh(\mathcal{C}_\mathbb{T}, J_{\mathbb{T}'})$.

Conversely, any Grothendieck topology $J$ on $ \mathbb{C}_{\mathbb{T}} $ corresponds to a geometric extension of $ \mathbb{T}$ whose axioms mirror covering of finitely presented objects:

\[  \mathbb{T}_{J} = \mathbb{T}_{\mathcal{B}} \cup \bigg{\{} \phi(\overline{x}) \vdash \bigvee_{i \in I} \exists \overline{y}_i( \psi_i(\overline{y}_i) \wedge \theta_{f_i}(\overline{y}_i,\overline{x}))  \bigg{\}}_{(f_i)_{i \in I} \in J(K_\phi ) \atop \{  \overline{x}, \phi \} \in \mathcal{C}_\mathbb{T} } \]

exhibiting covers as disjunctions of cases for witnesses of domain formulas.

Again we can generalize the notion of model of a geometric extension $ \mathbb{T}'$ for arbitrary Grothendieck toposes: for a Grothendieck topos $\mathcal{E}$, a  $\mathbb{T}'$ model in $\mathcal{E}$ is a lex $J_{\mathbb{T}'}$-continuous functor $ \mathcal{C}_{\mathbb{T}} \rightarrow \mathcal{E}$, while a morphism of $\mathbb{T}'$-models in $\mathcal{E}$ is a natural transformation. Beware that, contrarily to models of finite limit theories, models of arbitrary geometric theories in a topos of sheaves $\Sh(\mathcal{C},J)$ not be sheaves of set-valued models over $(\mathcal{C},J)$. In fact, they are in particular sheaves of set-valued models of their finite limit part, but the additional geometric axioms cannot be enforced objectwisely. However, this can be tested at points: for a point of a topos $ p : \mathcal{S} \rightarrow \mathcal{E}$ and $ E$ in $\mathbb{T}'[\mathcal{E}]$, the composite $ p^*E^* : \mathcal{C}_\mathbb{T} \rightarrow \mathcal{S}$ is $J_\mathbb{T'}$-continuous, hence the stalk $ Fp$ is in $\mathbb{T}'[\mathcal{S}]$.

We have

\[ \mathbb{T}'[\mathcal{E}] = \Lex_{J_{\mathbb{T}'}}[\mathcal{C}_\mathbb{T}, \mathcal{E}] \simeq \Geom[\mathcal{E}, \Sh(\mathcal{C}_\mathbb{T},J_{\mathbb{T}'})] \]

where the last equivalence follows from Diaconescu. Moreover, this equivalence is natural in the sense that for any geometric morphism $f : \mathcal{E} \rightarrow \mathcal{F}$, post-composition with $f$ induces a functor

\[ \mathbb{T}'[\mathcal{F}] \stackrel{f^*}{\longrightarrow} \mathbb{T}'[\mathcal{E}] \]

and the same is true for 2-cell of $\GTop$. This exhibits $ \Sh(\mathcal{C}_\mathbb{T},J_{\mathbb{T}'})$ as a representing object for the indexed category

\[ \GTop^{\op} \stackrel{\mathbb{T}'[-] }{\longrightarrow} \Cat \]

For this reason the sheaf topos $\Sh(\mathcal{C}_\mathbb{T},J_{\mathbb{T}'})$ is called the classifying topos of $\mathbb{T}'$ and denoted $ \mathcal{S}[\mathbb{T}']$.

In particular we have a subtopos $ \mathcal{S}[\mathbb{T}'] \hookrightarrow \mathcal{S}[\mathbb{T}]$ from the inclusion

\[ \Sh(\mathcal{C}_\mathbb{T},J_{\mathbb{T}'}) \hookrightarrow \widehat{\mathcal{C}_\mathbb{T}} \]

and for any topos $\mathcal{E}$, the 2-functor $ \Geom[\mathcal{E}, -]$ sends this inclusion of topos into the full inclusion of the categories of models $ \mathbb{T}'[\mathcal{E}] \hookrightarrow \mathbb{T}[\mathcal{E}]$.

To conclude those prerequisites, observe that the latter inclusion of category has no longer a left adjoint, contrarily to what we had for extension of finite limit theories. This means that a geometric extension does not enjoy a free construction. Finally, observe also that the inclusion above was full, and in general, it is not possible to axiomatize a non full subcategory of a category of model of a finite limit theory by a geometric extension. However, our situations of interest will precisely involve both a geometric extension and a choice of maps from a factorization system related by some condition. The next section is devoted to the factorization aspects.

§.§ Left generated factorization system

In this part we recall basic facts about factorization systems. We also give some elements of Anel presentation of the small object argument as presented in <cit.>, which is a process to construct a factorization system. In this case, we need for our purpose a left generated factorization system, where the left maps can be constructed as filtered colimits of left maps of finite presentation.

If $\mathcal{C}$ is a category endowed with a factorization system $ (\mathcal{L}, \mathcal{R})$, then the left and right classes enjoy the following properties :

* $\mathcal{L} $ is closed under composition

* $\mathcal{L}$ contains all isomorphisms

* $\mathcal{L}$ is right-cancellative: for any triangle

\[\begin{tikzcd}
{C_1} & {C_2} \\
& {C_3}
\arrow["{l_1}", from=1-1, to=1-2]
\arrow["{f}", from=1-2, to=2-2]
\arrow["{l_2}"', from=1-1, to=2-2]
\end{tikzcd}\]

with $l_1, \; l_2 $ in $\mathcal{L}$, then $f$ also is in $\mathcal{L}$

* $\mathcal{L}$ is closed under colimits in $ \overrightarrow{\mathcal{C}}$

* $\mathcal{R} $ is closed under composition

* $\mathcal{R}$ contains all isomorphisms

* $\mathcal{R} $ is left-cancellative: for any triangle

\[\begin{tikzcd}
{C_1} & {C_2} \\
& {C_3}
\arrow["{f}", from=1-1, to=1-2]
\arrow["{r_1}", from=1-2, to=2-2]
\arrow["{r_2}"', from=1-1, to=2-2]
\end{tikzcd}\]

with $r_1, \; r_2 $ in $\mathcal{R}$, then $f$ also is in $\mathcal{R}$

* $\mathcal{R} $ is closed under limits in $ \overrightarrow{\mathcal{C}}$

Let $\mathcal{B}$ be a locally finitely presentable category, with $ \mathbb{T}$ the underlying finite limit theory. If we are given a factorization system $(\mathcal{L},\mathcal{R})$ in $ \mathcal{B}$ then define the class of finitely presented left maps as $ \mathcal{L}_{fp} = \mathcal{L} \cap \overrightarrow{\mathcal{B}_{fp}} $. This class inherits some closure properties of $ \mathcal{L}$: it contains identities, is still right-cancellative, and is stable by pushout along any morphism of finite presentation. Those properties can be axiomatized into the following notion:

A saturated class in $\mathcal{B}$ is a class $ \mathcal{V}$ of maps of finite presentation such that:

* $ \mathcal{V}$ is closed under composition

* $ \mathcal{V}$ contains all isomorphisms

* $ \mathcal{V}$ is right-cancellative

* $ \mathcal{V}$ is closed under pushouts along arbitrary morphisms in $\overrightarrow{\mathcal{B}_ {\fp}}$

* $ \mathcal{V}$ is closed under finite colimits in $\overrightarrow{\mathcal{B}}$.

For any class $ \mathcal{V}_0$ of finitely presented morphisms, we can always consider the closure of $ \mathcal{V}_0$ under the axioms above. Therefore the class $ \mathcal{V}$ can be chosen already saturated and it will be so by the following.

For any object $B$ in $\mathcal{B}$, define the etale generator at $B$ as full subcategory $ \mathcal{V}_B$ of $ B \downarrow \mathcal{B}$ consisting of morphisms $ n : B \rightarrow C$ such that there exists some $ l : K \rightarrow K'$ in $\mathcal{V}$ and $ a : K \rightarrow B$ exhibiting $ n$ as the pushout

\[\begin{tikzcd}
{K} & {K'} \\
{B} & {C}
\arrow["{a}"', from=1-1, to=2-1]
\arrow["{l}", from=1-1, to=1-2]
\arrow[from=1-2, to=2-2]
\arrow["{n}"', from=2-1, to=2-2]
\arrow["\lrcorner"{very near start, rotate=180}, from=2-2, to=1-1, phantom]
\end{tikzcd}\]

Before going further, we should make mention of the following useful technical lemma from <cit.>[sub-lemma 12], which is important for our <ref> and is also involved in the proof of <ref> and <ref>:

Let be a diagram as below

\[\begin{tikzcd}
{K_0} & {K_0'} & K \\
B & D
\arrow["k", from=1-1, to=1-2]
\arrow["n"', from=2-1, to=2-2]
\arrow["a", from=1-3, to=2-2]
\arrow["{a_0}"', from=1-1, to=2-1]
\arrow[from=1-2, to=2-2]
\arrow["\lrcorner"{anchor=center, pos=0.125, rotate=180}, draw=none, from=2-2, to=1-1]
\end{tikzcd}\]

with $ K,\,K_0, \, K_0' $ $\alpha$-presented and $ k $ in $\mathcal{V}$: then there exist a factorization

\[\begin{tikzcd}
{K_0} & {K_1} \\
& B
\arrow["{a_2}", from=1-1, to=1-2]
\arrow["{a_1}", from=1-2, to=2-2]
\arrow["{a_0}"', from=1-1, to=2-2]
\end{tikzcd}\]

such that $a$ factorizes through the following pushout

\[\begin{tikzcd}
{K_0} & {K_0'} \\
{K_1} & {{a_2}_*K_0'} & K \\
B & D
\arrow["k", from=1-1, to=1-2]
\arrow["{a_2}"', from=1-1, to=2-1]
\arrow["{a_1}"', from=2-1, to=3-1]
\arrow[from=1-2, to=2-2]
\arrow["{{a_2}_*k}", from=2-1, to=2-2]
\arrow["n"', from=3-1, to=3-2]
\arrow[from=2-2, to=3-2]
\arrow["\lrcorner"{anchor=center, pos=0.125, rotate=180}, draw=none, from=2-2, to=1-1]
\arrow["\lrcorner"{anchor=center, pos=0.125, rotate=180}, draw=none, from=3-2, to=2-1]
\arrow["a", from=2-3, to=3-2]
\arrow[dashed, from=2-3, to=2-2]
\end{tikzcd}\]

Now we turn to the properties of the etale generator at a given object.

The etale generator $ \mathcal{V}_B$ is closed under finite colimits in the cocomma $B\downarrow\mathcal{B}$. Hence the category $ \Ind(\mathcal{V}_B)$ is locally finitely presented.

For a proof of this statement, this for instance <cit.>[Proposition 3.11]

Let be $(n_i, a_i)_{i \in I}$ with

\[\begin{tikzcd}
{K_i} & {K_i'} \\
\arrow["{a_i}"', from=1-1, to=2-1]
\arrow["{l_i}", from=1-1, to=1-2]
\end{tikzcd}\]
a finite diagram in $\mathcal{V}_B$. We can use the fact that $ \mathcal{V}$ is closed under finite colimit to compute the finite colimit of $F$ as

\[ % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAGkB9LEAX1PSZc+QigBM5KrUYs2XLAHI+AkBmx4CRAIyktU+s1aIQAHRNMwsAE5wYOYFgAEZrGEcBJXsADGEBlgBbXjNSR3llQXURIgBmXX0ZI1NzSxgbOwdnE1cPL19-IJCwhW4I1SENURJSMQTDNgBBPikYKABzeCJQADMrCACkMhAcCCQAFmoDWWM6UuoGOgAjGAYABQro4yssNoALHDLe-qQJYdHECel64zMFsDaGGEdZpzMrOnvHzkyXN09DvoDRBDEYnSaJNguCA4F6OEDzJYrdZRTTGVzYWAA47A6igxA6K7TEAAa1K-B6gLBZyQcUJSTMFmstnsr2yfzyfkCvCyoVJPHJICOQIJeNpU3p2WhLyUvAovCAA
\begin{tikzcd}
K_i \arrow[dd, "a_i"'] \arrow[rd, "q_i " description] \arrow[rr, "l_i"] &                                                                                & K_i' \arrow[rd, "q_i'"] &                                    \\
& {\underset{i \in I}{\colim}\, K_i} \arrow[ld, "\langle a_i \rangle_{i \in I}"] \arrow[rr, "{\underset{i \in I}{\colim} \, l_i}"'] &                             & {\underset{i \in I}{\colim}\, K'_i} \\
B                                                                           &                                                                                &                             &
\end{tikzcd} \]

where $(q_i, q_i') : n_i \rightarrow \colim_{i \in I} n_i$ is the inclusion in the colimit computed in the category of arrows and $ \colim_{i \in I}l_i$ is still in $\mathcal{V}$. Then by commutation of pushouts with colimits we have

\[ \langle a_i {\rangle_{i \in I}}_* \underset{i \in I}{\colim} \, l_i =  \underset{i \in I}{\colim} \, {a_i}_*l_i \]

and this map is still in $\mathcal{V}_B$.

Before recalling how we can construct a factorization system from the data above, we give here the following result, which relates the canonical cone of the codomain of an arbitrary etale arrow and the etale generators through a pseudocolimit construction. Recall first an arbitrary arrow $l : B \rightarrow C$ in $ \Ind(\mathcal{V}_B)$ defines a left exact functors $ \hirayo_l : \mathcal{V}_B^{op} \rightarrow \mathcal{S} $. Then the category of elements $ \int \! \hirayo_l$ is cofiltered.
Moreover, recall that, as a standard fact, we have an equivalence of categories

\[ \int \hirayo_l \simeq \mathcal{V}_B \downarrow l \]

for the category of elements has pairs $ (n,a)$ with $ n$ in $\mathcal{V}_B$ and $ a \in \hirayo_l(n) = \Ind(\mathcal{V}_B)[n,l]$ is an object of $ \mathcal{V}_B\downarrow l$, while an arrow $(n_1,a_1) \rightarrow (n_2, a_2)$ consists in some $ m : n_1 \rightarrow n_2$ in $\mathcal{V}_B$ such that $ a_2m=a_1$. Moreover, observe that we have a pseudofunctor

\[ \mathcal{V}_{\cod(-)} : \int \hirayo_l^{\op} \rightarrow \mathcal{C}at  \]

which sends any $(n,a) \in \int \! \hirayo_l $ to $ \mathcal{V}_{\cod(n)} $ and any $ m : (n_1,a_1) \rightarrow (n_2,a_2)$ to the pushout functor

\[ \mathcal{V}_{\cod(n_1)} \stackrel{m_*}{\longrightarrow} \mathcal{V}_{\cod(n_2)} \]

associating to a map $ n : \cod(n_1) \rightarrow D$ in the etale generator $\mathcal{V}_{\cod(n_1)}$ the pushout map $ m_*n : \cod(n_2) \rightarrow m_*D$. The latter is indeed in $\mathcal{V}_{\cod(n_2)}$, as if $ n$ was induced from a map $ k : K \rightarrow K'$ in $\mathcal{V}$ along some $ a : K \rightarrow \cod(n_1)$, then in the following diagram

\[\begin{tikzcd}
K & {\cod(n_1)} & {\cod(n_2)} \\
{K'} & D & {m_*D}
\arrow["k"', from=1-1, to=2-1]
\arrow["n"{description}, from=1-2, to=2-2]
\arrow["a", from=1-1, to=1-2]
\arrow[from=2-1, to=2-2]
\arrow["m", from=1-2, to=1-3]
\arrow["{m_*n}"{description}, from=1-3, to=2-3]
\arrow[from=2-2, to=2-3]
\arrow["\lrcorner"{anchor=center, pos=0.125, rotate=180}, draw=none, from=2-2, to=1-1]
\arrow["\lrcorner"{anchor=center, pos=0.125, rotate=180}, draw=none, from=2-3, to=1-2]
\end{tikzcd}\]

the outer square is also a pushout.

First, recall that the Grothendieck construction of a pseudofunctor is its oplax colimit, and that the pseudolimit is constructed by localizing the oplax colimit at the opcartesian morphisms. For generalities about pseudocolimits of categories, see <cit.>[Proposition 6.5] and also <cit.>.

For any $ l : B \rightarrow C $ in $ \Ind(\mathcal{V}_B)$, we have a pseudocolimit
\[ \mathcal{V}_C \simeq \underset{(n,a) \in \int \! \hirayo_l}{\underrightarrow{\colim}} \mathcal{V}_{\cod(n)} \]

$ \Ind(\mathcal{V}_B)$ is a locally finitely presentable category where $ l \simeq \colim \; \mathcal{V}_B \downarrow l$; this colimit being filtered, it is preserved by the codomain functor, so that $ C \simeq \colim_{(n,a) \in \int \! \hirayo_l} \cod(n)$.
From this we are going to produce a pseudococone

\[ (\mathcal{V}_{\cod(n)} \stackrel{a_*}{\longrightarrow} \mathcal{V}_C)_{(n,a) \in \int \! \hirayo_l} \]

where the $a_* $ are the pushout functors; the pseudocommutativity of this diagram can be seen as follows: for a morphism $ m : (n_1,a_1) \rightarrow (n_2,a_2)$, that is such that $ n_2=mn_1$ and $ a_2m=a_1$, and for an object $ n : \cod(n_1) \rightarrow D$ in $\mathcal{V}_{\cod(n_1)}$ we have by composition of pushouts the following diagram

\[\begin{tikzcd}
& {\cod(n_1)} & D \\
B & {} & {\cod(n_2)} & {m_*D} \\
& C & {{a_1}_*D}
\arrow["{n_1}", from=2-1, to=1-2]
\arrow["{n_2}"{description, pos=0.3}, from=2-1, to=2-3]
\arrow["m"{description}, from=1-2, to=2-3]
\arrow["l"', from=2-1, to=3-2]
\arrow["{a_2}"{description}, from=2-3, to=3-2]
\arrow["n", from=1-2, to=1-3]
\arrow[from=1-3, to=2-4]
\arrow[from=2-4, to=3-3]
\arrow["{{a_1}_*n}", from=3-3, to=3-2]
\arrow[no head, from=1-3, to=2-3]
\arrow[from=2-3, to=3-3]
\arrow["{m_*n}"{description}, from=2-3, to=2-4]
\arrow["\lrcorner"{anchor=center, pos=0.125, rotate=180}, shift left=2, draw=none, from=2-4, to=1-3]
\arrow[""{name=0, anchor=center, inner sep=0}, "{a_1}"{description}, no head, from=1-2, to=2-2]
\arrow[from=2-2, to=3-2]
\arrow["\lrcorner"{anchor=center, pos=0.125, rotate=180}, draw=none, from=3-3, to=0]
\end{tikzcd}\]

where the canonical isomorphism $ a_1*n \simeq {a_2}_*m_*n$ provides the value at $n$ of the natural isomorphism ensuring the pseudocommutativity of the triangle

\[\begin{tikzcd}[row sep=small]
{\mathcal{V}_{\cod(n_1)}} \\
& {\mathcal{V}_C} \\
\arrow[""{name=0, anchor=center, inner sep=0}, "{m_*}"', from=1-1, to=3-1]
\arrow["{{a_1}_*}", from=1-1, to=2-2]
\arrow["{{a_2}_*}"', from=3-1, to=2-2]
\arrow["\simeq"{description}, Rightarrow, draw=none, from=0, to=2-2]
\end{tikzcd}\]

Hence the universal property of the pseudocolimit returns a functor

\[\begin{tikzcd}[column sep=huge]
{\underset{(n,a) \in \int \! \hirayo_l}{\colim} \mathcal{V}_{\cod(n)}} & {\mathcal{V}_C}
\arrow["{\langle a_*\rangle_{(n,a) \in \int \! \hirayo_l}}", from=1-1, to=1-2]
\end{tikzcd}\]

where the pseudocolimit $ \colim_{(n,a) \in \int \! \hirayo_l}\mathcal{V}_{\cod(n)}$ is equivalent to the localization of the oplax colimit $ \oplaxcolim_{(n,a) \in \int \! \hirayo_l}\mathcal{V}_{\cod(n)}$ at the class of opcartesian morphisms which are the morphisms $ (m,1_{}) : ((n_1,a_1),m_1) \rightarrow ((n_2,a_2),m_2)$ coding for pushout squares exhibiting $ m_2 $ as $ m_2=m_*m_1$ as below

\[\begin{tikzcd}
& C \\
{\cod(n_1)} && {\cod(n_2)} \\
{D_1} && {D_2}
\arrow["{m_1}"', from=2-1, to=3-1]
\arrow[""{name=0, anchor=center, inner sep=0}, "m"{description}, from=2-1, to=2-3]
\arrow["{m_2}", from=2-3, to=3-3]
\arrow["{a_1}", from=2-1, to=1-2]
\arrow["{a_2}"', from=2-3, to=1-2]
\arrow["{{m_1}_*m}"', from=3-1, to=3-3]
\arrow["\lrcorner"{anchor=center, pos=0.125, rotate=180}, draw=none, from=3-3, to=0]
\end{tikzcd}\]

We are going to prove this functor to be both essentially surjective and full and faithful. Let be an object $ m$ in $\mathcal{V}_C$ is induced as a pushout from some finitely presented etale map $ m_0$ in $\mathcal{V}$

\[\begin{tikzcd}
K & C \\
{K'} & D
\arrow["m", from=1-2, to=2-2]
\arrow["b", from=1-1, to=1-2]
\arrow["{m_0}"', from=1-1, to=2-1]
\arrow[from=2-1, to=2-2]
\arrow["\lrcorner"{anchor=center, pos=0.125, rotate=180}, draw=none, from=2-2, to=1-1]
\end{tikzcd}\]

As $K$ is finitely presentable, it factorizes through some $ a$ for $ (n,a) \in \int \! \hirayo_l$ as $ b = ac$, and then we can exhibit $ m  $ as arising as the pushout of $ c_*m_0$ along $a$ by right cancellation of pushouts

\[\begin{tikzcd}
& K & {K'} \\
B & {} & {\cod(n)} & {c_*K'} \\
& C & D
\arrow["n"{description, pos=0.3}, from=2-1, to=2-3]
\arrow["c"{description}, from=1-2, to=2-3]
\arrow["l"', from=2-1, to=3-2]
\arrow["a"{description}, from=2-3, to=3-2]
\arrow["{m_0}", from=1-2, to=1-3]
\arrow[from=1-3, to=2-4]
\arrow[from=2-4, to=3-3]
\arrow["m"', from=3-2, to=3-3]
\arrow[no head, from=1-3, to=2-3]
\arrow[from=2-3, to=3-3]
\arrow["\lrcorner"{anchor=center, pos=0.125, rotate=180}, shift left=2, draw=none, from=2-4, to=1-3]
\arrow["{c_*m_0}"{description}, from=2-3, to=2-4]
\arrow[""{name=0, anchor=center, inner sep=0}, "b"{description}, shorten >=1pt, no head, from=1-2, to=2-2]
\arrow[from=2-2, to=3-2]
\arrow["\lrcorner"{anchor=center, pos=0.125, rotate=180}, draw=none, from=3-3, to=0]
\end{tikzcd}\]

That is, $m = b_*m_0 = a_*c_*m_0$; moreover, any such two lifts of $b$ are identified in the canonical cone of $C$, hence so are the induced pushouts of $m_0$. Hence the functor $ \langle a_*\rangle_{(n,a) \in \int \! \hirayo_l} $ is essentially surjective on objects.

Now we want for any $ ((n_1,a_1),m_1), \; ((n_2,a_2),m_2)$ we want an isomorphism

\[  {\underset{(n,a) \in \int \! \hirayo_l}{\colim} \mathcal{V}_{\cod(n)}} [((n_1,a_1),m_1), \, ((n_2,a_2),m_2)] \simeq   \mathcal{V}_C[{a_1}_*m_1, {a_2}_*m_2] \]

In one direction, to any $ (m, s) :((n_1,a_1),m_1) \rightarrow ((n_2,a_2),m_2) $ we can associate the unique map $ t_{(m,s)}$ induced as below from the universal property of the pushout

\[\begin{tikzcd}[row sep=small, column sep=small]
&& C \\
{\cod(n_1)} &&&& {\cod(n_2)} \\
& {{a_1}_*D_1} && {{a_2}_*D_2} \\
{D_1} &&&& {D_2}
\arrow["{m_1}"', from=2-1, to=4-1]
\arrow["{m_2}", from=2-5, to=4-5]
\arrow["{a_1}", from=2-1, to=1-3]
\arrow["{a_2}"', from=2-5, to=1-3]
\arrow["s"', from=4-1, to=4-5]
\arrow[from=1-3, to=3-2]
\arrow[from=1-3, to=3-4]
\arrow["{t_{(m,s)}}"', dashed, from=3-2, to=3-4]
\arrow[from=4-1, to=3-2]
\arrow[from=4-5, to=3-4]
\arrow["\lrcorner"{anchor=center, pos=0.125, rotate=180}, draw=none, from=3-2, to=2-1]
\arrow["\lrcorner"{anchor=center, pos=0.125, rotate=90}, draw=none, from=3-4, to=2-5]
\arrow["m"{description}, crossing over, from=2-1, to=2-5]
\end{tikzcd}\]

Observe by the way that this map is also obtained as the pushout $ t_{(m,s)} = ((m_*m_2)_*a_2)_*\langle s, m_2\rangle$ as below

\[\begin{tikzcd}
& {} && C \\
& {\cod(n_2)} & {{a_1}_*D_1} & {} & {{a_2}_*D_2} \\
{D_1} && {D_2}
\arrow["{a_2}"', from=2-2, to=1-4]
\arrow[from=1-4, to=2-3]
\arrow[from=1-4, to=2-5]
\arrow[from=3-1, to=2-3]
\arrow[from=3-3, to=2-5]
\arrow["{m_*m_1}"', from=2-2, to=3-1]
\arrow["{\langle s, m_2\rangle}"{description}, from=3-1, to=3-3]
\arrow["{t_{(m,s)}}"{description}, from=2-3, to=2-5]
\arrow["\lrcorner"{anchor=center, pos=0.125, rotate=180}, shift left=3, draw=none, from=2-5, to=1-4]
\arrow["{m_2}", from=2-2, to=3-3, crossing over]
\end{tikzcd}\]

where right cancellation of pushouts makes the bottom square a pushout and $ \langle s, m_2 \rangle $ is the map uniquely induced by $ s$.

Suppose that we have $ (m,s), (m',s')$ such that $ t_{(m,s)} = t_{(m',s')}$. Then for $ \int \! \hirayo_l$ is filtered, there is some $ (n_3, a_3)$ in $ \int \! \hirayo_l$ such that $ m''m = m''m'$. Then we have that $ {m_2}_*m'' s m_1 = {m''}_*m_s m'' m = {m''}_*m_s m'' m' =  {m_2}_*m'' s' m_1  $ as seen in the following diagram

\[\begin{tikzcd}
&& C \\
{\cod(n_1)} &&&& {\cod(n_2)} & {\cod(n_3)} \\
& {{a_1}_*D_1} && {{a_2}_*D_2} \\
{D_1} &&&& {D_2} & {m''_*D_2}
\arrow["{m_1}"', from=2-1, to=4-1]
\arrow["{a_1}", from=2-1, to=1-3]
\arrow["{a_2}"{description}, from=2-5, to=1-3]
\arrow["s", shift left=1, from=4-1, to=4-5]
\arrow[from=1-3, to=3-2]
\arrow[from=1-3, to=3-4]
\arrow["{t_{(m,s)} = t_{(m',s')}}"', from=3-2, to=3-4]
\arrow[from=4-1, to=3-2]
\arrow[from=4-5, to=3-4]
\arrow["\lrcorner"{anchor=center, pos=0.125, rotate=180}, draw=none, from=3-2, to=2-1]
\arrow["\lrcorner"{anchor=center, pos=0.125, rotate=90}, draw=none, from=3-4, to=2-5]
\arrow["{m''}", from=2-5, to=2-6]
\arrow["{s'}"', shift right=1, from=4-1, to=4-5]
\arrow["{a_3}"', curve={height=10pt}, from=2-6, to=1-3]
\arrow["{m''_*m_2}", from=2-6, to=4-6]
\arrow["\lrcorner"{anchor=center, pos=0.125, rotate=180}, draw=none, from=4-6, to=2-5]
\arrow[curve={height=6pt}, from=4-6, to=3-4]
\arrow["{m_2}"{description}, from=2-5, to=4-5, crossing over]
\arrow["{{m_2}_*m''}"', from=4-5, to=4-6]
\arrow["m", shift left=1, from=2-1, to=2-5, crossing over]
\arrow["{m'}"', shift right=1, from=2-1, to=2-5, crossing over]
\end{tikzcd}\]

and therefore $(m,s)$ and $ (m',s')$ are equalized in $ \oplaxcolim_{(n,a) \in \int \!\hirayo_l}\mathcal{V}_{\cod(n)}$ by the opcartesian morphism $ (m'', {m_2}_*m'')$. But the later becomes invertible in the pseudocolimit, where the morphisms $ (m,s)$ and $(m',s')$ are hence identified. Hence the faithfulness.

Now in a a situation as below

\[\begin{tikzcd}
& {\cod(n_1)} & C & {\cod(n_2)} \\
{D_1} & {{a_1}_*D_1} && {{a_2}_*D_2} & {D_2}
\arrow[""{name=0, anchor=center, inner sep=0}, "{m_1}"', from=1-2, to=2-1]
\arrow[""{name=1, anchor=center, inner sep=0}, "{m_2}"{description}, from=1-4, to=2-5]
\arrow["{a_1}", from=1-2, to=1-3]
\arrow["{a_2}"', from=1-4, to=1-3]
\arrow[from=1-3, to=2-2]
\arrow[from=1-3, to=2-4]
\arrow["t"{description}, from=2-2, to=2-4]
\arrow[from=2-1, to=2-2]
\arrow[from=2-5, to=2-4]
\arrow["\lrcorner"{anchor=center, pos=0.125, rotate=180}, draw=none, from=2-2, to=0]
\arrow["\lrcorner"{anchor=center, pos=0.125, rotate=90}, draw=none, from=2-4, to=1]
\end{tikzcd}\]

if $ m_1$ is induced through pushouts from map $ k : K \rightarrow K'$ in $\mathcal{V}$ as below, we get a composite arrow

\[\begin{tikzcd}
& K & {\cod(n_1)} & C & {\cod(n_2)} \\
{K'} & {D_1} & {{a_1}_*D_1} & {{a_2}_*D_2} & {D_2}
\arrow["{m_2}"{description}, from=1-5, to=2-5]
\arrow["{a_2}"', from=1-5, to=1-4]
\arrow[from=1-4, to=2-4]
\arrow[from=2-5, to=2-4]
\arrow["\lrcorner"{anchor=center, pos=0.125, rotate=90}, draw=none, from=2-4, to=1-5]
\arrow["t"', from=2-3, to=2-4]
\arrow[from=1-4, to=2-3]
\arrow[from=2-1, to=2-2]
\arrow[from=2-2, to=2-3]
\arrow["{a_1}", from=1-3, to=1-4]
\arrow[""{name=0, anchor=center, inner sep=0}, "k"', from=1-2, to=2-1]
\arrow["b", from=1-2, to=1-3]
\arrow[""{name=1, anchor=center, inner sep=0}, "{m_1}"{description}, from=1-3, to=2-2]
\arrow["\lrcorner"{anchor=center, pos=0.125, rotate=180}, draw=none, from=2-3, to=1]
\arrow["\lrcorner"{anchor=center, pos=0.125, rotate=180}, draw=none, from=2-2, to=0]
\end{tikzcd}\]

Now by Anel lemma there exists $ (a_3,n_3) $ in $ \int \! \hirayo_l$ and $ l_3 : (n_2, a_2) \rightarrow (n_3,a_3)$ such that we have a factorization

\[\begin{tikzcd}
& K & {\cod(n_1)} & C & {\cod(n_3)} & {\cod(n_2)} \\
{K'} & {D_1} & {{a_1}_*D_1} & {{a_2}_*D_2} & {{l_3}_*D_2} & {D_2}
\arrow["{m_2}", from=1-6, to=2-6]
\arrow[from=1-4, to=2-4]
\arrow["t"', from=2-3, to=2-4]
\arrow[from=1-4, to=2-3]
\arrow[from=2-1, to=2-2]
\arrow[from=2-2, to=2-3]
\arrow["{a_1}", from=1-3, to=1-4]
\arrow[""{name=0, anchor=center, inner sep=0}, "{m_1}"{description}, from=1-3, to=2-2]
\arrow[""{name=1, anchor=center, inner sep=0}, "k"', from=1-2, to=2-1]
\arrow["b", from=1-2, to=1-3]
\arrow["{{l_3}_*m_2}"{description}, from=1-5, to=2-5]
\arrow[from=1-6, to=1-5, "l_3"']
\arrow[from=1-5, to=1-4, "a_3"']
\arrow[from=2-5, to=2-4]
\arrow[from=2-6, to=2-5]
\arrow["c"{description}, curve={height=20pt}, dashed, from=2-1, to=2-5]
\arrow["\lrcorner"{anchor=center, pos=0.125, rotate=90}, draw=none, from=2-4, to=1-5]
\arrow["\lrcorner"{anchor=center, pos=0.125, rotate=90}, draw=none, from=2-5, to=1-6]
\arrow["\lrcorner"{anchor=center, pos=0.125, rotate=180}, draw=none, from=2-3, to=0]
\arrow["\lrcorner"{anchor=center, pos=0.125, rotate=180}, draw=none, from=2-2, to=1]
\end{tikzcd}\]

Then again by filteredness of $ \int \! \hirayo_l$, there exists $(n_3,a_3)$ and a span

\[\begin{tikzcd}
& C \\
{\cod(n_1)} & {\cod(n_4)} & {\cod(n_3)}
\arrow["{a_1}", from=2-1, to=1-2]
\arrow["{a_4}"{description}, from=2-2, to=1-2]
\arrow["{a_3}"', from=2-3, to=1-2]
\arrow["{l_1}"{description}, from=2-1, to=2-2]
\arrow["{l_2}"{description}, from=2-3, to=2-2]
\end{tikzcd}\]

along which we can consider pushouts

\[\begin{tikzcd}
& {\cod(n_1)} && {\cod(n_2)} \\
{D_1} && {\cod(n_4)} && {{l_3}_*D_2} \\
& {{l_1}_*D_1} & C & {{l_3}_*D_3} \\
& {{a_1}_*D_1} && {{a_2}_*D_2}
\arrow["{l_1}", from=1-2, to=2-3]
\arrow["{l_2}"', from=1-4, to=2-3]
\arrow["{a_4}"{description}, from=2-3, to=3-3]
\arrow["{m_1}"', from=1-2, to=2-1]
\arrow[from=2-3, to=3-2]
\arrow["{{a_1}_*m_1}"{description}, from=3-3, to=4-2]
\arrow["{{a_2}_*m_2}"{description}, from=3-3, to=4-4]
\arrow[from=2-3, to=3-4]
\arrow["{{l_3}_*m_2}", from=1-4, to=2-5]
\arrow[from=2-1, to=3-2]
\arrow[from=3-2, to=4-2]
\arrow[from=3-4, to=4-4]
\arrow[from=2-5, to=3-4]
\arrow["t"', from=4-2, to=4-4]
\arrow["\lrcorner"{anchor=center, pos=0.125, rotate=135}, draw=none, from=3-2, to=1-2]
\arrow["\lrcorner"{anchor=center, pos=0.125, rotate=135}, draw=none, from=3-4, to=1-4]
\arrow["\lrcorner"{anchor=center, pos=0.125, rotate=90}, draw=none, from=4-2, to=2-3]
\arrow["\lrcorner"{anchor=center, pos=0.125, rotate=180}, draw=none, from=4-4, to=2-3]
\arrow[curve={height=-12pt}, from=2-5, to=4-4]
\arrow[curve={height=12pt}, from=2-1, to=4-2]
\arrow["{a_1}"{description}, curve={height=12pt}, from=1-2, to=3-3, crossing over]
\arrow["{a_2}"{description}, curve={height=-12pt}, from=1-4, to=3-3, crossing over]
\end{tikzcd}\]

But now for $ {l_1}_*D_1 = {l_1b}_*K'$, we have a universal map $ \langle c, \, {l_2l_3}_*m_2 \rangle : {l_1}_*D_1 \rightarrow {l_3}_*D_3 $, and moreover in the diagram below

\[\begin{tikzcd}
& {\cod(n_4)} \\
& C \\
{{l_1}_*D_1} && {{l_2}_*D_2} \\
{{a_1}_*D_1} && {{a_2}_*D_2}
\arrow["{a_4}"{description}, from=1-2, to=2-2]
\arrow["{{l_1}_*m_1}"', from=1-2, to=3-1]
\arrow["{{l_2l_3}_*m_2}", from=1-2, to=3-3]
\arrow[from=3-1, to=4-1]
\arrow[from=3-3, to=4-3]
\arrow["t"', from=4-1, to=4-3]
\arrow["\lrcorner"{anchor=center, pos=0.125, rotate=90}, draw=none, from=4-1, to=1-2]
\arrow["\lrcorner"{anchor=center, pos=0.125, rotate=180}, draw=none, from=4-3, to=1-2]
\arrow["{\langle c, {l_2l_3}_*m_2 \rangle}"{description}, from=3-1, to=3-3]
\arrow["{{a_1}_*m_1}"{description, pos=0.3}, from=2-2, to=4-1, crossing over]
\arrow["{{a_2}_*m_2}"{description, pos=0.3}, from=2-2, to=4-3, crossing over]
\end{tikzcd}\]

the bottom square is a forced to be a pushout by right cancellation. This exhibits $ t$ as the induced maps $ t_{(1_{n_4}, \langle c, \, {l_2l_3}_*m_2 \rangle)}$, and to conclude, observe that $((n_1,a_1), m_1)$ and $ ((n_4,a_4), {l_1}_*m_1)$ are related by an opcartesian morphism, while on the other hand $ ((n_2,a_2), m_2)$ and $ ((n_4,a_4), {l_2l_3}_*m_2)$ are also related by an opcartesian morphism, and are hence identified in the pseudocolimit. Hence the pair $(1_{n_4}, \langle c, \, {l_2l_3}_*m_2 \rangle)$ can be seen as an antecedent of $ t$ in the homset above. Hence the fullness.

Hence the functor $ \colim_{ (n,a) \in \int \!\hirayo_l} \mathcal{V}_{\cod(n)} \rightarrow \mathcal{V}_C$, being at the same time essentially surjective, full and faithful, defines an equivalence of category as desired.

Observe that the condition of right cancellation of the maps in $\mathcal{V}$ seems of no use in the result above, which could apply actually to any class of maps that is closed under pushout along arbitrary map. This theorem will be useful to get an expression of the spectral site of an abritrary etale arrow in <ref>.

Now we recall the construction of the factorization system as done in <cit.> and also <cit.>. We saw that the etale generator at any object is closed under finite colimits. Then for any $ f : B \rightarrow C$, the category $ \mathcal{V}_B \downarrow f$ of diagonally universal morphisms under $B$ above $f$ is filtered. Moreover, recall that the codomain functor $ B \downarrow \mathcal{B}$ preserves filtered colimits. Now we can construct the factorization of any arrow $f$ in $\mathcal{B}$:

An arbitrary arrow $ f : B \rightarrow C$ in $ \mathcal{B}$ admits a factorization

\[ % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAGMQBfU9TXfIRQAmclVqMWbAMLQAFADMAlN14gM2PASIBGUjvH1mrRBwgMsAWwAEAHVulrsqHOsARAPrs7ttFgB6Cl4qXOIwUADm8ESgCgBOEJZIZCA4EEh6EsZsCqqxCUmIKWlIollSpuzmVj6Ont72foFeINQMdABGMAwACvxaQiBxWBEAFjh5IPGJpdQliJlGFUMeuW2d3X2agmzDYxOhXEA
\begin{tikzcd}
B \arrow[rr, "f"] \arrow[rd, "{\colim \, \mathcal{D}_B\downarrow f }"'] &                                                 & C \\
& {\underset{\mathcal{D}_B\downarrow f}{\colim} \,C} \arrow[ru, "r_f"'] &
\end{tikzcd} \]

with $ l_f$ an axiomatizable diagonally universal morphism and $ r_f$ is in $\mathcal{V}^\perp$.

See <cit.>[section 2.3] which contains the full proof of the statement [theorem 14], and also <cit.> - which however did not provided proof of it.

A factorization system is left generated if it is of the form $ (\Ind({\mathcal{V}}), \mathcal{V}^\perp)$. In particular, the factorization of an arrow is obtained as a filtered colimit as in <ref>.

For a proof the the next statement, see <cit.>[Proposition 3.13] for instance:

The factorization hence obtained is left generated, that is, $ \Ind(\mathcal{V}) = ^\perp(\mathcal{V}^{\perp})$.

Saturated classes and factorization systems on a locally finitely presentable category $ \mathcal{B}$ form posets $ Sat_\mathcal{B}$ and $ {F}act_\mathcal{B}$ with inclusion as wide subcategory of $ \mathcal{B}_{\fp}$
for $ Sat_\mathcal{B}$ and inclusion of the left class for $ {F}act_\mathcal{B}$. Saturated class and factorization systems in locally finitely presentable categories are related through an adjunction:

\[\begin{tikzcd}[row sep=tiny]
{Sat_\mathcal{B}} && {Fact_\mathcal{B}} \\
{\mathcal{V}} && {{(\Ind({\mathcal{V}}), \mathcal{V}^\perp)}} \\
{\mathcal{L} \cap \overrightarrow{\mathcal{B}_{\fp}}} && {{(\mathcal{L},\mathcal{R})}}
\arrow[""{name=0, inner sep=0}, from=1-1, to=1-3, curve={height=-12pt}, hook']
\arrow[""{name=1, inner sep=0}, from=1-3, to=1-1, curve={height=-12pt}]
\arrow[from=2-1, to=2-3, shorten <=10pt, shorten >=10pt, maps to]
\arrow[from=3-3, to=3-1, shorten <=11pt, shorten >=11pt, maps to]
\arrow["\dashv"{rotate=-90}, from=0, to=1, phantom]
\end{tikzcd}\]

where $ Sat_\mathcal{B} \hookrightarrow Fact_\mathcal{B}$ is a mono because for any saturated class $ \mathcal{V} = Ind( {\mathcal{V}})$. On the converse, it is not true that a factorization system is generated from a saturated class, so that in general we only have
\[ ( Ind({{\mathcal{L} \cap \overrightarrow{\mathcal{B}_{\fp}}
}), (\mathcal{L} \cap \overrightarrow{\mathcal{B}_{\fp}}
})^\perp) \leq (\mathcal{L}, \mathcal{R}) \]

so that in particular $ \mathcal{R} \subseteq (\mathcal{L} \cap \overrightarrow{\mathcal{B}_{\fp}}
)^\perp$. However in the following we are going to restrict our attention to the factorizations systems having this property ; the general case will however be met again when dealing with Diers contexts.

Left-generated factorizations systems in locally finitely presentable category are in some sense factorization systems that can be axiomatized by a finite limit theory. Indeed, from the axioms of a saturated class, the category $ \mathcal{V}$ is closed under finite colimits. But as a consequence its inductive completion $\Ind(\mathcal{V})$ is a locally finitely presentable category. In fact the inclusion

\[ \mathcal{V} \hookrightarrow \overrightarrow{\mathcal{B}_{\fp}} \]

preserves finite colimits; but for $ \overrightarrow{\mathcal{B}} \simeq \Ind(\overrightarrow{\mathcal{B}_{\fp}})$
we have locally finitely presentable functor

\[ \mathcal{B} \rightarrow \Ind(\mathcal{V}) \]

sending each arrow $ f$ to its left part $ l_f$. Hence the category of left maps in a left generated system is a locally finitely presentable category; the corresponding finite limit theory has the category $ \mathcal{V}^{\op}$ as syntactic site, and $ \widehat{\mathcal{V}^{\op}}$ as classifying topos.

In section 3 we shall see that the class of local maps $\Loc$ is also axiomatizable and admits a classifying topos. But for now, we turn to the notion of admissibility in the case of set-valued models.

§ ADMISSIBILITY STRUCTURE

The central condition that enables the construction of the spectrum is a relation entangling factorization and geometric extension. It can be presented in several equivalent ways. A first explicit axiomatization was

An admissibility structure in the sense of <cit.>[definition 6.57] is the data of

* a finite limit theory $ \mathbb{T}$

* a geometric extension $ \mathbb{T} \rightarrow \mathbb{T}'$

* a 2-functor $ \Loc: \GTop^{\op} \rightarrow \Cat$ such that for each $ \mathcal{E}$ one has $ \Loc[\mathcal{E}] \hookrightarrow \overrightarrow{\mathbb{T}[E]}$ and $ \Loc[\mathcal{E}]$ contains isomorphisms and enjoys left-cancellability.

* and such that we have the condition that for any $ \mathcal{E}$ in $\GTop$ and any arrow $ f: F \rightarrow E $ in $\mathbb{T}[\mathcal{E}]$ with $ E$ in $\mathbb{T}'[\mathcal{E}]$ there exists a factorization in $\mathbb{T}[\mathcal{E}]$

\[\begin{tikzcd}
{F} && {E} \\
& {H_f}
\arrow["{f}", from=1-1, to=1-3]
\arrow["{n_f}"', from=1-1, to=2-2]
\arrow["{u_f}"', from=2-2, to=1-3]
\end{tikzcd}\]

with $ u_f$ in $\Loc[\mathcal{E}]$ and $ H_\phi$ in $\mathbb{T}'[\mathcal{E}]$, which is moreover initial amongst all factorizations of $f$ with an arrow in $\Loc[\mathcal{E}]$.

However this definition, while encapsulating in the last condition the key feature that will produce geometry, may seem complicated at first sight. In this section, we are going to prove that it is achieved by the following data, which have been variously called admissibility triples in <cit.> and geometry in <cit.> (in the context of $\infty$-categories, and with geometric conventions); we shall retain this later name for its evocative virtue:

A geometry is the data of

* a finite limit theory $\mathbb{T}$

* a saturated class $\mathcal{V}$ in $ \mathbb{T}[\mathcal{S}]$

* a Grothendieck topology $J$ on $\mathcal{C}_\mathbb{T}$ whose covers are generated by duals of maps in $\mathcal{V}$.

In a close manner the notion of Nisnevich contextwas introduced in <cit.>, although this definition does not rely exactly on the same data and is slightly less constricted (beside it is formulated also in the geometric convention, while we follow here the algebraic convention):

A Nisnevich context is the data of a factorization system $(\Et, \Loc)$ on a locally finite presentable category $\mathcal{B}$ and a class of objects $ \mathcal{L}$. Then one defines the $\mathcal{L}$-localizing families as the coarsest system of covers $ (f_i : B \rightarrow B_i)_{i \in I}$ exhibiting $\mathcal{L}$ as local objects.

A Nisnevich context is said to be compatible if $\mathcal{L}$ is exactly the class of $J_\mathcal{L}$-local objects.

Finally, a Nisnevich context is said to be good if it is compatible and the factorization system is left generated.

Beware that there is in general no condition for the localizing topology to be generated from etale maps: this generalization was used for instance to capture geometric situation as the Nisnevich topologies that are not generated from factorization data.

In some sense, Nisnevich contexts arise when the local objects are given before any axiomatization through a geometric extension, and it may happen that the least geometric extension sufficient to axiomatize them has more models than the local objects we started with. In section 2.2 we will meet similar situation, but from the point of view of the fourth way to axiomatize those situations, that are Diers context. However, we do not give their definition right now for we shall devote section 2.2 to their relation with those approaches from which they differ significantly.

§.§ Admissibility condition from a geometry

In the following we deploy the information encoded into the notion of geometry and how it is sufficient to achieve admissibility; in this subsection we focus on the set-valued models and explain how the topological interpretation of left and right map, and of the points of the topology.

Throughout this section, we fix a geometry $ (\mathbb{T}, \mathcal{V}, J)$. Following <cit.> terminology, we shall call maps in $\Ind(\mathcal{V})$ etale maps, and denote their class as $ \Et$, while the maps in $\mathcal{V}^\perp$ will be called local maps and their class denoted as $\Loc$. They form a factorization system  as $(\Et, \Loc)$. Models of the geometric extension $\mathbb{T}_J$ of $\mathbb{T}$ corresponding to $J$ will be often qualified of local objects.

We begin this section by proving that we have the factorization property of admissibility in $\mathbb{T}[\mathcal{S}]$.

Recall that a model of $\mathbb{T}_J$ is an object in $\mathbb{T}[\mathcal{S}]$ which is local relatively to the dual in $\mathbb{T}[\mathcal{S}]_{\fp}$ of covering families in $J$. But the families dual to cover in $J$ can be extended to the whole category $ \mathbb{T} [\mathcal{S}]$ as follows:

Define the generalized $J$-covers as consisting, for each $ B$ in $\mathbb{T}[\mathcal{S}]$, of the families $({n_i} : B {\rightarrow} B_i)_{i \in I}$ such that there exists some $ a : K \rightarrow B$ and some family $(l_i : K \rightarrow K_i)_{i \in I}$ dual to a $J$-cover such that for each $i \in I$ one has a pushout
\[ \begin{tikzcd}
B \arrow[]{d}[swap]{f_i} & \arrow[]{l}{} K  \arrow[]{d}{k_i \in \mathcal{V}} \\ B_i \arrow[phantom, very near start]{ru}{\urcorner} & \arrow[]{l}{} K_i
\end{tikzcd} \]

For we supposed $J$ to be a Grothendieck topology, it will be closed under pullback in $\mathcal{C}_\mathbb{T}$. Hence in $\mathbb{T}[\mathcal{S}]_{\fp}$, pushouts of duals of $J$-covers are still $J$-covers, hence the generalized coverage does not create new covering families under finitely presented objects.

Then the property of injectiveness of local objects relatively to $J$-covers extends automatically to those extended covers:

An object $ A$ in $\mathbb{T}[\mathcal{S}]$ is $J$-local if and only if for any object $ B$ in $\mathbb{T}[\mathcal{S}]$, any arrow $ f : B \rightarrow A$ and any generalized $J$-cover $ (n_i : B \rightarrow B_i)_{i \in I}$, there is a factorization of $f$ for some $i \in I$

\[ \begin{tikzcd}
& B \arrow[]{ld}[swap]{n_i} \arrow[]{rd}{n_j} \arrow[]{rr}{f} & & A \\ B_i & ... & B_j \arrow[dashed]{ru}[swap]{\exists}
\end{tikzcd} \]

In fact $A$ is $J$-local if and only if for any generalized cover under it $ (n_i : A \rightarrow B_i)_{i \in I}$, one has a retraction

\[ \begin{tikzcd}
& A \arrow[]{ld}[swap]{n_i} \arrow[]{rd}{n_j} \arrow[equal]{rr}{} & & A \\ B_i & ... & B_j \arrow[dashed]{ru}[swap]{\exists}
\end{tikzcd} \]

In one sense it is obvious that $A$ is injective relatively to the generalized covers, then in particular it is injective.
The converse is a consequence of the property of the pushout: if $ A$ is $J$-local, then for any generalized family induced from a $J$ cover $ (n_i)_{i \in I}$ we have a factorization through some $n_i$ as below

\[\begin{tikzcd}
{K} & {B} & {A} \\
{K_i} & {B_i}
\arrow["{a}", from=1-1, to=1-2]
\arrow["{a_*n_i}", from=1-2, to=2-2]
\arrow["{n_i}"', from=1-1, to=2-1]
\arrow[from=2-1, to=2-2]
\arrow["{f}", from=1-2, to=1-3]
\arrow["\lrcorner"{very near start, rotate=180}, from=2-2, to=1-1, phantom]
\arrow[from=2-1, to=1-3, curve={height=30pt}]
\end{tikzcd}\]

and then an arrow $ B_i \rightarrow A$ factorizing $f$ by the property of the pushout. Now, if for any generalized cover of $A$, $A$ is a retract of a member of this cover, then in particular for any $(n_i : K \rightarrow K_i)_{i \in I}$ in $J$ and any $ a : K \rightarrow A$, the pushout of the $n_i$ under $A$ give a generalized cover of $A $ and we have a factorization for some $i \in I$

\[\begin{tikzcd}
{K} & {A} & {A} \\
{K_i} & {B_i}
\arrow["{a}", from=1-1, to=1-2]
\arrow["{a_*n_i}" description, from=1-2, to=2-2]
\arrow["{n_i}"', from=1-1, to=2-1]
\arrow[from=2-1, to=2-2]
\arrow[Rightarrow, from=1-2, to=1-3, no head]
\arrow["\lrcorner"{very near start, rotate=180}, from=2-2, to=1-1, phantom]
\arrow[from=2-2, to=1-3, dashed]
\end{tikzcd}\]

so that $ A$ is $J$-local.

For an object $ B$ in $\mathbb{T}[\mathcal{S}]$, a local form of $B$ is an etale map $ n : B \rightarrow A$ toward a $J$-local object. Beware that local forms are not required to be finitely presented in general.

Etale maps will play the role of saturated compacts of the spectral topology, while finitely presented etale maps will play the role of basic compact opens from which we are going to construct the spectral topology. While this is not apparent in $\mathbb{T}[\mathcal{S}]$ which is “on the algebraic side", this is more intuitive on the opposite category $\mathbb{T}[\mathcal{S}]^{ \op}$, whose objects should now be thought as spaces, where the etale morphisms could be seen as generalized inclusions, and the generalized covers induced from $J$ as cover over objects.

Local objects are like focal spaces, that is, spaces with a least point in the specialization order. For instance, in a topological space $X$ and a point $x \in X$, the focal component of $X$ in $x$ is the intersection of all neighborhoods of $x$, and this is the upset $ \uparrow x$ in the specialization order. Local forms behave like inclusions of the form $ \uparrow_\sqsubseteq x \hookrightarrow X$ as such upsets are unreachable by open covering: indeed, in a cover of $\uparrow_\sqsubseteq x$, one open must contain $x$ itself. But as open are up-sets for the specialization order, this open is the whole $\uparrow_\sqsubseteq x$. Hence maximal points, as they do not admit non trivial local forms, are alike those $ x $ such that $ \uparrow_\sqsubseteq x = \{ x \}$.

In particular, triangles between local forms

\[\begin{tikzcd}
{B} & {A_1} \\
& {A_2}
\arrow["{x_1}", from=1-1, to=1-2]
\arrow["{n}", from=1-2, to=2-2]
\arrow["{x_2}"', from=1-1, to=2-2]
\end{tikzcd}\]

should be seen as coding for specialization order between the corresponding minimal point $ x_1 \leq x_2$. Then in $\mathbb{T}[\mathcal{S}]^{ \op}$ this will be turned into an inclusion of focal component $ \uparrow x_2 \subseteq \uparrow x_1$.

Any object $A$ admitting local morphism into a local object $ u : A \rightarrow A_0 $ is itself a local object.

If $ (n_i :A {\longrightarrow} B_i)_{i \in I}$ is a point covering family of $A$ then its pushout along $u$ is a point covering family for $ A_0$ hence admits a lifting $r$ for some $i$, so we have a square that diagonalizes because $u $ is local and $ n_i$ is in $ \mathcal{V}_A$ :

\[ \begin{tikzcd}
A \arrow[equal]{r}{} \arrow[]{d}[swap]{n_i} & A \arrow[]{d}{u } \\ B_i \arrow[]{r}[swap]{r} \arrow[dashed]{ru}{\exists} & A_0
\end{tikzcd} \]

For any arrow $f : B \rightarrow A$ in $\mathbb{T}[\mathcal{S}]$ with $ A$ a $J$-local object, the $ (\Et, \Loc)$ factorization of $f$

\[\begin{tikzcd}
{B} && {A} \\
& {A_f}
\arrow["{f}", from=1-1, to=1-3]
\arrow["{n_f}"', from=1-1, to=2-2]
\arrow["{u_f}"', from=2-2, to=1-3]
\end{tikzcd}\]

returns a $J$-local object $A_f$

Now we want to understand how this admissibility condition induced from a geometry is inherited in arbitrary toposes.

For a geometry $ (\mathbb{T}, \mathcal{V}, \mathcal{J})$, with $(\Et, \Loc)$ the associated factorization system, the class of finitely presented etale maps $ \mathcal{V}$ in $\mathbb{T}[\mathcal{S}]_{\fp}$ is dual to a class of morphisms in the syntactic category $ \mathcal{C}_\mathbb{T}$. That is, an arrow $ n : K_\phi \rightarrow K_\psi$ (with $ \phi$, $\psi$ the presentation formulas of the domain and codomains) in $\mathcal{V}$ corresponds to an arrow $ [\theta_n(\overline{x}, \overline{y})]_\mathbb{T} : \{ \overline{y}, \psi \} \rightarrow  \{ \overline{x}, \phi \}$, which, as a $\mathbb{T}$-provably functional formula, should be seen as a function symbol coding for an operation.

Moreover, from the definition of a saturated class, the category $ \mathcal{V}^{\op}$ has finite limits, hence codes for a finite limit theory which admits as classifying topos $ \widehat{\mathcal{V}^{op}}= [ \mathcal{V}, \mathcal{S}]$ which we denote as $\mathcal{S}[\Et] $. In particular this allows us to define for each Grothendieck topos $\mathcal{E}$ a class of arrows $ \Et[\mathcal{E}]$ in $\mathbb{T}[\mathcal{E}]$ as

\[   \Et[\mathcal{E}]\simeq \Geom\big{[}\mathcal{E}, \mathcal{S}[\Et] \big{]} \simeq \Lex[\mathcal{V}^{\op}, \mathcal{E}] \]

Let us look at the counterpart of this for the right class $\Loc$ of local maps.

An arrow $ u : A \rightarrow B$ is in $\Loc$ if and only if it satisfies the following condition:

\[ \textrm{if } \Bigg{ \{ }  \begin{array}{l}
\forall n : K_\phi \rightarrow K_\psi \textrm{ in } \mathcal{V} \\
\forall \overline{a} \in A(\{ \overline{x}, \phi \})  \\
\forall \overline{b}  \in B( \{ \overline{y}, \psi \}) \textrm{ such that } B \models \theta_f(\overline{u(a)},\overline{b})

\end{array}  \textrm{ then } \exists!\overline{c} \in A( \{ \overline{y}, \psi \}) \; \Bigg{\{}\begin{array}{l}
A \models \theta_f(\overline{a},\overline{c})\\
\overline{u(c)} = \overline{b}

\end{array}   \]

First, note that the situation expressed by the conditions in the characterization just amounts to the commutativity of the following square

\[  \begin{tikzcd}
K_\phi \arrow[]{d}[swap]{n} \arrow[]{r}{\ulcorner a \urcorner} & A \arrow[]{d}{u} \\ K_\psi \arrow[]{r}[swap]{\ulcorner b \urcorner} & B
\end{tikzcd} \]

as the composite $ u \circ \ulcorner a \urcorner $ corresponds to the element $\overline{u(a)}$ while the composite $ \ulcorner b \urcorner \circ n$ expresses the fact that

\[ B \models \phi_f( \tau_1[\sfrac{\overline{b}}{\overline{y}}], ... , \tau_n [\sfrac{\overline{b}}{\overline{y}}]) \]

and the condition $B \models
\theta_f(\overline{u(a)},\overline{b}) $ just means that
\[ g(a_1) = \tau_1[\sfrac{\overline{b}}{\overline{y}}] \wedge ... \wedge g(a_n) = \tau_n[\sfrac{\overline{b}}{\overline{y}}] \]

hence that the square commutes.

Consequently, if $ u$ is $ \mathcal{V}$ local, such a square always diagonalizes uniquely: the diagonal provides us with a witness $ \overline{c} $ of $ \psi_f$ in $A$, and the commutation of the down-right triangle expresses the equality $ \overline{u(c)} = \overline{b}$ while the up-left one expresses that in $A$ we have

\[ a_1 = \tau_1[\sfrac{\overline{c}}{\overline{y}}] \wedge ... \wedge a_n = \tau_n[\sfrac{\overline{c}}{\overline{y}}] \]

If conversely $ u $ satisfies the syntactic characterization, then the map $g$ defined on the generators as $ g(y_1)= c_1,...,g(y_m)=c_m $ provides a diagonalization

\[\begin{tikzcd}
{K_\phi} & {A} \\
{K_\psi} & {B}
\arrow["{\ulcorner a \urcorner}", from=1-1, to=1-2]
\arrow["{\ulcorner b \urcorner}"', from=2-1, to=2-2]
\arrow["{n}"', from=1-1, to=2-1]
\arrow["{u}", from=1-2, to=2-2]
\arrow["{g}" description, from=2-1, to=1-2, dashed]
\end{tikzcd}\]

In this context, as explained by Coste, local morphisms are those which “reflect" elements produced through the operation coded by finitely presented etale arrows. While morphisms in $ \mathcal{V}$ “produce witnesses of their codomain formula from witnesses of their domain formula", local morphisms do not add new witness of propositions presenting the codomain, and reflect the one that already exists.

Observe that the condition above defined a cartesian sequent for the existential in the syntactic characterization was unique. This suggests that local arrows are definable by a finite limit theory.

The process above allows to define a theory for local maps, proving they are axiomatized not only by a geometric, but in fact a cartesian theory.

The condition above says in particular by Yoneda lemma that

\[\begin{tikzcd}
{\mathbb{T}[\mathcal{S}]\big[ K_\psi, A \big]} & {\mathbb{T}[\mathcal{S}]\big[ K_\psi, B \big]} \\
{\mathbb{T}[\mathcal{S}]\big[ K_\phi, A \big]} & {\mathbb{T}[\mathcal{S}]\big[ K_\phi, B \big]}
\arrow[from=1-1, to=2-1]
\arrow[from=1-1, to=1-2]
\arrow[from=1-2, to=2-2]
\arrow[from=2-1, to=2-2]
\arrow["\lrcorner"{very near start, rotate=0}, from=1-1, to=2-2, phantom]
\end{tikzcd}\]

But observe that
\[ {\mathbb{T}[\mathcal{S}]\big[ K_\psi, A \big]} \simeq \Lex[\mathcal{C}_\mathbb{T}, \mathcal{S}]\big[ \hirayo^*_{\{ \overline{y}, \psi \} }, A \big] \simeq A(\{ \overline{y}, \psi \}) \]

and similarly for the other homset. This last formulation still makes sense even if we consider left exact functors with value in other toposes than $ \mathcal{S}$: and as we are going to deal with sheaves of local objects in different Grothendieck toposes, this motivates the following generalization of admissibility amongst $\mathbb{T}[\mathcal{S}]$ in arbitrary Grothendieck toposes.

For a Grothendieck topos $\mathcal{E}$, a local arrow in $\mathcal{E}$ is a natural transformation

\[\begin{tikzcd}
{\mathcal{V}^{\op}} && {\quad \mathcal{E}}
\arrow["{F}"{name=0}, from=1-1, to=1-3, curve={height=-12pt}]
\arrow["{E}"{name=1, swap}, from=1-1, to=1-3, curve={height=12pt}]
\arrow[Rightarrow, "{u}", from=0, to=1, shorten <=2pt, shorten >=2pt]
\end{tikzcd}\]
in $\Lex[\mathcal{V}^{op},\mathcal{E}]$ whose naturality square at a morphism $ [\theta_n(\overline{x}, \overline{y})]_\mathbb{T} : \{ \overline{y}, \psi \} \rightarrow  \{ \overline{x}, \phi \}$ dual of a morphism $n$ in $\mathcal{V}$ is a pullback in $\mathcal{E}$

\[\begin{tikzcd}
{F(\{ \overline{y}, \psi \})} & {E(\{ \overline{y}, \psi \})} \\
{F(\{ \overline{x}, \phi \})} & {E(\{ \overline{x}, \phi \})}
\arrow["{F([\theta_n(\overline{x}, \overline{y})]_\mathbb{T})}"', from=1-1, to=2-1]
\arrow["{u_{\{ \overline{y}, \psi \}}}", from=1-1, to=1-2]
\arrow["{E([\theta_n(\overline{x}, \overline{y})]_\mathbb{T})}", from=1-2, to=2-2]
\arrow["{u_{\{ \overline{x}, \phi \}}}"', from=2-1, to=2-2]
\arrow["\lrcorner"{very near start, rotate=0}, from=1-1, to=2-2, phantom]
\end{tikzcd}\]

In the following we denote as $ \Loc[\mathcal{E}]$ the class of local morphisms in $\mathcal{E}$.

Local morphisms are stable under inverse image: any geometric morphism $ f : \mathcal{F} \rightarrow \mathcal{E}$ induces a functor $ f^* : \Loc[\mathcal{E}] \rightarrow \Loc[\mathcal{F}]$.

The inverse image $ f^* : \mathcal{E} \rightarrow \mathcal{F}$ is lex, hence preserves pullback. Then at each morphism $ n$ in $\mathcal{V}$, $f^* $ sends the pullback above in $\mathcal{E}$ to a pullback in $\mathcal{F}$. Hence $f*u$ is local.

Now, recall form section 1 that models of finite limit theories in sheaf toposes were sheaves of set-valued models over the base site. In particular this suggests the following:

If $ \mathcal{E} \simeq \Sh (\mathcal{C}_\mathcal{E}, J_\mathcal{E})$, then a transformation $ u : F \rightarrow E$ in $ \mathcal{E}$ is a local transformation if and only if  for any object $c $ in $ \mathcal{C}$, $ u(c) : F(c) \rightarrow E(c)$ is in $\Loc$.

For each $c$ then $ u(c) : F(c) \rightarrow E(c)$ is an arrow in $ \mathcal{B}$ and thus can be seen as a natural transformation between Ind-objects. Conversely for any $f : A \rightarrow B$ of finite presentation, $ \alpha_f : F(A) \rightarrow F(B)$ is a morphism of sheaves on $(\mathcal{C}_\mathcal{E}, J_\mathcal{E})$ ; as evaluation of sheaves preserves limits, each $ u(c)$ is local as a morphism in $ \mathcal{B}$ if and only if the following square is a pullback

\[\begin{tikzcd}
{F(\{ \overline{y}, \psi \})(c)} & {E(\{ \overline{y}, \psi \})(c)} \\
{F(\{ \overline{x}, \phi \})(c)} & {E(\{ \overline{x}, \phi \})(c)}
\arrow["{F([\theta_n(\overline{x}, \overline{y})]_\mathbb{T})_c}"', from=1-1, to=2-1]
\arrow["{{u_{\{ \overline{y}, \psi \}}}_c}", from=1-1, to=1-2]
\arrow["{E([\theta_n(\overline{x}, \overline{y})]_\mathbb{T})_c}", from=1-2, to=2-2]
\arrow["{{u_{\{ \overline{x}, \phi \}}}_c}"', from=2-1, to=2-2]
\arrow["\lrcorner"{very near start, rotate=0}, from=1-1, to=2-2, phantom]
\end{tikzcd}\]

But then for evaluation creates limits, this proves we have a pullback in the category of sheaves.

Now it appears that the factorization structure $ (\Et, \Loc)$ generated from $ \mathcal{V} $ in $\mathbb{T}[\mathcal{S}]$ is inherited in the category $\mathbb{T}[\mathcal{E}]$ in any Grothendieck topos, and moreover in a functorial way:

For any Grothendieck topos $\mathcal{E}$ of the form $ \Sh (\mathcal{C}_\mathcal{E}, J_\mathcal{E})$, we have a factorization system $(\Et[\mathcal{E}], \Loc[\mathcal{E}]) $ in $\mathbb{T}[\mathcal{E}]$. Moreover, for any $ f: \mathcal{F} \rightarrow \mathcal{E}$ we have adjunctions

\[\begin{tikzcd}
{\Et[\mathcal{E}]} && {\Et[\mathcal{F}]}
\arrow["{f_*}"{name=0, swap}, from=1-1, to=1-3, curve={height=12pt}]
\arrow["{f^*}"{name=1, swap}, from=1-3, to=1-1, curve={height=12pt}]
\arrow["\dashv"{rotate=-90}, from=1, to=0, phantom]
\end{tikzcd} \quad \quad \begin{tikzcd}
{\Loc[\mathcal{E}]} && {\Loc[\mathcal{F}]}
\arrow["{f_*}"{name=0, swap}, from=1-1, to=1-3, curve={height=12pt}]
\arrow["{f^*}"{name=1, swap}, from=1-3, to=1-1, curve={height=12pt}]
\arrow["\dashv"{rotate=-90}, from=1, to=0, phantom]
\end{tikzcd}  \]

In fact the desired factorization is pointwise relatively to the definition site of $ \mathcal{E}$. Let be $ f : F \rightarrow E$ in $\mathbb{T}[\mathcal{E}]$. For any $ c $ in $ \mathcal{C}_{\mathcal{E}}$, we have a morphism $ f_c$ which admits a factorization in $\mathbb{T}[\mathcal{E}]$

\[ \begin{tikzcd}[row sep=small]
F(c) \arrow[]{rr}{f_c} \arrow[]{rd}[swap]{n_{f_c}} & & F'(c) \\
& H_{f_c} \arrow[]{ru}[swap]{u_{f_c}}
\end{tikzcd} \]

With $ n_{f_c}$ etale and $ u_{f_c}$ local in $ \mathbb{T}[\mathcal{S}]$. As this defined a functorial factorization structure, this process is itself functorial and defines a presheaf over $ \mathcal{C}_{\mathcal{E}}$. The desired object is just its sheafification

\[ H_{f} = a_{J_\mathcal{E}}(c \mapsto H_{f,c})  \]

while the data of $ (n_{f_c})_{c \in \mathcal{C}_\mathcal{E}}$, $ (u_{f_c})_{c \in \mathcal{C}_\mathcal{E}}$ defines the desired etale and local transformation in an unique way.

The functoriality of this result might look surprising at first sight. If it is expected that inverse image, as left adjoints, preserves etales maps for they are a left class, and dually, that direct image, as right adjoints, preserve local maps for they are a right class, the preservation of local maps by etale maps by inverse image and etale map by direct image are consequence of the specific fact that we are in a left generated factorization system, exhibiting both etale and local maps as models of finite limit theories, hence stable either under direct and inverse images.

Hence the factorization structure is inherited by the category of models in any Grothendieck topos. But the admissibility structure itself is inherited.

Let be $\mathcal{E}$ a Grothendieck topos and $ u : F \rightarrow E$ in $\Loc[\mathcal{E}]$ with $ E$ in $\mathbb{T}_J[\mathcal{E}]$: then $F$ itself is in $\mathbb{T}_J[\mathcal{E}]$.

The inverse image part $F$ is in particular a Lex functor from $(\mathcal{B}_{\fp})^{op} $ into $ \mathcal{E}$. Hence we have to prove that $ F$ is also $J$ continuous. Let $ (n_i : \{ \overline{x}_i, \phi_i \} \rightarrow \{ \overline{x}, \phi \})_{ i \in I} \in J$: those are finitely presented maps in the saturated class $ \mathcal{V}$, hence as $f$ is a local transformation, the naturality square in each $ k_i$ expresses each $ F^*(\{ \overline{x}_i, \phi_i \})$ as a pullback

\[\begin{tikzcd}[column sep=large]
{F(\{ \overline{x}_i, \phi_i \})} & {F(\{ \overline{x}, \phi \})} \\
{E(\{ \overline{x}_i, \phi_i \})} & {E(\{ \overline{x}, \phi \})}
\arrow["{F([\theta_i(\overline{x}, \overline{x}_i)]_\mathbb{T})}", from=1-1, to=1-2]
\arrow["{u_{\{ \overline{x}_i, \phi_i \}}}"', from=1-1, to=2-1]
\arrow["{E([\theta_i(\overline{x}, \overline{x}_i)]_\mathbb{T})}"', from=2-1, to=2-2]
\arrow["{u_{\{ \overline{x}, \phi \}}}", from=1-2, to=2-2]
\arrow["\lrcorner"{very near start, rotate=0}, from=1-1, to=2-2, phantom]
\end{tikzcd}\]

As those inverse image lands in the topos $ \mathcal{E}$, where, as in any Grothendieck topos, colimits are stable by pullback, one has

\begin{equation*}
\begin{split}
\coprod_{i \in I} \; F(\{ \overline{x}_i, \phi_i \}) &\simeq \coprod_{i \in I} \; \Big( E(\{ \overline{x}_i, \phi_i \}) \times_{E(\{ \overline{x}, \phi \})} F(\{ \overline{x}, \phi \}) \Big)\\
& \simeq \Big(\coprod_{i \in I} \; E(\{ \overline{x}_i, \phi_i \}) \Big) \times_{E(\{ \overline{x}, \phi \})} F(\{ \overline{x}, \phi \})
\end{split}
\end{equation*}

But $ E$, as a local object, transforms covers into jointly epimorphic families. Hence the lower arrow of the corresponding pullback square is an epimorphism

\[
\begin{tikzcd}[column sep=large]
\underset{i \in I}{\coprod}\; F(\{ \overline{x}_i, \phi_i \}) \arrow[]{r}{F([\theta_i(\overline{x}, \overline{x}_i)]_\mathbb{T})}  \arrow[]{d}[swap]{\langle u_{\{ \overline{x}_i, \phi_i \}} \rangle_{i \in I}} \arrow[phantom, very near start]{rd}{\lrcorner} & F(\{ \overline{x}, \phi \}) \arrow[]{d}{u_{\{ \overline{x}, \phi \}}} \\
\underset{i \in I}{\coprod}\; E(\{ \overline{x}_i, \phi_i \}) \arrow[two heads]{r}[swap]{E([\theta_i(\overline{x}, \overline{x}_i)]_\mathbb{T})} & E(\{ \overline{x}, \phi \})
\end{tikzcd} \]

But in Grothendieck topos, pullback of epi are epi, then the upper arrow in this square in also an epimorphism and this confirms that $F$ also is a local object.

This says that for any Grothendieck topos $\mathcal{E}$, the category of $\mathbb{T}$-models in $ \mathcal{E}$ inherits the admissibility structure defined by the geometry $ (\mathbb{T},\mathcal{V}, J)$

Let be $\mathcal{E}$ be a Grothendieck topos: then for any $ f: F \rightarrow E$ in $\mathbb{T}[\mathcal{S}]$ with $ E$ in $\mathbb{T}_J[\mathcal{E}]$, then in the $(\Et[\mathcal{E}], \Loc[\mathcal{E}])$-factorization

\[\begin{tikzcd}
{F} && {E} \\
& {H_f}
\arrow["{f}", from=1-1, to=1-3]
\arrow["{n_f}"', from=1-1, to=2-2]
\arrow["{u_f}"', from=2-2, to=1-3]
\end{tikzcd}\]

the middle term is in $\mathbb{T}_J[\mathcal{E}]$.

This proves that a geometry as defined produces an admissibility structure. However the converse is not clear, for the functor $ \Loc[-]$ may not be deduced to be representable from the hypothesis.

Now some precision on the geometric aspects of local objects: in fact they behave exactly as expected in the sense that they form sheaves of $ \mathcal{B}$ objects over the definition site of their base topos and have set valuated local objects as stalk:

The stalks of a local object $ F$ are set-valuated local objects. Evaluated at points, local transformations between local objects return admissible maps between set valuated local objects.

All of this justifies the terminology for local objects and local transformations as our objects of interest behave locally, at points, as such objects.

Now just an observation in the vain of 1.3.9 that will reveal useful when investigating algebraic aspects of the spectral construction:

In any topos $\mathcal{E}$, a retract of a $J$-local object still is a local object.

For lex functors $ F, F_0 : \mathcal{B}^{op}_{\fp} \rightarrow \mathcal{E}$ such that $F_0$ is moreover $ J$-local, then by naturality a situation of retraction as below induces for any cover in $J$ a roof as the following:

\[ \begin{tikzcd}[row sep=small, column sep=small]
& F_0 \arrow[]{rd}{r} & \\ F \arrow[equal]{rr}{} \arrow[]{ru}{s} & & F
\end{tikzcd} \quad
\begin{tikzcd}[row sep=small]
& \coprod F_0(K_i) \arrow[two heads]{rd}{\coprod r_{K_i}} \arrow[two heads]{rr}{\langle F_0(k_i)\rangle} & & F_0(K) \arrow[two heads]{rd}{r_K} & \\
& & \coprod F(K_i) \arrow[]{rr}{\langle F(k_i)\rangle} & & F(K) \\
\coprod F(K_i) \arrow[]{ruu}{\coprod s_{K_i}} \arrow[equal]{rru}{} \arrow[]{rr}[near end]{\langle F(k_i)\rangle} & & F(K) \arrow[crossing over]{ruu}[swap, near start]{s_K} \arrow[equal]{rru}{}& &
\end{tikzcd}\]

Where the top arrow is epic as $F_0$ is local, the $r_k$ is epic as a retraction, while $ \coprod r_{K_i}$ also is epic as epi are stable by colimits in $Set$, so that the middle horizontal arrow also is epic, exhibiting $F$ as a local object.

As for set-valued models, we also have that retract of local maps between local objects still are local, as it can be tested point-wisely.

Topos-wise stable inclusion

Retract of local are local

§.§ Stable functors and multi-reflective subcategories

In this section we focus on an alternative way to encapsulate admissibility, relating in particular the factorization condition involved in <ref> to a more general class of conditions,

A local unit for a functor $ U : \mathcal{A} \rightarrow \mathcal{B}$ is a morphism $ n : B \rightarrow U(A)$ such that for any square of the following form there exists an unique morphism $ w : A \rightarrow A_1$  such that $ U(w)$ diagonalizes uniquely the square and the left triangle already commutes in $ \mathcal{A}$

\[
\begin{tikzcd}[column sep=large]
B \arrow[]{r}{f} \arrow[]{d}[swap]{n} & U(A_1) \arrow[]{d}{U(v)} & &  & A_1 \arrow[]{d}{v}  \\ U(A) \arrow[]{r}[swap]{U(u)} \arrow[]{ru}{ U(w)} & U(A_2)  & & A \arrow[dashed]{ru}{\exists ! w} \arrow[]{r}{u} & A_2
\end{tikzcd}\]

A functor $ U : \mathcal{A} \rightarrow \mathcal{B}$ is stable when any morphism $ f : B \rightarrow U(A)$ factorizes uniquely through the range of $ U $ as

\[ % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRACEQBfU9TXfIRQAmclVqMWbAKoAKAIIBKbrxAZseAkQCMpbePrNWiEHPkB9AGbKu4mFADm8IqEsAnCAFskZEDghIuhJGbJYqrh7eiL7+SKLBUiZgViDUDHQARjAMAAr8mkIgblgOABY44SDuXnHUsYhBhommskxWymmZ2XkagmzFZRW2XEA
\begin{tikzcd}
B \arrow[rr, "f"] \arrow[rd, "n_f"'] &                              & U(A) \\
& U(A_f) \arrow[ru, "U(u_f)"'] &
\end{tikzcd} \]

where $n_f : B \rightarrow U(A_f) $ is a candidate. We refer to this factorization as the stable factorization of $f$ and to $ n_f$ as the candidate of $f$.

A right multi-adjoint is a stable functor $U : \mathcal{A} \rightarrow \mathcal{B}  $ such that for each object $B$, the set of local units $ \eta : B \rightarrow U(A)$ under $B$ is small. Then we denote as $X_B $ this set of local units.

A multireflective subcategory is a (non necessarily full) subcategory $ U : \mathcal{A} \hookrightarrow \mathcal{B} $ whose inclusion functor is a right multi-adjoint.

A typical example of non full multireflection is the following as pointed out in <cit.>:

For any unique factorization system $(\mathcal{L}, \mathcal{R})$, seeing $ \mathcal{R}$ as a category from it is stable by composition and contains isomorphisms, the inclusion $\mathcal{R} \hookrightarrow \mathcal{B}$ is a stable functor. Conversely any stable functor which is surjective on objects and faithful is of this form.

If we have a factorization system, any morphism $ f : B \rightarrow \iota_\mathcal{R}(A) $ factorizes uniquely as $ \iota_\mathcal{R}(r)l$ and $ l$ is orthogonal to any morphism in $ \mathcal{R}$ so we can see it as the desired candidate.

If $ U : \mathcal{A} \rightarrow \mathcal{B} $ is stable, faithful and surjective on objects, any object is some $ U(A)$ and any $ f : B \rightarrow U(A)$ factorizes through a candidate which is left orthogonal to the morphisms in the range of $U$, hence the class of candidates constitutes the left part and the morphisms in $ U(\overrightarrow{A})$ the right part.

The following terminology was suggested by Anel, and was also identified in <cit.>[part 4] amongst condition to produce a spectral construction:

Let be $\mathcal{R}$ a class of maps in a category and $ \mathcal{A}$ a class of objects. We say that $\mathcal{A}$ has the glidding property relatively to $\mathcal{R}$ if for any arrow $ l : B \rightarrow A $ in $\mathcal{R}$ with $ A$ a n object of $\mathcal{A}$, then $ B$ must also be in $\mathcal{A}$.

Let $\mathcal{B}$ be a category equiped with a factorization system $(\mathcal{L}, \mathcal{R})$ and $\mathcal{A}$ be a class of objects of $\mathcal{B}$ with the gliding condition relative to $\mathcal{R}$, that is, such that any object $A$ admitting an arrow in $\mathcal{R}$ toward an object of $\mathcal{A}$ is itself in $\mathcal{A}$. Then the inclusion $\mathcal{A}^\mathcal{R} \hookrightarrow \mathcal{B}$ of $ \mathcal{A}$ objects equiped with arrows of $\mathcal{R} $ between them defines a relatively full and faithful stable functor.

This follows from the previous proposition: for any $B$ in $ \mathcal{B}$ and any arrow $ f : B \rightarrow A$ with $A $ an object in $\mathcal{A}$, as there exists a unique factorization $ B \stackrel{n_f}{\rightarrow} A_f \stackrel{u_f}{\rightarrow} A$ with $n_f$ in $ \mathcal{E}$ and $ u_f $ in $\mathcal{R}$, then $A_f$ is also an object in $\mathcal{A}$; this factorization is initial amongst those through a morphism in $\mathcal{R}$ on the right. Moreover $ n_f$ is a morphism in $ \mathcal{L}$ with an objects in $\mathcal{A}$ as codomain, and such arrows are exactly the candidates for the inclusion as for any square with $ A_0, A_1,A_2$ objects in $\mathcal{A}$, we have the diagonalization

\[
\begin{tikzcd}
B \arrow[]{d}[swap]{n} \arrow[]{r}{} & A_1 \arrow[]{d}{u } \\ A_0 \arrow[dashed]{ru}{\exists d} \arrow[]{r}[swap]{u' } &A_2
\end{tikzcd}
\]

But recall that $ \mathcal{R}$ is left-cancellative as any right class, so that $ u $ must itself be in $\mathcal{R}$. Hence in the factorization above $ n_f$ is a candidate and the inclusion is stable; left-cancellativity of $\mathcal{R}$ also enforces that this inclusion is relatively full and faithful.

Let be $ (\mathbb{T}, \mathcal{V},J)$ a geometry. Then we have a stable inclusion
\[ \mathbb{T}_J[\mathcal{S}]^\Loc \hookrightarrow \mathbb{T}[\mathcal{S}] \]

More generally, for any Grothendieck topos, we have a stable inclusion
\[ \mathbb{T}_J[\mathcal{E}]^{\Loc[\mathcal{E}]} \hookrightarrow \mathbb{T}[\mathcal{E}] \]

In both case, this inclusion is relatively full and faithful.

Hence geometries define stable inclusions, and moreover this stability property is inherited to categories of models in arbitrary toposes, producing topos-wise stable inclusions.

Now let us examine the converse process. We start from a stable functor $ U : \mathcal{A} \rightarrow \mathcal{B}$ with $ \mathcal{B}$ locally finitely presentable. We are going to construct a geometric axiomatization of a stable functor by inducing a left generated factorization system and a geometric extension of the finite limit theory $\mathbb{T}$ underlying $\mathcal{B}$, or equivalently, by defining a Grothendieck topology on $\mathcal{B}_{\fp}^{\op}$. This will produce a notion of local objects and local maps in $\mathcal{B}$ encompassing the objects and maps in the essential image of $U$, and a notion of etale maps by an orthogonality process.

First we construct the factorization data associated to a stable functor. This process is already implicit in <cit.> and we recalled it in <cit.>[section 3] using <cit.>.

A  diagonally universal morphism for $U$ is a morphism $ f : B \rightarrow C$ in $ \mathcal{B}$ such that we have a lifting in any square of the form

\[ \begin{tikzcd}
B \arrow[]{r}{g} \arrow[]{d}[swap]{f} & U(A) \arrow[]{d}{U(u)} \\ C \arrow[]{r}[swap]{h} \arrow[dashed]{ru}{\exists !} & U(A') \end{tikzcd} \]

We denote as $\Diag_U$ the class of diagonally universal morphisms for $U$. We have $ \Diag_U = ^\perp\!U(\overrightarrow{\mathcal{A}}) $.

Note that diagonally universal morphisms with codomain in the range of $U$ are not necessarily candidates as the factorization needs not come from a (unique) morphism in $ \mathcal{A}$. However, if $ U$ is relatively full and faithful, that is if all its restrictions $ U/_A$ are full and faithful, then such a diagonal necessarily comes from a convenient morphism in $\mathcal{A}$. Then candidates coincide with diagonally universal morphisms with codomain in the range of $U$.

Now we have an orthogonality structure $(\Diag_U, \Diag_U^{\perp})$ in $\mathcal{B}$: however this is not yet a factorization system, for the class of diagonally universal morphisms in not necessarily accessible. To construct a factorization system, we must restrict ourselves to diagonally universal morphisms that are filtered colimits of diagonally universal morphisms of finite presentation.

Observe that in the context of a stable functor, the class $ ^\perp U(\overrightarrow{\mathcal{A}}) \cap \mathcal{B}_{\fp}$ is already a saturated class from it inherits the closure properties of $ U(\overrightarrow{\mathcal{A}})$ which is a left class in an orthogonality structure.

We define the class $ \Et_U$ of $U$-etale morphisms as

\[ \Et_U = \Ind(\Diag_U \cap \overrightarrow{\mathcal{B}_{\fp}}) \]

(they were called “axiomatizable diagonally universal morphisms" in <cit.>). In particular $ \Et_U \subseteq \Diag_U$. We define the class $\Loc $ of $U$-local morphisms as
\[ \Loc_U = (^\perp U(\mathcal{\overrightarrow{\mathcal{A}}}) \cap \overrightarrow{\mathcal{B}_{\fp}})^\perp \]

In particular $ U( \overrightarrow{\mathcal{A}}) \subseteq \Diag_U^\perp \subseteq \Loc_U$.

By section 1.3 we know that $ (\Et_U, \Loc_U)$ is a factorization system on $\mathcal{B}$.

In <cit.> were isolated conditions enabling the construction of a point-set notion of spectrum. We sum up those conditions under the following notion:

Define a Diers context as the data of

* a locally finitely presentable category $\mathcal{B}$

* a right multi-adjoint functor $ U : \mathcal{A} \rightarrow \mathcal{B} $ with $ \mathcal{B}$

* satisfying Diers condition: any local unit is the filtered colimit of diagonally universal morphism of finite presentation above it.

Diers condition implies in particular that the stable factorization of an arrow $f$ through its local unit coincides with the $(\Et_U, \Loc_U)$-factorization of $f$.

We must understand how a Diers context induces a situation of admissibility. We have already seen how it produces a left generated factorization system; now we have to construct canonically a topology associated to it.

For $ \mathcal{C}$ a class of cones

* The cone-injectivity class for $ \mathcal{C}$ is the class $ \mathcal{C}^{{\perp}}$ of objects $A$ such that for any cone $(f_i : B \rightarrow B_i)_{i \in I} $ in $ \mathcal{C}$, any $ f : B \rightarrow A$ from the submit of the cone factorizes through some arrow $f_i$ in the cone.

* The cone-orthogonality class for $\mathcal{C}$ is the class $\mathcal{C}^{{inj}}$ of objects $A$ such that for any cone $(f_i : B \rightarrow B_i)_{i \in I} $ in $ \mathcal{C}$, any $ f : B \rightarrow A$ from the submit of the cone factorizes uniquely through exactly one $f_i$ in the cone.

As detailed in <cit.>, if $ \mathcal{C}$ is a class of cones made of arrows of finite presentation, then the corresponding injectivity class is an accessibly embedded, accessible full subcategory of $\mathcal{B}$, while the corresponding orthogonality class is a full multireflexive subcategory of $ \mathcal{B}$.

Moreover, it should be pointed out that in an admissibility structure, the full subcategory of local objects, which are the models of a geometric extension, have no reason in general to be multireflective in the ambient locally finitely presentable category, though it should be closed under filtered colimits. It can be shown that the full, accessible embedded multireflective subcategories of locally finitely presentable categories are models of disjunctive extensions, that are theories whose non cartesian sequents are made at most from strict disjunctions. Though it encompasses certain prominent examples, our situations of interests are not restricted to disjunctive extensions, hence multireflectivity of the class of local objects should in general appear only after restriction to local maps.

In fact we can exhibit models of a geometric extension as class of cone-injectivity, while models of strictly disjunctive theory form class of cone-orthogonality.

Now, if we consider a geometric extension $\mathbb{T}_\mathcal{C}$ of $ \mathbb{T}_{\mathcal{B}}$ (respectively, a geometric extension equivalent to a strictly disjunctive theory), that is a theory whose axioms are given as the following from a given class $\mathcal{C}$ of cocones in $ \mathbb{C}_{\mathbb{T}_{\mathcal{B}}}$ whose tops range over all objects:

\[   \{ \phi(\overline{x}) \vdash \bigvee_{i \in I} \exists \overline{y}_i (\psi_i(\overline{y}_i) \wedge \theta_i(\overline{y}_i,\overline{x})) \mid \; (\theta_i(\overline{y}_i,\overline{x}) : \psi_i(\overline{y}_i) \rightarrow  \phi(\overline{x}))_{i \in I} \in \mathcal{C} \} \]

(respectively with strict disjonctions),
then $ \mathcal{C}$ generates a Grothendieck topology  the models $B$ of $\mathbb{T}_{\mathcal{C}}$ are exactly the objects that are cone injective (respectively cone-orthogonal) to $\mathcal{C}$:

\[ \mathbb{T}_\mathcal{C}[\mathcal{S}et] =   \mathcal{C}^{inj} \]

Indeed, for any covering $ (f_i : \sfrac{\langle \overline{x} \rangle_\Sigma}{\phi(\overline{x})} \rightarrow \sfrac{\langle \overline{y}_i \rangle_\Sigma}{\psi_i(\overline{y}_i)})_{i \in I}  $, an arrow $ g : \sfrac{\langle \overline{x} \rangle_\Sigma}{\phi(\overline{x})} \rightarrow B $ just defines some $ \overline{b} \in B$ such that $ B \models \phi(\overline{b})$, so being a model of $ \mathbb{T}_\mathcal{V}$ there is one (respectively exactly one) $ i \in I$ and a $ \overline{b}_i \in B$ such that $ B \models \psi_i(\overline{b}_i)$ and $ B \models \theta_{f_i}(\overline{b}_i,\overline{b}))$, so that the corresponding arrow
\[ \ulcorner \overline{b}_i \urcorner : \sfrac{\langle \overline{y}_i \rangle_\Sigma}{\psi_i(\overline{y}_i)} \rightarrow B \]
factorizes uniquely $g$, and conversely.

On the converse, is it true that for a class of cone of finite presentation, the cone injectivity class

However in a Diers context the considered stable functor usually does not define a full subcategory. It is so exactly when the specialization order between local objects is trivial. But in the general case, we have to distinguish the considered non full subcategory defined by the essential image of $U$ from the full multireflexive subcategory given by the class of cone orthogonality it is expected to generate.

Consider the class of cones of local units under arbitrary objects:

\[ \mathcal{C}_U = \{ (B \stackrel{\eta_A(f)}{\rightarrow} UL_A(f))_{B \stackrel{f}{\rightarrow} U(A) \in B\downarrow U} \mid B \in \mathcal{B}  \} \]

If we denote $ (B \stackrel{\eta_i}{\rightarrow} U(A_i))_{i \in I_B}$ the cone of (isomorphism classes of) all local units of $B $, then it generates in some way a cocoverage in $ \mathcal{B}$ in the sense that for any $f : B_1 \rightarrow B_2$, one has for any $ i \in I_{B_2}$ the following factorization

\[ \begin{tikzcd}
B_1 \arrow[]{r}{f} \arrow[]{d}[swap]{n_{\eta^2_if}} & B_2 \arrow[]{d}{\eta^2_i} \\
U(A_{\eta^2_if}) \arrow[]{r}[swap]{U(u_{\eta^2_if})} & U(A_i)
\end{tikzcd} \]

However the cones of local units are not stable by pushout: indeed, for $i \in I_{B_1}$ even though the $ f_*\eta^1_i : B_2 \rightarrow B_2 +_{B_1} U(A_i)$ is still diagonally universal, its codomain needs not be in the range of $U$. But pushouts of cones of local units are still covering families in the sense that "they still cover the local units" of the codomain. By this we mean that all its local units factorize through one of those pushouts at least. This results just from the property of pushout as visualized in the following:

\[ \begin{tikzcd}
B_1 \arrow[]{r}{f} \arrow[]{d}[swap]{n_{\eta^2_if}} & B_2 \arrow[]{d}{f_*\eta_i^1} \arrow[bend left=20]{rdd}{\eta^2_i} &  \\
U(A_{\eta^2_if}) \arrow[bend right=10]{rrd}[swap]{U(u_{\eta^2_if})} \arrow[]{r}{} &   B_2 +_{B_1} U(A_i) \arrow[dashed]{rd}{\exists} \arrow[phantom, very near start]{ul}{\ulcorner} & \\
& & U(A_i)
\end{tikzcd} \]

This must be generalized by considering cones of etales maps jointly factorizing local units under a given objects. Those were considered in <cit.>.

For a stable functor $ U : \mathcal{A} \rightarrow \mathcal{B}$, we can consider the $U$-localizing families as the families of $U$-etale morphisms jointly factorizing local units of $U$, that are all the families of the form  $ (B \stackrel{n_i}{\rightarrow} B_i)_ {i \in I} $ with $  n_i $ in $ \Et_B $  such that for each local unit $ x \in X_B$ there is some $ i \in I$ providing a factorization
\[ \begin{tikzcd}[row sep=small]
B \arrow[]{rr}{\eta_x} \arrow[]{rd}[swap]{n_i} & & U(A_x) \\ & B_i \arrow[dashed]{ru}[swap]{\exists}
\end{tikzcd}  \]

In particular one can define $U$-local objects as those $A$ such that for any $B$ and any $(n_i : B \rightarrow B_i)_{i \in I}$ in $\Et_U$ such that for any $ f : B \rightarrow A$ and any $ (f_i : B \rightarrow B_i)_{i \in I}$ $U$-localizing family we have a factorization of $f$ through some $ f_i$.

For any object $ A$ in $\mathcal{B}$, the following are equivalent:

* $ A$ is $ U$-local,

* $ A$ is a retract of an object in the range of $U$,

* $ A $ is in $\mathcal{C}_U^{inj}$

* The $(\Et_U, \Loc_U)$ factorization of  any $ f : B \rightarrow A$ returns a retract of an object in the essential image of $U$.

First, observe that if $A$ is $U$-local, then in particular $ 1_A$ lifts through its own unit cone: that is we have a retraction for some $i \in I_A$:

\[
\begin{tikzcd}
A \arrow[]{d}[swap]{n_i} \arrow[equal]{r}{} & A \\
U(A_i) \arrow[dashed]{ru}[swap]{\exists}
\end{tikzcd}
\]

Now if $A$ is a retract of some $U(A_0)$ with $ 1_A = rs$, for any $f : B \rightarrow A$ the factorization of $sf$ provides a factorization of $f$ itself through $ rU(u_{sf})$:

\[
\begin{tikzcd}
B \arrow[]{r}{f} \arrow[]{d}[swap]{\eta_{sf}} & A \arrow[]{rd}{s} \arrow[equal]{rr}{1_A} && A\\
U(A_{sf}) \arrow[]{rr}[swap]{U(u_{sf})} & & U(A_0) \arrow[]{ru}{r} &
\end{tikzcd}
\]

So that $A$ is cone injective.

Now, if $A$ is cone injective and $ ( B \rightarrow B_i)_{i \in I} $ is a cover, any $ f : B \rightarrow A $ lifts through some unit $n_i : B \rightarrow U(A_i)$ which is itself factorized by some member of the cover, thus so is $f$, exhibiting $A$ as a $U$-local object.

Next, for $A$ a $U$-local object, let

\[ \begin{tikzcd}
B \arrow[]{rr}{f} \arrow[]{rd}[swap]{n_f} & & A \\
& A_f \arrow[]{ru}[swap]{u_f}&
\end{tikzcd} \]

be the $(\Et_U, \Loc_U)$ factorization of $f : B \rightarrow A$; then $A_f$ itself possesses a cone of local units under it, which is in particular trivially a $U$-localizing family: hence the local part $u_f$ of the factorization itself lifts through one of them as $A $ is $U$-local. But now the factorization of the lift returns the same object as the factorization of $f$ by uniqueness of factorization, as we have a chain of diagonally universal maps on the left:

\[ \begin{tikzcd}
B \arrow[]{rr}{f} \arrow[]{rd}[swap]{n_f} && A && A_\exists \arrow[]{ll}[swap]{u_\exists} \\
& A_f \arrow[equal, crossing over]{rrru}{ } \arrow[]{ru}[swap]{u_f} \arrow[]{rr}[swap]{n_i \in Diag_U} && U(A_i) \arrow[]{ru}[swap]{n_\exists} \arrow[dashed]{lu}[swap]{\exists}
\end{tikzcd} \]

Hence $ A_f$ is a retract of some object in the range of $U$.

We can also prove directly that the second and the fourth item are equivalent. If $ A $ is a retract of an object in the range of $U$, then for $ f : B \rightarrow A$ we can compare the left part of the factorization of $A$ with the local unit

\[ % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBoBGAXVJADcBDAGwFcYkQAhEAX1PU1z5CKAEwVqdJq3YBBHnxAZseAkQAs4mgxZtEIOb37KhRAMykRE7dL0BVABQyA+sQCU8o4NUpypYlaldEABhJwAzD0UBFWFkX0stQPYHZ2A4MO53bgkYKABzeCJQMIAnCABbJF8QHAgkMRBGLDAgqAhmACNGNhoACxh6KCQwZkZGGhx6LEZ2SBbI0oqqibrEc0kddgQaRnoOmEYABWiTPWbsWAWyyrWV+sTNvRKQHb2D4+NvEBKsPN6cK5LRBkGqrDQbGyNcKAm7g2rLCFBZjQwwgRY3EHwxAAVgekIAOviYJMnGkMi9Gm8jicvj8-gDUeikLjQUh1tYgg5kWTMhTdvtqZ9hN9fv8YUhMatyIzrkg4asWU15nooPQ4P0hnigoTifRSXBkeTstwgA
\begin{tikzcd}
& C_f \arrow[rd, "u_f"] \arrow[dd, "\eta_{su_f}", dashed, near start] &                                                      &                         &   \\
B \arrow[ru, "l_f"] \arrow[rd, "\eta_{sf}"'] \arrow[rr, "f", crossing over, near start] &                                                         & A \arrow[rr, equal] \arrow[rd, "s" description] &                         & A \\
& U(A_{sf}) \arrow[rr, "U(u_{sf})"']                      &                                                      & U(A_0) \arrow[ru, "r"'] &
\end{tikzcd} \]

but now the following square below admits some filler

\[ % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAGEB9AMxAF9S6TLnyEUARnJVajFmy68BQ7HgJEy46fWatEIAKoAKAIKdgcbnwCU-QSAwrRRSZura5e9v2kwoAc3giUG4AJwgAWyQyEBwIJEkQBiwwXRAoCCYAIwZWagALGDooJDAmBgZqHDosBjZIFNtgsMjEaNikACY3WVSAHV6YKrM4Jh4+EGoGOkyYBgAFYVUxEBCsPzycRpBQiPjKuMQAZm6dNlHFOx2WrpiD45lTvRCjUfNLG0np2YXHNSe1jZbK6dfZ7RLJVJQOhwArFT4zeaLJz-dabPgUPhAA
\begin{tikzcd}
C_f \arrow[r, equal] \arrow[d, "\eta_{su_f}"']   & C_f \arrow[d, "u_f"] \\
U(A_{sf}) \arrow[r, "rU(u_{sf})"'] \arrow[ru, dashed] & C
\end{tikzcd} \]

exhibiting $ C_f$ as a retract of $U(A_{sf})$. Proving that the last item implies the second is trivial by applying the result the identity of $A$

Finally, suppose that any arrow toward $A$ factorizes through a retract of an object in the range of $U$. Then for any $f : B \rightarrow A$ and any $U$-localizing family $(n_i)_{i \in I}$ of $B$, we have a factorization as below

\[  % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRACEQBfU9TXfIRQBGUsKq1GLNgGEA+gDNuvEBmx4CRAMxiJ9Zq0Qh5Snn3WCiAFnJ6phkAEFl5gZpQAmUh7sG2AVQAKRzliAEoXVX4NIRJvX2kjeSxuCRgoAHN4IlAFACcIAFskMhAcCCQdST8jUxV8opLqcqRREAY6ACMYBgAFaMsjLDBsWBBqfUTwRUiG4sQ2lsQvduGHKAgmToZWagALGDooJDAmBgZmuiwGNkgwVjMQOdbmisQbaqmEag7uvoH3CBhqMHvUCvMPksVpMHHlxu0uj1+hZAXksBk9jhZuCkCsllUYWwmDMfoj-iihCA0RisY9nohSksAKwTey3OQpUl-ZFuSnUzHYxqIFllN4fBhrNhQOhwA7HLgULhAA
\begin{tikzcd}[row sep=small]
B \arrow[rrrr, "f"] \arrow[rd, "n_f" description] \arrow[dd, "n_i"'] &                                                        &                         &                        & A \\
& C_f \arrow[rr, equal] \arrow[rd, "s" description] &                         & C_f \arrow[ru, "u_f"'] &   \\
C_i \arrow[rr, dashed]                                               &                                                        & U(A_0) \arrow[ru, "r"'] &                        &
\end{tikzcd}\]

ensuring that $A $ is $U$-local.

We have a similar property of local maps:

* A map between retract of objects in the essential image of $U$ is $U$-local (that is right orthogonal to the maps in $\Diag_U$) if and only if it is a retract of a map in the range of $U$.

* In particular, any object sending a local map toward a retract of an object in the range of $U$ is itself such a retract, as is the local map.

For a retract of a map in the range of $U$ and an arbitrary square as in the following with $n$ in $ ^\perp(U(\overrightarrow{A}))$:

\[\begin{tikzcd}[row sep=small]
C \arrow[]{dd}{n} \arrow[]{r}[swap]{} & A \arrow[equal]{rr}{} \arrow[]{rd}[swap]{s} \arrow[]{dd}{f} & & A \arrow[]{dd}{f} \\
& & U(A_0) \arrow[]{dd}[near start]{U(u)} \arrow[]{ru}{r} & \\
C \arrow[]{r}{} & A' \arrow[equal]{r}{} \arrow[]{rd}[swap]{s'} & \arrow[equal]{r}{} & A' \\
& & U(A_0') \arrow[]{ru}[swap]{r'} &
\end{tikzcd}\]

the composite square with $ n$ on the left and $ U(u)$ on the right admits a lifting, whose composite with $r$ constitutes a lifting of the leftmost square.

For the converse we have to prove the second item. Consider a local map whose codomain is a retract, then consider the factorization of the composite map, as in the following:

\[
\begin{tikzcd}[row sep=small]
A \arrow[equal]{rr}{} \arrow[]{dd}[swap]{f} \arrow[]{rd}[swap, near end]{n_{s'f}} & & A \arrow[]{dd}{f} \\
& U(n_{s'f}) \arrow[]{dd}[near start]{U(u_{s'f})} & \\
A' \arrow[equal]{r}{} \arrow[]{rd}[swap]{s'} & \arrow[equal]{r}{} & A' \\ & U(A_0) \arrow[]{ru}[swap]{r'} &
\end{tikzcd}
\]

This provides us with the following square

\[
\begin{tikzcd}[column sep=large]
A \arrow[equal]{r}{} \arrow[]{d}[swap]{n_{s'f}} & A \arrow[]{d}{f} \\
U(A_{s'f}) \arrow[]{r}[swap]{r'U(u_{s'f})} & A'
\end{tikzcd}\]

whose diagonalization exhibits both $A$ as a retract of $ U(A_{s'f})$ and $ f$ as a retract of $ U(u_{s'f})$.

The $U$-local objects and $U $-local maps between them form a (non full) multireflexive subcategory

\[ U\hy\Loc^{\Loc_U} \hookrightarrow \mathcal{B} \]

This means that for any stable functor, the closure by retracts of its essential image image is multireflexive. Its full closure in $\mathcal{B}$ is the cone injectivity class $\mathcal{C}_U^{inj}$.

However, beware that the class thus obtained is not yet adequate to provide an admissibility: in fact, we cannot yet see the $U$-local objects as all the models of a geometric theory, for the injectivity condition involved are too constraining. Indeed, the $ U$-localizing families were defined under arbitrary objects, and injectivity had to be tested hence relatively to a large class of cover; similarly, the cone-injectivity condition was tested relatively to a large family of cone outside the generator of finitely presented objects, and hence we cannot ensure the resulting class to be the class of models of a geometric extension.

To produce the desired admissibility structure, we have to restrict the localness test relatively to finitely presented object. Hence we have to restrict the $ U$-localizing families to the finitely presented objects and the finitely presented etale morphisms under them:

Define the $U$-localizing topology $J_U$ on $\mathcal{B}_{\fp}^{\op}$ as having as covering families the duals of the families $ (n_i : K \rightarrow K_i)_{i \in I}$ in $\Et_{\fp}$ such that for each $ x \in X_K$, there is some $i \in I$ and a factorization

\[ \begin{tikzcd}[row sep=small]
K \arrow[]{rr}{\eta_x} \arrow[]{rd}[swap]{n_i} & & U(A_x) \\ & K_i \arrow[dashed]{ru}[swap]{\exists}
\end{tikzcd}  \]

Then from what was said in 1.2, $J_U$ codes for a geometric extension of the finite limit theory whose category of set-valued models is $\mathbb{T}$:

We denote as $\mathbb{T}_U$ the corresponding geometric extension, and call it the geometric axiomatization of $U$.

Observe that any $U$-local object is trivially $ J_U$. hence is a model of the $\mathbb{T}_U$. This is because a $J_U$-local object lifts in particular the basic cover of finitely presented objects.
However, as pointed out in Anel[], the converse is not true in general: there may be $\mathbb{T}_U$-models that fail to be $J_U$-local. In that sense the notion defined in <cit.> is not yet sufficient to axiomatize our local data in an universal way. Indeed, while $ J_U$-local objects will be points o

The rest of this section is devoted to prove that the data of $ (J_U, \Et_U)$ define an admissibility structure with $ \mathbb{T}_{U}[\mathcal{S}et]$ as the local objects.

Any object in the essential image of $U$ is in particular a model of $ \mathbb{T}$. However, not only there are additional $U$-local objects, but there are even more $\mathbb{T}_U$-models. We have the following:

We have the following:

* The subcategory $ \mathbb{T}_U[\mathcal{S}et]$ is closed in $\mathcal{B}$ by retracts.

* $\mathbb{T}_U$-models have the gliding property relatively to local maps.

For the first item, the proof is the same as for retract of objects in the range of $U$: if $A$ is a retract of a $\mathbb{T}_U$-model $A_0$ then for any $f : K \rightarrow A$ and covering family of $K$ in $ J_U \mid_{\fp}$ one has

\[ \begin{tikzcd}
K \arrow[]{r}{f} \arrow[]{d}[swap]{\eta_{sf}} & A \arrow[]{rd}{s} \arrow[equal]{rr}{1_A} && A\\
K_i \arrow[]{rr}[swap]{\exists} & & A_0 \arrow[]{ru}{r} &
\end{tikzcd} \]

So that $A$ itself is $ J_U \mid_{\fp}$-flat by Yoneda lemma, hence is a $ \mathbb{T}_U$-model.

For the second item, let $ u: A \rightarrow A_0$ be local with $ A_0$ a $\mathbb{T}_U$-object. Then for any $f : K \rightarrow A$ and covering family of $K$ in $ J_U \mid_{\fp}$ the composite $ uf : K \rightarrow A_0$ extends through some $n_i$; but as those $ n_i$ are in $Diag_U$, this extension induces a diagonalization

\[  \begin{tikzcd}
K \arrow[]{r}{f} \arrow[]{d}{n_i} & A \arrow[]{d}{u} \\ K_i \arrow[dashed]{ru}{} \arrow[]{r}{\exists } & A_0
\end{tikzcd}  \]

Hence $A$ is a $\mathbb{T}_U$-model.

$\mathbb{T}_U$-models and local maps between them form altogether a (non full) multireflective subcategory

\[  \mathbb{T}_U[\mathcal{S}et]^{Loc} \hookrightarrow \mathcal{B} \]

In particular the full subcategory $\mathbb{T}_U[\mathcal{S}et]$ can be seen as a geometric envelope of the functor $U$.

Observe that $\mathcal{A}$ is full in $J_U-Loc^{Loc}$ and $ \mathbb{T}_U^{Loc}$ as for a local map $ u : U(A_1) \rightarrow U(A_2)$ the diagonally universal part is trivial while the left part is in the range of $U$, so is $u$.

To sum up, we have the following factorization of $U$, where all the faithful functors into $\mathcal{B}$ are also multireflective

\[\begin{tikzcd}
& {U(\mathcal{A})} \\
{\mathcal{A}} & {U\hy \Loc^{\Loc_U}} & {\mathcal{B}} \\
& {\mathbb{T}_U[\mathcal{S}]^{\Loc_U}}
\arrow["{eso, full}", from=2-1, to=1-2]
\arrow[from=1-2, to=2-2, hook]
\arrow[from=2-2, to=3-2, hook]
\arrow[from=3-2, to=2-3, hook]
\arrow[from=1-2, to=2-3, hook]
\arrow[from=2-2, to=2-3, hook]
\arrow[from=2-1, to=2-2]
\arrow[from=2-1, to=3-2]
\end{tikzcd}\]

More generally, one could ask, for a topology $J$, why it is worth to restrict us to a certain class of local maps between $J$-local objects rather that considering just general maps. The reason is that etale maps would have to be iso, although cover generating $J$ are not, so this would not define an admissibility structure.

From what was said, it seems we can formalize the correspondence between Diers contexts and admissibility structures.

For a fixed locally finitely presentable category $ \mathcal{B}$, we can either define a poset of Diers contexts on $ \mathcal{B}$ and a poset of admissibility structures on $\mathcal{B}$ equiped with the respective orders:

* $(U_1 : \mathcal{A}_1 \rightarrow \mathcal{B}) \leq (U_2 : \mathcal{A}_2 \rightarrow \mathcal{B})$ if  $\begin{tikzcd}
\mathcal{A}_1 \arrow[]{r}{U_1} \arrow[hook]{d}{} & \mathcal{B} \\ \mathcal{A}_2 \arrow[]{ru}[swap]{U_2}
\end{tikzcd}$

* $(\mathcal{V}_1, J_1) \leq (\mathcal{V}_2, J_2) $ if $ \mathcal{V}_2 \subseteq \mathcal{V}_1$ (so that $ Et_{\mathcal{V}_2} \subseteq Et_{\mathcal{V}_1}$ and $ Loc_{\mathcal{V}_1} \subseteq Loc_{\mathcal{V}_2}$) and $J_2 \leq J_1$. Hence in particular $ (\mathcal{V}_1, J_2)$ is an admissibility structure.

We denote by $ \mathfrak{Diers}_\mathcal{B}$ and $\mathfrak{Coste}_\mathcal{B}$ those respective poset.

We have an adjunction between Coste and Diers contexts on $\mathcal{B}$

\[ \begin{tikzcd}[row sep=tiny, column sep=small]
\mathfrak{Coste}_\mathcal{B} \arrow[bend right=17]{rr}{} & \perp & \arrow[bend right=17]{ll} \mathfrak{Diers}_\mathcal{B} \\
(\mathcal{V}, J) & \longmapsto & \mathbb{T}_J[Set]^{Loc_\mathcal{V}} \hookrightarrow \mathcal{B}\\
(\Diag_U \cap \overrightarrow{\mathcal{B}_{\fp}}, J_U) & \longmapsfrom &  \mathcal{A} \stackrel{U}{\rightarrow} \mathcal{B}
\end{tikzcd} \]

Where the closure operation on Diers contexts corresponded to the unit of this adjunction.

Now we want precision on the counit. We may ask is if the stable inclusion induced by a given admissibility context itself induces the same admissibility by this process. However it is obvious that such a thing can be true only if $ J$ has enough set-valued models, otherwise the induced stable inclusion may be void.

If $ (\mathcal{V},J)$ is an admissibility where $J$ has enough points, then, the stable inclusion

\[ \mathbb{T}_J[Set]^{Loc} \hookrightarrow \mathcal{B} \]

induces an admissibility equivalent to $ (\mathcal{V}, J)$.

§ (LOCALLY) MODELLED TOPOSES AND THE SPECTRAL ADJUNCTION

In the previous section, we have seen that an admissibility structure generates a toposwise stable functor which can be seen as a situation of local adjunction. In the following we fix an admissibility structure $(\mathcal{V},J)$ for a finite limit theory $\mathbb{T}$.

We have a pseudofunctor, which is representable by $\mathcal{S}[\mathbb{T}]$

\[\begin{tikzcd}
{\GTop^{\op}} & {\Cat} & {}
\arrow["{\mathbb{T}[-]}", from=1-1, to=1-2]
\end{tikzcd}\]

and is defined as follows:

* for 0-cells, it returns the category $\mathbb{T}[\mathcal{E}]$ of $\mathbb{T}$-models

* for 1-cell $ f : \mathcal{F} \rightarrow \mathcal{E}$, it returns the inverse image functor

\[ \mathbb{T}[\mathcal{E}] \stackrel{f^*}{\longrightarrow} \mathbb{T}[\mathcal{F}] \]

which is moreover lex and cocontinuous, for it is left adjoint to the direct image functor

\[ \mathbb{T}[\mathcal{F}] \stackrel{f_*}{\longrightarrow} \mathbb{T}[\mathcal{E}] \]

* on 2-cells $ \alpha : f \Rightarrow g$ it returns the natural transformation also denoted as $\alpha$

\[\begin{tikzcd}
{\mathbb{T}[\mathcal{E}] } && {\mathbb{T}[\mathcal{F}] }
\arrow["{f^*}"{name=0}, from=1-1, to=1-3, curve={height=-12pt}]
\arrow["{g^*}"{name=1, swap}, from=1-1, to=1-3, curve={height=12pt}]
\arrow[Rightarrow, "{\alpha}", from=0, to=1, shorten <=2pt, shorten >=2pt]
\end{tikzcd}\]

This defines a 2-fibration with 1-truncated fibers

\[\begin{tikzcd}
{\int\mathbb{T}[-]} & {\GTop}
\arrow["{p_\mathbb{T}}", from=1-1, to=1-2]
\end{tikzcd}\]

This bicategory can be considered as a category of all models of $\mathbb{T}$ regardless of their base topos. An object in this category is a modelled topos, that is, a pair $( \mathcal{E},E)$ with $ \mathcal{E}$ a Grothendieck topos and $ E $ in $\mathbb{T}[E]$. However in the following, we choose to work with an algebraic convention in the sense that we want morphisms between modelled toposes to have the orientation of the morphisms of models rather than the orientation of the underlying geometric morphism. To this end, we use the following, which is nothing but the direct fibration associated to $ \mathbb{T}[-]$:

The bicategory $ \mathbb{T}\hy\GTop$ of $\mathbb{T}$-modelled toposes has

* for 0-cells, modelled toposes $ (\mathcal{E}, E)$

* for 1-cells, $(f, \phi) : (\mathcal{F}, F) \rightarrow (\mathcal{E}, E)$ with $ f: \mathcal{E} \rightarrow \mathcal{F}$ a geometric morphism and $ \phi$ consisting of a pair $ (\phi^\flat, \phi^\sharp)$ with

\[ f^*F \stackrel{f^\flat}{\longrightarrow} E  \quad \textrm{ and } \quad F \stackrel{f^\sharp}{\longrightarrow } f_*E \]

mates along the adjunction $f^* \dashv f_*$

* for 2-cells $ \alpha : (f,\phi) \rightarrow (g,\psi)$, 2-cell $ \alpha : f \rightarrow g$ in $\GTop$.

From its construction, this bicategories is equiped with an obfibration $ p_\mathbb{T} : \mathbb{T}-\GTop \rightarrow \GTop^{\op}$

Now the local data associated to $ (\mathcal{V},J)$ also define an pseudofunctor

\[\begin{tikzcd}
{\GTop^{op}} & {\Cat}
\arrow["{\mathbb{T}_J[-]^{\Loc}}", from=1-1, to=1-2]
\end{tikzcd}\]

and again we can consider the associated direct 2-fibration; but for each topos $\mathcal{E}$, we have a non full inclusion of category