aa r X i v : . [ m a t h . C T ] F e b The general construction of Spectra
Axel Osmond
Abstract
We provide a synthesis of different topos-theoretical approaches of the general constructionof spectra, exploring jointly its purely categorical, model theoretic and geometric aspects. Wedetail also the exact relation with Diers notion of spectrum. Then we give a more detailedaccount of Cole’s original construction from bicategorical universal properties. We also providenew formulas to compute the spectrum of arbitrary etale maps as pseudolimits, as well asgeneralizations of results on fibered topos we then apply to compute spectrum of a modelledtopos.
Contents
Introduction
Many prominent dualities in mathematics are instances of a common construction centered onthe notion of spectral functor . Roughly stated, one starts with a finite limit theory, equiped witha geometric extension whose models are tough as local objects encoding point-like data, togetherwith a factorization system (
Etale maps , Local maps ), where the etale maps behave as duals ofdistinguished continuous maps. Several manners of axiomatizing the correct relation between thoseingredients have been proposed, either through topos theoretic methods by “localizing” local ob-jects with a Grothendieck topology generated by etale maps, or in an alternative (though tightlyrelated) way based on the notion of stable functor. Moreover those data make still sense in cat-egories of models of the ambient finite limit theory in arbitrary toposes. Then the spectrum of agiven object is constructed as a topos classifying etale maps under this given object toward localobjects, equiped with a structural sheaf playing the role of the “free local object” under it. Thisdefines a spectral functor, left adjoint of a inclusion of a category of locally modelled toposes intoa category of modelled spaces.The history of this construction goes back to [18], where the classifying topos of the theory oflocal rings was introduced as well as the notion of “(locally) ringed topos” and a first generaliza-tion of the notion of locally ringed spaces in the context of algebraic geometry. This work inspiredthen [7], where a first systematic construction was provided by means of 2-dimensional universal1roperties and the ingredients of the general situation were isolated, while a first axiomatization ofthe notion was introduced in [19]. [8] then gave a first syntactic approach of the process from thepoint of view of categorical model theory. On his side, [11] had provided an alternative approachwhich does not make use of toposes and processes in a more point-set and concrete way. Later, [24]provided an ∞ -categorical version of the construction from a purely geometric point of view, while[3] 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 fromalgebraic geometry, as for instance [5], [6], [12].In this paper, we sum up and compare different approaches, especially [7], [8], [3] and [11].Beside the expository aspects of this work, we provide comparative results expressing how thosedifferent 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 thedifferent 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 fromfunctorial, 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 thepaper will require to switch from one to another with fluidity.The second section revolves around the notion of geometry and admissibility . The latter is acondition entangling the data above into a factorization condition, which can also be captured bythe 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 inthose situations, set-valued data happen to be sufficient to induce the same structure on arbitrarytoposes, 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 sta-ble 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 thespectrum from a geometry by means of abstract 2-dimensional properties. Though the processwas suggested in [7], we try to be as explicit as possible and give some new insight and precisionon the matter.In the fourth section, we recall [8] construction of the corresponding spectral site, either for aset valued object or a modelled topos, and a pseudolimit formula theorem 4.1.10 generalizing thecorrespondence between etale maps and etale geometric morphisms; after introducing a new formof a classical result from [29] about fibered toposes, we prove that the spectrum of a modelled toposis a topos of continuous section of a canonical fibered topos, giving also new fibrational aspects ofthe construction.We end with two examples, spectral accounts of Stone duality and Jipsen and Moshier. In this first part we introduce the three ingredients involved in the construction of a spec-trum, the ambient data, the local data and the factorization data, and how they are entangled toconstitute a geometry that will be deployed into the spectrum.
This first section just contains very standard facts about finite limit theories and locally finitelypresentable categories. Its purpose is to recall all the equivalent ways to describe a finite limit the-ory, and how one can toggle between the logical, functorial and toposic way of seeing models. Weemphasize the way the data of a finite limit theory, its syntactic category, its classifying topos and2ts class of set valued models are mutually determined by given as much as possible of the involvedequivalence of categories, for we will need to have in mind those translations to make more intuitivethe rest of the paper.While all the well known examples of the spectral construction are defined relatively to al-gebraic categories, which are the object of interest of universal algebra and are axiomatized byLawvere 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 finitelypresentable categories and their syntactic counterparts, the finite limit theories . The reason is thateach Grothendieck topos admits a standard presentation , exhibiting it as a category of sheavesover a small site with finite limits, and that the extensions theorem as Diaconescu are formulatedrelatively to finite limits. In the following, we call functors preserving finite limits lex functors anddenote as
Lex the 2-category of small categories with finite limits, lex functors between them, andnatural transformations between lex functors; for concision we also call small categories with finitelimits lex categories . For a lex category C and a Grothendieck topos E we denote as Lex [ C , E ] thecategory of lex functors and natural transformations between them. We also denote as S the toposof sets.In the following we fix a first order language L . For a theory T in L , a sequent is said to be T -provable if it can be deduced from the axioms of T and the structural rules of first order logic,and we denote as ⊢ T the relation of T -provability. The relation of T -equivalence is the equivalencerelation ⊣⊢ T on formulas in L defined as φ ( x ) ⊣⊢ T ψ ( x ) iff φ ( x ) ⊢ T ψ ( x ) , ψ ( x ) ⊢ T φ ( x )Its classes are denoted as [ φ ( x )] T .We recall that a finite limit theory in L is a first order theory in L whose sequents consistof cartesian formulas , that are formulas built from atomic formulas with finite conjunctions andexistentials with proof of uniqueness. A T -provably functional formula in L is a formula θ [ x, y ]such that the following sequents are T -provable: − θ ( x, y ) ⊢ T ψ ( y ) − φ ( x ) ⊢ T ∃ yθ ( x, y ) − θ ( x, y ) ∧ θ ( x, y ′ ) ⊢ T y = y ′ A provably functional formula θ [ x, y ] should be thought as a statement of equality exhibiting y as afunction of x . The first condition says that this image of this function should satisfy the codomainformula; the second says that this function is defined over witnesses of the domain formula, andthe third is the condition of functionality, ensuring that an object has a unique image through thisfunction.To any first order theory T we can associate a category C T , the syntactic category of T , whose − objects are formulas in context { x, φ } in the language of T − morphisms { x, φ } → { y, ψ } are equivalence classes [ θ ( x, y )] T for the relation of T -provabilityof functional formulas.If T is cartesian then C T is a small category with finite limits. Conversely, any lex category C isthe C T of some cartesian theory T . However in the following we shall not need to explicitly givea syntactic presentation of the finite limit theory associated to a lex category, so we do not recallthe process here.A set-valued model of T is a lex functor F : C T → S et . The intuition behind this definition isthat such a functor associates to any formula in context its set of witnesses in the correspondingmodel F ( { x, φ } ) = { a ∈ F | F | = φ ( a ) } and to any morphism of the syntactic category, the application F ( { x, φ } ) F ([ θ ( x,y )] T ) −→ F ( { y, ψ } )3ssociating to each a in F ( { x, φ } ) the unique b in F ( { y, ψ } ) such that F | = θ [ a, b ].A morphism of L -structures between T -models F, G is a natural transformation F ⇒ G in Lex [ C T , S ]. Set-valued T -models and morphisms of L -structures between them form a category Lex [ C T , S ] which we denote as T [ S ].Now we turn to the general properties of the categories of set-valued models of finite limitstheories. A locally finitely presentable category B is a category with: − filtered colimits − small limits − a small generator B fp of finitely presented objects such that any object B is the filteredcolimit of B fp ↓ B In particular any locally finitely presentable category also has small colimits. Moreover, the fullsubcategory B fp of finitely presented objects is closed in B under finite limits, and the inclusion B fp ֒ → B exhibits B as the inductive completion B ≃
Ind( B fp )where Ind( B fp ) is the full subcategory of [ B op fp , S ] consisting of functors that are filtered limits ofrepresentable.A morphism of locally finitely presentable category is a functor F : B → B preserving filteredcolimits and small limits. It can be shown by variation of Freyd adjoint functor theorem that anysuch functor has a left adjoint F ∗ sending finitely presented objects of B to finitely presentedobjects of B . We denote as LFP the 2-category of locally finitely presentable categories, withnatural 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:
Lex op ≃ LFPC = C T Lex [ C , S ] B op fp ← [ B where the left to right process associates to any locally finitely presentable category the lex categorymade of the opposite category of the small subcategory of finitely presented objects. Objects of thesmall generator of finitely presented objects in Lex [ C T , S ] are the corepresentable functors, whichwe denote as ょ ∗{ x,φ } = C T [ { x, φ } , − ]. Moreover, for any object B in B , the restriction to B fp of thecorresponding representable functor ょ B = B [ − , B ] defines a lex functor B op fp ょ B −→ S This defines an equivalence of category
B ≃
Lex [ B op fp , S ].For a language L and any string of variables x , we can construct the free L -structure h x i L consisting of all terms in L over the variables x . For a finite limit theory T , this can be completedinto the free model of T by forcing the equality of terms that are T -provable in T , and we denoteit as h x i T .For a formula in context { x, φ } in C T , the corresponding finitely presented set-valued models of T can be seen as the quotient h x i T /φ ( x ). We denote this finitely presented model as K φ , and seenas an object of Lex [ C T , S ], K φ coincides with the corepresentable functor ょ ∗{ x,φ } .Because any lex category is the syntactic category of a finite limit theory, Gabriel-Ulmer dualitysays that any locally finitely presentable B category is the category of models of a finite limit theory T such that T [ S ] ≃ Lex [ C T , S ]. Then it can be shown that any finitely presented model K in B is uniquely determined by a presentation formula { x, φ } with φ a formula in the language of T , exhibiting K as h x i T /φ ( x ). Similarly, any map of finite presentation f : K → K ′ such that K = h x i T /φ ( x ) and K ′ = h y i T /ψ ( y ) can be presented by a formula θ f ( y, x ) encoding the expression4f the image by f of the generators of K in terms of the generators of K ′ : that is, if x = x , ..., x n one has f ( x i ) = τ i [ y ] for i = 1 , ..., n , then θ f can be chosen as the formula θ f ( y, x ) ⇔ ^ i =1 ,...,n x i = τ i [ y ]This correspondence is part of an equivalence of category C T ≃ B op fp In the following we shall denote K φ = h x i /φ the finitely presented model presented by the formula φ and f θ the finitely presented arrow presented by θ .Because K φ corresponds to the representable functor ょ ∗{ x,φ } in B ≃
Lex [ C , S ], Yoneda lemmatells us that any arrow K φ → B in B is the name of some witness in B of the formula φ ( x ): B [ K φ , B ] ≃ B ( { x, φ } ) = { a ∈ B | B | = φ ( a ) } In the following, if a is a witness in B of some φ ( x ), then we denote as p a q : K φ → B the corre-sponding arrow in B .As well as we considered set-valued models of a finite limit theory T , we can also considermodels in arbitrary Grothendieck toposes: those are the Lex functors T [ E ] ≃ Lex [ C T , E ]with natural transformations between them. In the case of finite limit theories, a T -model in asheaf topos E ≃ Sh ( C , J ) is the same as a sheaf of T -models over ( C , J ). Indeed, if F : C T → E isa T -model, then not only for each { x, φ } we have a sheaf F ( { x, φ } ) over ( C , J ), sending an object c to a set F ( { x, φ } )( c ), but for any c in C , we have functor C T F ( − )( c ) −→ S sending any { x, φ } to F ( { x, φ } )( 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 T .Now we turn to the classifying topos for T . For a category C , we denote as b C the presheaf topos[ C op , S ]. For any Grothendieck topos E , we have an equivalence of category Geom [ E , b C ] ≃ Lex [ C , E ]sending any lex functor F : C → E to its
Yoneda extension , that is, the left Kan extension alongthe Yoneda embedding
C E b C F lan ょ F ょ ≃ where the canonical transformation F ⇒ lan ょ F ょ is invertible for ょ being full and faithful. More-over the functor lan ょ F is left adjoint to the functor F ∗ : E → b C . Moreover lan ょ F is still lex,hence it defines a geometric morphism lan ょ F ⊣ F ∗ . In the following we will denote F ∗ = lan ょ F .The functoriality of this process is ensured by the universal property of the left Kan extension.Conversely any geometric morphism F ∗ ⊣ F ∗ induces a lex functor F ∗ ょ : C → E .Hence in particular for any finite limit theory T , we have an equivalence of category T [ E ] ≃ Geom [ E , c C T ]. For this reason, we call the presheaf topos c C T the classifying topos of T and denoteit S [ T ]. In particular the set-valued models of T are the points of the classifying topos S [ T ], thatis T [ S ] ≃ Ind( C op T ) ≃ Lex [ C T , S ] ≃ Geom [ S , c C T ]If one starts with a locally finitely presentable category B this gives the following equivalenceof categories B ≃
Ind( B fp ) ≃ Lex [ B opfp , S et ] ≃ Geom [ S et, d B opfp ]5e also have the following fact for finite limit theories: if a Grothendieck topos E is equivalentto the sheaf topos Sh ( C , J ), then T -models in E are exactly sheaves of set-valued T -models over( C , J ), that are functors F : C op → T [ S ] such that for any covering family ( s i : c i → c ) i ∈ I in J therestrictions maps of F at the s i exhibits F ( c ) as a limit in T [ S ] F ( c ) ≃ lim (cid:18) Y i ∈ I F ( c i ) ⇒ Y i,j ∈ I F ( c i × c c j ) (cid:19) This comes from the fact that evaluation at an object preserves limits, hence in particular finitelimits.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 categoryIn the following we will fix once for all a finite limit theory T , and denote as B the associatedcategory of set-valued models, and S [ T ] its classifying topos.Observe that, from what was said, a finite limit theory can be recovered (up to Morita equiv-alence) from its categories of models in S : this is a situation where we have “enough set-valuedmodels”, in the sense that the points of the classifying topos are jointly conservative . In somesense, 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, locallyfinitely presentable categories also have a notion of free object. For a finite limit theory T , a finitelimit extension T ′ of T is a finite limit theory in the same language L and whose axioms containthe axioms of T . Then models of T ′ are in particular models of T and it can be checked that theinclusion functor ι T , T ′ : T ′ [ S ] ֒ → T [ S ] is a morphism of locally finitely presentable category: henceit admits a left adjoints ι T , T ′ , assigning to each T -model B a T ′ -model ι ∗ T , T ′ ( B ) which we can seeas the free T ′ -model under B . However, for a arbitrary extension of T that is not a finite limittheory, it is no longer true that such a free construction exists in S . As we shall see, spectra willprovide a way to fix this. In this section, we introduce what will play the role of the local data as announced in theintroduction. The previous section was devoted to “well behaved” theories. In this section we givegeneralities about the more general class of theories whose models will be used as local objects inthe 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 C and a Grothendieck topos E , a functor F : C → E is said to be J -continuous if for any J -covering family ( u i : c i → c ) i ∈ I , the induced arrow ` i ∈ I F ( c i ) F ( c ) h F ( u i ) i i ∈ I is an epimorphism.As well as the Yoneda extension of a lex functor produced the inverse image part of a geometricmorphism in the context of presheaf toposes, lex continuous functors also enjoy an extensionproperty, as stated in Diaconescu theorem : for a lex site ( C , J ) and a Grothendieck topos E , wehave an equivalence of category Lex J [ C , E ] ≃ Geom [ E , Sh ( C , J )]6here Lex J [ C , E ] denotes the category of lex J -continuous functors. Indeed it can be shownthat a geometric morphism F : E → b C factorizes through Sh ( C , J ) if and only if the restriction F ∗ ょ : C → E is J -continuous. Hence, even when the induced functor a J ょ : C → Sh ( C , J ) isnot full and faithful, any J -continuous lex functor coincides with the restriction of its extensionalong a J ょ . When J is subcanonical, a J is full and faithful and this becomes true for any functor.In particular, we have points of Sh ( C , J ), which are the geometric functors S → Sh ( C , J ), areobtained as Lex J [ C , S ] ≃ Geom [ S , Sh ( C , J )]A Geometric theory is a theory whose sequents involve formulas built from atomic formulasusing finite ∧ , arbitrary ∃ and small W . Beware that the existentials are no longer supposed tohave a proof of uniqueness, and the disjunction may be infinite as long as they are indexed by a set.For a geometric theory T , the finite limit part of T is the finite limit T theory made of allthe sequents of T involving formulas containing only conjunctions and existentials with proof ofuniqueness. Conversely, for T a finite limit theory, a geometric extension of T is a geometric theory T ′ whose finite limit part is T .Suppose we have a geometric extension T → T ′ . Then T ′ can be shown to be built from T byadding sequents of the form φ ( x ) ⊢ _ i ∈ I ∃ y i (cid:0) ψ ( y i ) ∧ θ i ( y i , x ) (cid:1) with each θ i ( y i , x ) T -provably functional from ψ ( y i ) to φ ( x ).Then on the cartesian site C T , one can define a topology J T ′ whose covering families consist ofthe corresponding families (cid:0) { y i , ψ i } { x, φ } [ θ i ( y i ,x )] T (cid:1) i ∈ I Then a model of T ′ in a Grothendieck topos E is a lex functor F : C T → S which is moreover J T ′ -continuous. In particular, this is a lex functor: this means that we have a full inclusion ofcategories T ′ [ S ] ֒ → T [ S ]It will be relevant in the following to characterize models of T ′ amongst models of T . The con-dition of being a model of the geometric extension can be reexpressed by an injectivity condition .Observe that, if B is the locally finitely presentable category T [ S ], those families correspond in B op fp to families of cone ( f θ i : K φ → K ψ i ) i ∈ I under the corresponding finitely presented objects. Byabuse of notation, we also call J the data of all the families ( f θ i ) i ∈ I in B .For a family J of cones ( f i : K → K i ) i ∈ I in B , a J -local object (or also J -injective ) is an object A such that for any cone ( f i : K → K i ) i ∈ I in J and any arrow f : K → A , f admits a factorizationthrough some of the f i K AK i ... K jk i k j f ∃ Then for a geometric extension T ′ of a finite limit theory T , T ′ -models are exactly the J T ′ -localobjects, that is we have L oc J T ′ ≃ T ′ [ S ]Indeed, in T [ S ] = Lex [ C T , S ], we have T [ S ]( K φ , F ) ≃ Lex [ ょ ∗{ x,φ } , F ] which is F ( { x, φ } ) by Yonedalemma, hence requiring that J T ′ -local object exactly amounts to say that for any covering familyas above we have a surjection ` i ∈ I F ( { y i , ψ i } ) F ( { x, φ } ) h F ([ θ i ( y i ,x )] T ) i i ∈ I Syntactically, this should be understood as follows: for any covering ( f θ i : K φ → K ψ i ) i ∈ I , an arrow g : K φ → B just defines some b ∈ B such that B | = φ ( b ), so as it is a model of T V , there is some7 ∈ I and a b i ∈ B such that B | = ψ i ( b i ) and B | = θ i ( b i , b )). Hence we have the factorizationthrough the name of this witness of ψ i : K φ BK ψ i f θi g = p b q ∃ p b i q In the other direction, from the equivalence between arrows from finitely presented objects andnames of witnesses of their presentation formula in the codomain, it is clear that local objects aremodels of the geometric extension T J .In particular, we have that set-valued models of T ′ are the points of the sheaf topos Sh ( C T , J T ′ ).Conversely, any Grothendieck topology J on C T corresponds to a geometric extension of T whose axioms mirror covering of finitely presented objects: T J = T B ∪ (cid:26) φ ( x ) ⊢ _ i ∈ I ∃ y i ( ψ i ( y i ) ∧ θ f i ( y i , x )) (cid:27) ( fi ) i ∈ I ∈ J ( Kφ ) { x,φ }∈C T exhibiting covers as disjunctions of cases for witnesses of domain formulas.Again we can generalize the notion of model of a geometric extension T ′ for arbitrary Grothendiecktoposes: for a Grothendieck topos E , a T ′ model in E is a lex J T ′ -continuous functor C T → E ,while a morphism of T ′ -models in E is a natural transformation. Beware that, contrarily to modelsof finite limit theories, models of arbitrary geometric theories in a topos of sheaves Sh ( C , J ) notbe sheaves of set-valued models over ( C , J ). In fact, they are in particular sheaves of set-valuedmodels of their finite limit part, but the additional geometric axioms cannot be enforced object-wisely. However, this can be tested at points: for a point of a topos p : S → E and E in T ′ [ E ], thecomposite p ∗ E ∗ : C T → S is J T ′ -continuous, hence the stalk F p is in T ′ [ S ].We have T ′ [ E ] = Lex J T ′ [ C T , E ] ≃ Geom [ E , Sh ( C T , J T ′ )]where the last equivalence follows from Diaconescu. Moreover, this equivalence is natural in thesense that for any geometric morphism f : E → F , post-composition with f induces a functor T ′ [ F ] f ∗ −→ T ′ [ E ]and the same is true for 2-cell of GTop . This exhibits Sh ( C T , J T ′ ) as a representing object for theindexed category GTop op T ′ [ − ] −→ Cat
For this reason the sheaf topos Sh ( C T , J T ′ ) is called the classifying topos of T ′ and denoted S [ T ′ ].In particular we have a subtopos S [ T ′ ] ֒ → S [ T ] from the inclusion Sh ( C T , J T ′ ) ֒ → c C T and for any topos E , the 2-functor Geom [ E , − ] sends this inclusion of topos into the full inclusionof the categories of models T ′ [ E ] ֒ → T [ E ].To conclude those prerequisites, observe that the latter inclusion of category has no longer aleft adjoint, contrarily to what we had for extension of finite limit theories. This means that ageometric extension does not enjoy a free construction. Finally, observe also that the inclusionabove was full, and in general, it is not possible to axiomatize a non full subcategory of a categoryof model of a finite limit theory by a geometric extension. However, our situations of interestwill precisely involve both a geometric extension and a choice of maps from a factorization systemrelated by some condition. The next section is devoted to the factorization aspects.8 .3 Left generated factorization system In this part we recall basic facts about factorization systems. We also give some elements ofAnel presentation of the small object argument as presented in [3], which is a process to constructa 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.
Proposition 1.3.1. If C is a category endowed with a factorization system ( L , R ) , then the leftand right classes enjoy the following properties : − L is closed under composition − L contains all isomorphisms − L is right-cancellative: for any triangle C C C l fl with l , l in L , then f also is in L− L is closed under colimits in −→C − R is closed under composition − R contains all isomorphisms − R is left-cancellative: for any triangle C C C f r r with r , r in R , then f also is in R− R is closed under limits in −→C
Let B be a locally finitely presentable category, with T the underlying finite limit theory. If weare given a factorization system ( L , R ) in B then define the class of finitely presented left maps as L fp = L ∩ −−→B fp . This class inherits some closure properties of L : it contains identities, is still right-cancellative, and is stable by pushout along any morphism of finite presentation. Those propertiescan be axiomatized into the following notion: Definition 1.3.2. A saturated class in B is a class V of maps of finite presentation such that: − V is closed under composition − V contains all isomorphisms − V is right-cancellative − V is closed under pushouts along arbitrary morphisms in −→B fp − V is closed under finite colimits in −→B . Remark . For any class V of finitely presented morphisms, we can always consider the closureof V under the axioms above. Therefore the class V can be chosen already saturated and it willbe so by the following. Definition 1.3.4.
For any object B in B , define the etale generator at B as full subcategory V B of B ↓ B consisting of morphisms n : B → C such that there exists some l : K → K ′ in V and a : K → B exhibiting n as the pushout K K ′ B C a ln y Before going further, we should make mention of the following useful technical lemma from[3][sub-lemma 12], which is important for our theorem 1.3.7 and is also involved in the proof oftheorem 1.3.9 and theorem 4.1.10:
Lemma 1.3.5.
Let be a diagram as below K K ′ KB D kn aa y ith K, K , K ′ α -presented and k in V : then there exist a factorization K K B a a a such that a factorizes through the following pushout K K ′ K a ∗ K ′ KB D ka a a ∗ kn yy a Now we turn to the properties of the etale generator at a given object.
Proposition 1.3.6.
The etale generator V B is closed under finite colimits in the cocomma B ↓ B .Hence the category Ind( V B ) is locally finitely presented. For a proof of this statement, this for instance [27][Proposition 3.11]Before recalling how we can construct a factorization system from the data above, we give herethe following result, which relates the canonical cone of the codomain of an arbitrary etale arrowand the etale generators through a pseudocolimit construction. Recall first an arbitrary arrow l : B → C in Ind( V B ) defines a left exact functors ょ l : V opB → S . Then the category of elements R ょ l is cofiltered.Moreover, recall that, as a standard fact, we have an equivalence of categories Z ょ l ≃ V B ↓ l for the category of elements has pairs ( n, a ) with n in V B and a ∈ ょ l ( n ) = Ind( V B )[ n, l ] is anobject of V B ↓ l , while an arrow ( n , a ) → ( n , a ) consists in some m : n → n in V B such that a m = a . Moreover, observe that we have a pseudofunctor V cod( − ) : Z ょ op l → C at which sends any ( n, a ) ∈ R ょ l to V cod( n ) and any m : ( n , a ) → ( n , a ) to the pushout functor V cod( n ) m ∗ −→ V cod( n ) associating to a map n : cod( n ) → D in the etale generator V cod( n ) the pushout map m ∗ n :cod( n ) → m ∗ D . The latter is indeed in V cod( n ) , as if n was induced from a map k : K → K ′ in V along some a : K → cod( n ), then in the following diagram K cod( n ) cod( n ) K ′ D m ∗ D k na m m ∗ n y y the outer square is also a pushout.First, recall that the Grothendieck construction of a pseudofunctor is its oplax colimit, andthat the pseudolimit is constructed by localizing the oplax colimit at the opcartesian morphisms.For generalities about pseudocolimits of categories, see [29][Proposition 6.5] and also [9]. Theorem 1.3.7.
For any l : B → C in Ind( V B ) , we have a pseudocolimit V C ≃ colim −−−→ ( n,a ) ∈ R ょ l V cod( n ) roof. Ind( V B ) is a locally finitely presentable category where l ≃ colim V B ↓ l ; this colimit beingfiltered, it is preserved by the codomain functor, so that C ≃ colim ( n,a ) ∈ R ょ l cod( n ). From this weare going to produce a pseudococone( V cod( n ) a ∗ −→ V C ) ( n,a ) ∈ R ょ l where the a ∗ are the pushout functors; the pseudocommutativity of this diagram can be seen asfollows: for a morphism m : ( n , a ) → ( n , a ), that is such that n = mn and a m = a , and foran object n : cod( n ) → D in V cod( n ) we have by composition of pushouts the following diagramcod( n ) DB cod( n ) m ∗ DC a ∗ D n n ml a na ∗ n m ∗ n y a y where the canonical isomorphism a ∗ n ≃ a ∗ m ∗ n provides the value at n of the natural isomor-phism ensuring the pseudocommutativity of the triangle V cod( n ) V C V cod( n ) m ∗ a ∗ a ∗ ≃ Hence the universal property of the pseudocolimit returns a functorcolim ( n,a ) ∈ R ょ l V cod( n ) V C h a ∗ i ( n,a ) ∈ R ょ l where the pseudocolimit colim ( n,a ) ∈ R ょ l V cod( n ) is equivalent to the localization of the oplax colimitoplaxcolim ( n,a ) ∈ R ょ l V cod( n ) at the class of opcartesian morphisms which are the morphisms ( m,
1) :(( n , a ) , m ) → (( n , a ) , m ) coding for pushout squares exhibiting m as m = m ∗ m as below C cod( n ) cod( n ) D D m m m a a m ∗ m y We are going to prove this functor to be both essentially surjective and full and faithful. Letbe an object m in V C is induced as a pushout from some finitely presented etale map m in V K CK ′ D mbm y As K is finitely presentable, it factorizes through some a for ( n, a ) ∈ R ょ l as b = ac , and then wecan exhibit m as arising as the pushout of c ∗ m along a by right cancellation of pushouts K K ′ B cod( n ) c ∗ K ′ C D n cl am m y c ∗ m b y m = b ∗ m = a ∗ c ∗ m ; moreover, any such two lifts of b are identified in the canonicalcone of C , hence so are the induced pushouts of m . Hence the functor h a ∗ i ( n,a ) ∈ R ょ l is essentiallysurjective on objects.Now we want for any (( n , a ) , m ) , (( n , a ) , m ) we want an isomorphismcolim ( n,a ) ∈ R ょ l V cod( n ) [(( n , a ) , m ) , (( n , a ) , m )] ≃ V C [ a ∗ m , a ∗ m ]In one direction, to any ( m, s ) : (( n , a ) , m ) → (( n , a ) , m ) we can associate the unique map t ( m,s ) induced as below from the universal property of the pushout C cod( n ) cod( n ) a ∗ D a ∗ D D D m m a a st ( m,s ) y y m Observe by the way that this map is also obtained as the pushout t ( m,s ) = (( m ∗ m ) ∗ a ) ∗ h s, m i as below C cod( n ) a ∗ D a ∗ D D D a m ∗ m h s,m i t ( m,s ) y m where right cancellation of pushouts makes the bottom square a pushout and h s, m i is the mapuniquely induced by s .Suppose that we have ( m, s ) , ( m ′ , s ′ ) such that t ( m,s ) = t ( m ′ ,s ′ ) . Then for R ょ l is filtered, thereis some ( n , a ) in R ょ l such that m ′′ m = m ′′ m ′ . Then we have that m ∗ m ′′ sm = m ′′∗ m s m ′′ m = m ′′∗ m s m ′′ m ′ = m ∗ m ′′ s ′ m as seen in the following diagram C cod( n ) cod( n ) cod( n ) a ∗ D a ∗ D D D m ′′∗ D m a a st ( m,s ) = t ( m ′ ,s ′ ) y y m ′′ s ′ a m ′′∗ m y m m ∗ m ′′ mm ′ and therefore ( m, s ) and ( m ′ , s ′ ) are equalized in oplaxcolim ( n,a ) ∈ R ょ l V cod( n ) by the opcartesianmorphism ( m ′′ , m ∗ m ′′ ). But the later becomes invertible in the pseudocolimit, where the mor-phisms ( m, s ) and ( m ′ , s ′ ) are hence identified. Hence the faithfulness.Now in a a situation as belowcod( n ) C cod( n ) D a ∗ D a ∗ D D m m a a t y y if m is induced through pushouts from map k : K → K ′ in V as below, we get a composite arrow K cod( n ) C cod( n ) K ′ D a ∗ D a ∗ D D m a y ta k bm yy a , n ) in R ょ l and l : ( n , a ) → ( n , a ) such that we have afactorization K cod( n ) C cod( n ) cod( n ) K ′ D a ∗ D a ∗ D l ∗ D D m ta m k b l ∗ m l a c y y yy Then again by filteredness of R ょ l , there exists ( n , a ) and a span C cod( n ) cod( n ) cod( n ) a a a l l along which we can consider pushoutscod( n ) cod( n ) D cod( n ) l ∗ D l ∗ D C l ∗ D a ∗ D a ∗ D l l a m a ∗ m a ∗ m l ∗ m t y y y y a a But now for l ∗ D = l b ∗ K ′ , we have a universal map h c, l l ∗ m i : l ∗ D → l ∗ D , and moreoverin the diagram below cod( n ) Cl ∗ D l ∗ D a ∗ D a ∗ D a l ∗ m l l ∗ m t y y h c,l l ∗ m i a ∗ m a ∗ m the bottom square is a forced to be a pushout by right cancellation. This exhibits t as the inducedmaps t (1 n , h c, l l ∗ m i ) , and to conclude, observe that (( n , a ) , m ) and (( n , a ) , l ∗ m ) are relatedby an opcartesian morphism, while on the other hand (( n , a ) , m ) and (( n , a ) , l l ∗ m ) are alsorelated by an opcartesian morphism, and are hence identified in the pseudocolimit. Hence the pair(1 n , h c, l l ∗ m i ) can be seen as an antecedent of t in the homset above. Hence the fullness.Hence the functor colim ( n,a ) ∈ R ょ l V cod( n ) → V C , being at the same time essentially surjective,full and faithful, defines an equivalence of category as desired. Remark . Observe that the condition of right cancellation of the maps in V seems of no usein the result above, which could apply actually to any class of maps that is closed under pushoutalong arbitrary map. This theorem will be useful to get an expression of the spectral site of anabritrary etale arrow in theorem 4.1.10.Now we recall the construction of the factorization system as done in [3] and also [8]. We sawthat the etale generator at any object is closed under finite colimits. Then for any f : B → C ,the category V B ↓ f of diagonally universal morphisms under B above f is filtered. Moreover,recall that the codomain functor B ↓ B preserves filtered colimits. Now we can construct thefactorization of any arrow f in B : 13 roposition 1.3.9. An arbitrary arrow f : B → C in B admits a factorization B C colim D B ↓ f C f colim D B ↓ f r f with l f an axiomatizable diagonally universal morphism and r f is in V ⊥ . See [3][section 2.3] which contains the full proof of the statement [theorem 14], and also [8] -which however did not provided proof of it.
Definition 1.3.10.
A factorization system is left generated if it is of the form (Ind( V ) , V ⊥ ). Inparticular, the factorization of an arrow is obtained as a filtered colimit as in theorem 1.3.9.For a proof the the next statement, see [27][Proposition 3.13] for instance: Proposition 1.3.11.
The factorization hence obtained is left generated, that is,
Ind( V ) = ⊥ ( V ⊥ ) .Remark . Saturated classes and factorization systems on a locally finitely presentable cat-egory B form posets Sat B and F act B with inclusion as wide subcategory of B fp for Sat B andinclusion of the left class for F act B . Saturated class and factorization systems in locally finitelypresentable categories are related through an adjunction: Sat B F act B V (Ind( V ) , V ⊥ ) L ∩ −→B fp ( L , R ) ⊣ where Sat B ֒ → F act B is a mono because for any saturated class V = Ind ( V ). On the converse, itis not true that a factorization system is generated from a saturated class, so that in general weonly have ( Ind ( L ∩ −→B fp ) , ( L ∩ −→B fp ) ⊥ ) ≤ ( L , R )so that in particular R ⊆ ( L∩−→B fp ) ⊥ . However in the following we are going to restrict our attentionto the factorizations systems having this property ; the general case will however be met againwhen dealing with Diers contexts.Left-generated factorizations systems in locally finitely presentable category are in some sensefactorization systems that can be axiomatized by a finite limit theory. Indeed, from the axioms ofa saturated class, the category V is closed under finite colimits. But as a consequence its inductivecompletion Ind( V ) is a locally finitely presentable category. In fact the inclusion V ֒ → −→B fp preserves finite colimits; but for −→B ≃ Ind( −→B fp ) we have locally finitely presentable functor B →
Ind( V )sending each arrow f to its left part l f . Hence the category of left maps in a left generated systemis a locally finitely presentable category; the corresponding finite limit theory has the category V op as syntactic site, and d V op as classifying topos.In section 3 we shall see that the class of local maps Loc is also axiomatizable and admits aclassifying topos. But for now, we turn to the notion of admissibility in the case of set-valuedmodels.
The central condition that enables the construction of the spectrum is a relation entanglingfactorization and geometric extension. It can be presented in several equivalent ways. A firstexplicit axiomatization was 14 efinition 2.0.1. An admissibility structure in the sense of [19][definition 6.57] is the data of − a finite limit theory T − a geometric extension T → T ′ − a 2-functor Loc : GTop op → Cat such that for each E one has Loc [ E ] ֒ → −−→ T [ E ] and Loc [ E ]contains isomorphisms and enjoys left-cancellability. − and such that we have the condition that for any E in GTop and any arrow f : F → E in T [ E ] with E in T ′ [ E ] there exists a factorization in T [ E ] F EH ffn f u f with u f in Loc [ E ] and H φ in T ′ [ E ], which is moreover initial amongst all factorizations of f with an arrow in Loc [ E ].However this definition, while encapsulating in the last condition the key feature that willproduce geometry, may seem complicated at first sight. In this section, we are going to prove thatit is achieved by the following data, which have been variously called admissibility triples in [8]and geometry in [24] (in the context of ∞ -categories, and with geometric conventions); we shallretain this later name for its evocative virtue: Definition 2.0.2. A geometry is the data of − a finite limit theory T − a saturated class V in T [ S ] − a Grothendieck topology J on C T whose covers are generated by duals of maps in V .In a close manner the notion of Nisnevich context was introduced in [3], although this definitiondoes not rely exactly on the same data and is slightly less constricted (beside it is formulated alsoin the geometric convention, while we follow here the algebraic convention):
Definition 2.0.3. A Nisnevich context is the data of a factorization system ( Et , Loc ) on a locallyfinite presentable category B and a class of objects L . Then one defines the L -localizing families as the coarsest system of covers ( f i : B → B i ) i ∈ I exhibiting L as local objects.A Nisnevich context is said to be compatible if L is exactly the class of J L -local objects.Finally, a Nisnevich context is said to be good if it is compatible and the factorization systemis left generated. Remark . Beware that there is in general no condition for the localizing topology to begenerated from etale maps: this generalization was used for instance to capture geometric situationas the Nisnevich topologies that are not generated from factorization data.In some sense, Nisnevich contexts arise when the local objects are given before any axiom-atization through a geometric extension, and it may happen that the least geometric extensionsufficient to axiomatize them has more models than the local objects we started with. In section2.2 we will meet similar situation, but from the point of view of the fourth way to axiomatize thosesituations, that are
Diers context . However, we do not give their definition right now for we shalldevote section 2.2 to their relation with those approaches from which they differ significantly.
In the following we deploy the information encoded into the notion of geometry and how it issufficient to achieve admissibility; in this subsection we focus on the set-valued models and explainhow the topological interpretation of left and right map, and of the points of the topology.Throughout this section, we fix a geometry ( T , V , J ). Following [3] terminology, we shall callmaps in Ind( V ) etale maps , and denote their class as Et , while the maps in V ⊥ will be called localmaps and their class denoted as Loc . They form a factorization system as ( Et , Loc ). Models of15he geometric extension T J of 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 T [ S ].Recall that a model of T J is an object in T [ S ] which is local relatively to the dual in T [ S ] fp ofcovering families in J . But the families dual to cover in J can be extended to the whole category T [ S ] as follows: Definition 2.1.1.
Define the generalized J -covers as consisting, for each B in T [ S ], of the families( n i : B → B i ) i ∈ I such that there exists some a : K → B and some family ( l i : K → K i ) i ∈ I dual toa J -cover such that for each i ∈ I one has a pushout B KB i K if i k i ∈V q Remark . For we supposed J to be a Grothendieck topology, it will be closed under pullbackin C T . Hence in T [ S ] fp , pushouts of duals of J -covers are still J -covers, hence the generalizedcoverage does not create new covering families under finitely presented objects.Then the property of injectiveness of local objects relatively to J -covers extends automaticallyto those extended covers: Proposition 2.1.3.
An object A in T [ S ] is J -local if and only if for any object B in T [ S ] , anyarrow f : B → A and any generalized J -cover ( n i : B → B i ) i ∈ I , there is a factorization of f forsome i ∈ I B AB i ... B jn i n j f ∃ In fact A is J -local if and only if for any generalized cover under it ( n i : A → B i ) i ∈ I , one has aretraction A AB i ... B jn i n j ∃ Proof.
In one sense it is obvious that A is injective relatively to the generalized covers, then inparticular 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 ∈ I we have a factorizationthrough some n i as below K B AK i B ia a ∗ n i n i f y and then an arrow B i → A factorizing f by the property of the pushout. Now, if for any generalizedcover of A , A is a retract of a member of this cover, then in particular for any ( n i : K → K i ) i ∈ I in J and any a : K → A , the pushout of the n i under A give a generalized cover of A and we havea factorization for some i ∈ I K A AK i B ia a ∗ n i n i y so that A is J -local. Definition 2.1.4.
For an object B in T [ S ], a local form of B is an etale map n : B → A towarda J -local object. Beware that local forms are not required to be finitely presented in general. Remark . Etale maps will play the role of saturated compacts of the spectral topology, whilefinitely presented etale maps will play the role of basic compact opens from which we are goingto construct the spectral topology. While this is not apparent in T [ S ] which is “on the algebraic16ide”, this is more intuitive on the opposite category T [ S ] op , whose objects should now be thoughtas spaces, where the etale morphisms could be seen as generalized inclusions, and the generalizedcovers 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 ∈ X , the focal component of X in x is theintersection of all neighborhoods of x , and this is the upset ↑ x in the specialization order. Localforms behave like inclusions of the form ↑ ⊑ x ֒ → X as such upsets are unreachable by open cov-ering: indeed, in a cover of ↑ ⊑ x , one open must contain x itself. But as open are up-sets for thespecialization order, this open is the whole ↑ ⊑ x . Hence maximal points, as they do not admit nontrivial local forms, are alike those x such that ↑ ⊑ x = { x } .In particular, triangles between local forms B A A x nx should be seen as coding for specialization order between the corresponding minimal point x ≤ x .Then in T [ S ] op this will be turned into an inclusion of focal component ↑ x ⊆↑ x . Lemma 2.1.6.
Any object A admitting local morphism into a local object u : A → A is itself alocal object.Proof. If ( n i : A −→ B i ) i ∈ I is a point covering family of A then its pushout along u is a pointcovering family for A hence admits a lifting r for some i , so we have a square that diagonalizesbecause u is local and n i is in V A : A AB i A n i ur ∃ Corollary 2.1.7.
For any arrow f : B → A in T [ S ] with A a J -local object, the ( Et , Loc ) factorization of f B AA ffn f u f returns a J -local object A f Now we want to understand how this admissibility condition induced from a geometry is in-herited in arbitrary toposes.For a geometry ( T , V , J ), with ( Et , Loc ) the associated factorization system, the class of finitelypresented etale maps V in T [ S ] fp is dual to a class of morphisms in the syntactic category C T . Thatis, an arrow n : K φ → K ψ (with φ , ψ the presentation formulas of the domain and codomains) in V corresponds to an arrow [ θ n ( x, y )] T : { y, ψ } → { x, φ } , which, as a 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 V op has finite limits, hencecodes for a finite limit theory which admits as classifying topos d V op = [ V , S ] which we denote as S [ Et ]. In particular this allows us to define for each Grothendieck topos E a class of arrows Et [ E ]in T [ E ] as Et [ E ] ≃ Geom (cid:2) E , S [ Et ] (cid:3) ≃ Lex [ V op , E ]Let us look at the counterpart of this for the right class Loc of local maps.
Proposition 2.1.8.
An arrow u : A → B is in Loc if and only if it satisfies the followingcondition:if ( ∀ n : K φ → K ψ in V∀ a ∈ A ( { x, φ } ) ∀ b ∈ B ( { y, ψ } ) such that B | = θ f ( u ( a ) , b ) then ∃ ! c ∈ A ( { y, ψ } ) ( A | = θ f ( a, c ) u ( c ) = b roof. First, note that the situation expressed by the conditions in the characterization justamounts to the commutativity of the following square K φ AK ψ B n p a q u p b q as the composite u ◦ p a q corresponds to the element u ( a ) while the composite p b q ◦ n expresses thefact that B | = φ f ( τ [ b / y ] , ..., τ n [ b / y ])and the condition B | = θ f ( u ( a ) , b ) just means that g ( a ) = τ [ b / y ] ∧ ... ∧ g ( a n ) = τ n [ b / y ]hence that the square commutes.Consequently, if u is V local, such a square always diagonalizes uniquely: the diagonal providesus with a witness c of ψ f in A , and the commutation of the down-right triangle expresses theequality u ( c ) = b while the up-left one expresses that in A we have a = τ [ c / y ] ∧ ... ∧ a n = τ n [ c / y ]If conversely u satisfies the syntactic characterization, then the map g defined on the generatorsas g ( y ) = c , ..., g ( y m ) = c m provides a diagonalization K φ AK ψ B p a qp b q n ug Remark . In this context, as explained by Coste, local morphisms are those which “reflect” el-ements produced through the operation coded by finitely presented etale arrows. While morphismsin V “produce witnesses of their codomain formula from witnesses of their domain formula”, localmorphisms do not add new witness of propositions presenting the codomain, and reflect the onethat already exists.Observe that the condition above defined a cartesian sequent for the existential in the syntacticcharacterization 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 notonly by a geometric, but in fact a cartesian theory. Remark . The condition above says in particular by Yoneda lemma that T [ S ] (cid:2) K ψ , A (cid:3) T [ S ] (cid:2) K ψ , B (cid:3) T [ S ] (cid:2) K φ , A (cid:3) T [ S ] (cid:2) K φ , B (cid:3) y But observe that T [ S ] (cid:2) K ψ , A (cid:3) ≃ Lex [ C T , S ] (cid:2) ょ ∗{ y,ψ } , A (cid:3) ≃ A ( { y, ψ } )and similarly for the other homset. This last formulation still makes sense even if we considerleft exact functors with value in other toposes than S : and as we are going to deal with sheavesof local objects in different Grothendieck toposes, this motivates the following generalization ofadmissibility amongst T [ S ] in arbitrary Grothendieck toposes. Definition 2.1.11.
For a Grothendieck topos E , a local arrow in E is a natural transformation V op E FEu
Lex [ V op , E ] whose naturality square at a morphism [ θ n ( x, y )] T : { y, ψ } → { x, φ } dual of amorphism n in V is a pullback in E F ( { y, ψ } ) E ( { y, ψ } ) F ( { x, φ } ) E ( { x, φ } ) F ([ θ n ( x,y )] T ) u { y,ψ } E ([ θ n ( x,y )] T ) u { x,φ } y In the following we denote as
Loc [ E ] the class of local morphisms in E . Proposition 2.1.12.
Local morphisms are stable under inverse image: any geometric morphism f : F → E induces a functor f ∗ : Loc [ E ] → Loc [ F ] .Proof. The inverse image f ∗ : E → F is lex, hence preserves pullback. Then at each morphism n in V , f ∗ sends the pullback above in E to a pullback in F . Hence f ∗ u is local.Now, recall form section 1 that models of finite limit theories in sheaf toposes were sheaves ofset-valued models over the base site. In particular this suggests the following: Proposition 2.1.13. If E ≃ Sh ( C E , J E ) , then a transformation u : F → E in E is a localtransformation if and only if for any object c in C , u ( c ) : F ( c ) → E ( c ) is in Loc .Proof.
For each c then u ( c ) : F ( c ) → E ( c ) is an arrow in B and thus can be seen as a naturaltransformation between Ind-objects. Conversely for any f : A → B of finite presentation, α f : F ( A ) → F ( B ) is a morphism of sheaves on ( C E , J E ) ; as evaluation of sheaves preserves limits, each u ( c ) is local as a morphism in B if and only if the following square is a pullback F ( { y, ψ } )( c ) E ( { y, ψ } )( c ) F ( { x, φ } )( c ) E ( { x, φ } )( c ) F ([ θ n ( x,y )] T ) c u { y,ψ } c E ([ θ n ( x,y )] T ) c u { x,φ } c y 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 V in T [ S ] is inheritedin the category T [ E ] in any Grothendieck topos, and moreover in a functorial way: Proposition 2.1.14.
For any Grothendieck topos E of the form Sh ( C E , J E ) , we have a factorizationsystem ( Et [ E ] , Loc [ E ]) in T [ E ] . Moreover, for any f : F → E we have adjunctions Et [ E ] Et [ F ] f ∗ f ∗ ⊣ Loc [ E ] Loc [ F ] f ∗ f ∗ ⊣ Proof.
In fact the desired factorization is pointwise relatively to the definition site of E . Let be f : F → E in T [ E ]. For any c in C E , we have a morphism f c which admits a factorization in T [ E ] F ( c ) F ′ ( c ) H f c f c n fc u fc With n f c etale and u f c local in T [ S ]. As this defined a functorial factorization structure, this processis itself functorial and defines a presheaf over C E . The desired object is just its sheafification H f = a J E ( c H f,c )while the data of ( n f c ) c ∈C E , ( u f c ) c ∈C E defines the desired etale and local transformation in anunique way. Remark . The functoriality of this result might look surprising at first sight. If it is expectedthat inverse image, as left adjoints, preserves etales maps for they are a left class, and dually, thatdirect image, as right adjoints, preserve local maps for they are a right class, the preservation oflocal maps by etale maps by inverse image and etale map by direct image are consequence of thespecific fact that we are in a left generated factorization system, exhibiting both etale and localmaps as models of finite limit theories, hence stable either under direct and inverse images.19ence the factorization structure is inherited by the category of models in any Grothendiecktopos. But the admissibility structure itself is inherited.
Proposition 2.1.16.
Let be E a Grothendieck topos and u : F → E in Loc [ E ] with E in T J [ E ] :then F itself is in T J [ E ] .Proof. The inverse image part F is in particular a Lex functor from ( B fp ) op into E . Hence wehave to prove that F is also J continuous. Let ( n i : { x i , φ i } → { x, φ } ) i ∈ I ∈ J : those are finitelypresented maps in the saturated class V , hence as f is a local transformation, the naturality squarein each k i expresses each F ∗ ( { x i , φ i } ) as a pullback F ( { x i , φ i } ) F ( { x, φ } ) E ( { x i , φ i } ) E ( { x, φ } ) F ([ θ i ( x,x i )] T ) u { xi,φi } E ([ θ i ( x,x i )] T ) u { x,φ } y As those inverse image lands in the topos E , where, as in any Grothendieck topos, colimits arestable by pullback, one has a i ∈ I F ( { x i , φ i } ) ≃ a i ∈ I (cid:16) E ( { x i , φ i } ) × E ( { x,φ } ) F ( { x, φ } ) (cid:17) ≃ (cid:16) a i ∈ I E ( { x i , φ i } ) (cid:17) × E ( { x,φ } ) F ( { x, φ } )But E , as a local object, transforms covers into jointly epimorphic families. Hence the lower arrowof the corresponding pullback square is an epimorphism ` i ∈ I F ( { x i , φ i } ) F ( { x, φ } ) ` i ∈ I E ( { x i , φ i } ) E ( { x, φ } ) F ([ θ i ( x,x i )] T ) h u { xi,φi } i i ∈ I y u { x,φ } E ([ θ i ( x,x i )] T ) But in Grothendieck topos, pullback of epi are epi, then the upper arrow in this square in also anepimorphism and this confirms that F also is a local object.This says that for any Grothendieck topos E , the category of T -models in E inherits the admis-sibility structure defined by the geometry ( T , V , J ) Corollary 2.1.17.
Let be E be a Grothendieck topos: then for any f : F → E in T [ S ] with E in T J [ E ] , then in the ( Et [ E ] , Loc [ E ]) -factorization F EH ffn f u f the middle term is in T J [ E ] . This proves that a geometry as defined produces an admissibility structure. However theconverse is not clear, for the functor
Loc [ − ] may not be deduced to be representable from thehypothesis.Now some precision on the geometric aspects of local objects: in fact they behave exactly asexpected in the sense that they form sheaves of B objects over the definition site of their base toposand have set valuated local objects as stalk: Proposition 2.1.18.
The stalks of a local object F are set-valuated local objects. Evaluated atpoints, local transformations between local objects return admissible maps between set valuated localobjects. All of this justifies the terminology for local objects and local transformations as our objectsof interest behave locally, at points, as such objects.20 .2 Stable functors and multi-reflective subcategories
In this section we focus on an alternative way to encapsulate admissibility, relating in particularthe factorization condition involved in theorem 2.0.1 to a more general class of conditions,
Definition 2.2.1. A local unit for a functor U : A → B is a morphism n : B → U ( A ) such thatfor any square of the following form there exists an unique morphism w : A → A such that U ( w )diagonalizes uniquely the square and the left triangle already commutes in A B U ( A ) A U ( A ) U ( A ) A A fn U ( v ) vU ( u ) U ( w ) ∃ ! wu Definition 2.2.2.
A functor U : A → B is stable when any morphism f : B → U ( A ) factorizesuniquely through the range of U as B U ( A ) U ( A f ) fn f U ( u f ) where n f : B → 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 . Definition 2.2.3. A right multi-adjoint is a stable functor U : A → B such that for each object B , the set of local units η : B → U ( A ) under B is small. Then we denote as X B this set of localunits.A multireflective subcategory is a (non necessarily full) subcategory U : A ֒ → B whose inclusionfunctor is a right multi-adjoint.A typical example of non full multireflection is the following as pointed out in [31]: Proposition 2.2.4.
For any unique factorization system ( L , R ) , seeing R as a category from itis stable by composition and contains isomorphisms, the inclusion R ֒ → B is a stable functor.Conversely any stable functor which is surjective on objects and faithful is of this form.Proof. If we have a factorization system, any morphism f : B → ι R ( A ) factorizes uniquely as ι R ( r ) l and l is orthogonal to any morphism in R so we can see it as the desired candidate.If U : A → B is stable, faithful and surjective on objects, any object is some U ( A ) and any f : B → U ( A ) factorizes through a candidate which is left orthogonal to the morphisms in therange of U , hence the class of candidates constitutes the left part and the morphisms in U ( −→ A ) theright part.The following terminology was suggested by Anel, and was also identified in [11][part 4] amongstcondition to produce a spectral construction: Definition 2.2.5.
Let be R a class of maps in a category and A a class of objects. We say that A has the glidding property relatively to R if for any arrow l : B → A in R with A a n object of A , then B must also be in A . Theorem 2.2.6.
Let B be a category equiped with a factorization system ( L , R ) and A be a classof objects of B with the gliding condition relative to R , that is, such that any object A admitting anarrow in R toward an object of A is itself in A . Then the inclusion A R ֒ → B of A objects equipedwith arrows of R between them defines a relatively full and faithful stable functor.Proof. This follows from the previous proposition: for any B in B and any arrow f : B → A with A an object in A , as there exists a unique factorization B n f → A f u f → A with n f in E and u f in R , then A f is also an object in A ; this factorization is initial amongst those through a morphismin R on the right. Moreover n f is a morphism in L with an objects in A as codomain, and such21rrows are exactly the candidates for the inclusion as for any square with A , A , A objects in A ,we have the diagonalization B A A A n u ∃ du ′ But recall that R is left-cancellative as any right class, so that u must itself be in R . Hence inthe factorization above n f is a candidate and the inclusion is stable; left-cancellativity of R alsoenforces that this inclusion is relatively full and faithful. Corollary 2.2.7.
Let be ( T , V , J ) a geometry. Then we have a stable inclusion T J [ S ] Loc ֒ → T [ S ] More generally, for any Grothendieck topos, we have a stable inclusion T J [ E ] Loc [ E ] ֒ → T [ E ] In both case, this inclusion is relatively full and faithful.
Hence geometries define stable inclusions, and moreover this stability property is inherited tocategories of models in arbitrary toposes, producing topos-wise stable inclusions.Now let us examine the converse process. We start from a stable functor U : A → B with B locally finitely presentable. We are going to construct a geometric axiomatization of a stablefunctor by inducing a left generated factorization system and a geometric extension of the finitelimit theory T underlying B , or equivalently, by defining a Grothendieck topology on B op fp . Thiswill produce a notion of local objects and local maps in B encompassing the objects and maps inthe 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 alreadyimplicit in [11] and we recalled it in [27][section 3] using [ ]. Definition 2.2.8. A diagonally universal morphism for U is a morphism f : B → C in B suchthat we have a lifting in any square of the form B U ( A ) C U ( A ′ ) gf U ( u ) h ∃ ! We denote as
Diag U the class of diagonally universal morphisms for U . We have Diag U = ⊥ U ( −→A ). Remark . Note that diagonally universal morphisms with codomain in the range of U arenot necessarily candidates as the factorization needs not come from a (unique) morphism in 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 A . Then candidates coincidewith diagonally universal morphisms with codomain in the range of U .Now we have an orthogonality structure ( Diag U , Diag ⊥ U ) in B : however this is not yet a fac-torization system, for the class of diagonally universal morphisms in not necessarily accessible. Toconstruct a factorization system, we must restrict ourselves to diagonally universal morphisms thatare filtered colimits of diagonally universal morphisms of finite presentation. Remark . Observe that in the context of a stable functor, the class ⊥ U ( −→A ) ∩ B fp is al-ready a saturated class from it inherits the closure properties of U ( −→A ) which is a left class in anorthogonality structure. Definition 2.2.11.
We define the class Et U of U -etale morphisms as Et U = Ind( Diag U ∩ −→B fp )22they were called “axiomatizable diagonally universal morphisms” in [27]). In particular Et U ⊆ Diag U . We define the class Loc of U -local morphisms as Loc U = ( ⊥ U ( −→A ) ∩ −→B fp ) ⊥ In particular U ( −→A ) ⊆ Diag ⊥ U ⊆ Loc U .By section 1.3 we know that ( Et U , Loc U ) is a factorization system on B .In [11] were isolated conditions enabling the construction of a point-set notion of spectrum. Wesum up those conditions under the following notion: Definition 2.2.12.
Define a Diers context as the data of − a locally finitely presentable category B− a right multi-adjoint functor U : A → B with B− satisfying Diers condition : any local unit is the filtered colimit of diagonally universal mor-phism of finite presentation above it.
Remark . 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 alreadyseen how it produces a left generated factorization system; now we have to construct canonicallya topology associated to it. Definition 2.2.14.
For C a class of cones − The cone-injectivity class for C is the class C ⊥ of objects A such that for any cone ( f i : B → B i ) i ∈ I in C , any f : B → A from the submit of the cone factorizes through some arrow f i inthe cone. − The cone-orthogonality class for C is the class C inj of objects A such that for any cone( f i : B → B i ) i ∈ I in C , any f : B → A from the submit of the cone factorizes uniquelythrough exactly one f i in the cone. Remark . As detailed in [2], if 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 B , while the corresponding orthogonality class is a full multireflexive subcategory of B .Moreover, it should be pointed out that in an admissibility structure, the full subcategoryof local objects, which are the models of a geometric extension, have no reason in general to bemultireflective in the ambient locally finitely presentable category, though it should be closed underfiltered colimits. It can be shown that the full, accessible embedded multireflective subcategories oflocally finitely presentable categories are models of disjunctive extensions, that are theories whosenon cartesian sequents are made at most from strict disjunctions. Though it encompasses certainprominent examples, our situations of interests are not restricted to disjunctive extensions, hencemultireflectivity of the class of local objects should in general appear only after restriction to localmaps.However in a Diers context the considered stable functor usually does not define a full subcat-egory. It is so exactly when the specialization order between local objects is trivial. But in thegeneral case, we have to distinguish the considered non full subcategory defined by the essentialimage of U from the full multireflexive subcategory given by the class of cone orthogonality it isexpected to generate. Remark . Consider the class of cones of local units under arbitrary objects: C U = { ( B η A ( f ) → U L A ( f )) B f → U ( A ) ∈ B ↓ U | B ∈ B} If we denote ( B η i → U ( A i )) i ∈ I B the cone of (isomorphism classes of) all local units of B , thenit generates in some way a cocoverage in B in the sense that for any f : B → B , one has for any23 ∈ I B the following factorization B B U ( A η i f ) U ( A i ) fn η i f η i U ( u η i f ) However the cones of local units are not stable by pushout: indeed, for i ∈ I B even though the f ∗ η i : B → B + B U ( A i ) is still diagonally universal, its codomain needs not be in the rangeof U . But pushouts of cones of local units are still covering families in the sense that ”they stillcover the local units” of the codomain. By this we mean that all its local units factorize throughone of those pushouts at least. This results just from the property of pushout as visualized in thefollowing: B B U ( A η i f ) B + B U ( A i ) U ( A i ) fn η i f f ∗ η i η i U ( u η i f ) ∃ p This must be generalized by considering cones of etales maps jointly factorizing local unitsunder a given objects. Those were considered in [10].
Definition 2.2.17.
For a stable functor U : A → B , we can consider the U -localizing families asthe families of U -etale morphisms jointly factorizing local units of U , that are all the families ofthe form ( B n i → B i ) i ∈ I with n i in Et B such that for each local unit x ∈ X B there is some i ∈ I providing a factorization B U ( A x ) B iη x n i ∃ In particular one can define U -local objects as those A such that for any B and any ( n i : B → B i ) i ∈ I in Et U such that for any f : B → A and any ( f i : B → B i ) i ∈ I U -localizing family we have afactorization of f through some f i . Proposition 2.2.18.
For any object A in B , the following are equivalent:1. A is U -local,2. A is a retract of an object in the range of U ,3. A is in C injU
4. The ( Et U , Loc U ) factorization of any f : B → A returns a retract of an object in the essentialimage of U .Proof. 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 ∈ I A : A AU ( A i ) n i ∃ Now if A is a retract of some U ( A ) with 1 A = rs , for any f : B → A the factorization of sf provides a factorization of f itself through rU ( u sf ): B A AU ( A sf ) U ( A ) fη sf s A U ( u sf ) r
24o that A is cone injective.Now, if A is cone injective and ( B → B i ) i ∈ I is a cover, any f : B → A lifts through some unit n i : B → 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 B AA ffn f u f be the ( Et U , Loc U ) factorization of f : B → A ; then A f itself possesses a cone of local units underit, which is in particular trivially a U -localizing family: hence the local part u f of the factorizationitself lifts through one of them as A is U -local. But now the factorization of the lift returns thesame object as the factorization of f by uniqueness of factorization, as we have a chain of diagonallyuniversal maps on the left: B A A ∃ A f U ( A i ) fn f u ∃ u f n i ∈ Diag U n ∃ ∃ 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 retractof an object in the range of U , then for f : B → A we can compare the left part of the factorizationof A with the local unit C f B A AU ( A sf ) U ( A ) u f η suf l f η sf f sU ( u sf ) r but now the following square below admits some filler C f C f U ( A sf ) C η suf u f rU ( u sf ) exhibiting C f as a retract of U ( A sf ). Proving that the last item implies the second is trivial byapplying the result the identity of A Finally, suppose that any arrow toward A factorizes through a retract of an object in the rangeof U . Then for any f : B → A and any U -localizing family ( n i ) i ∈ I of B , we have a factorizationas below B AC f C f C i U ( A ) fn f n i s u f r ensuring that A is U -local. Proposition 2.2.19.
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 orthogonalto 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.Proof. For a retract of a map in the range of U and an arbitrary square as in the following with n in ⊥ ( U ( −→ A )): C A AU ( A ) C A ′ A ′ U ( A ′ ) n sf fU ( u ) rs ′ r ′ the composite square with n on the left and U ( u ) on the right admits a lifting, whose compositewith 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 aretract, then consider the factorization of the composite map, as in the following: A AU ( n s ′ f ) A ′ A ′ U ( A ) f n s ′ f fU ( u s ′ f ) s ′ r ′ This provides us with the following square
A AU ( A s ′ f ) A ′ n s ′ f fr ′ U ( u s ′ f ) whose diagonalization exhibits both A as a retract of U ( A s ′ f ) and f as a retract of U ( u s ′ f ). Corollary 2.2.20.
The U -local objects and U -local maps between them form a (non full) multire-flexive subcategory U - Loc
Loc U ֒ → B This means that for any stable functor, the closure by retracts of its essential image image ismultireflexive. Its full closure in B is the cone injectivity class C injU . 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 theinjectivity condition involved are too constraining. Indeed, the U -localizing families were definedunder 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 thegenerator of finitely presented objects, and hence we cannot ensure the resulting class to be theclass of models of a geometric extension.To produce the desired admissibility structure, we have to restrict the localness test relativelyto finitely presented object. Hence we have to restrict the U -localizing families to the finitelypresented objects and the finitely presented etale morphisms under them: Definition 2.2.21.
Define the U -localizing topology J U on B op fp as having as covering families theduals of the families ( n i : K → K i ) i ∈ I in Et fp such that for each x ∈ X K , there is some i ∈ I anda factorization K U ( A x ) K iη x n i ∃ J U codes for a geometric extension of the finite limit theorywhose category of set-valued models is T : Definition 2.2.22.
We denote as T U the corresponding geometric extension, and call it the geo-metric axiomatization of U . Remark . Observe that any U -local object is trivially J U . hence is a model of the T U . Thisis 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 T U -modelsthat fail to be J U -local. In that sense the notion defined in [10] is not yet sufficient to axiomatizeour local data in an universal way. Indeed, while J U -local objects will be points oThe rest of this section is devoted to prove that the data of ( J U , Et U ) define an admissibilitystructure with T U [ S et ] as the local objects.Any object in the essential image of U is in particular a model of T . However, not only there areadditional U -local objects, but there are even more T U -models. We have the following: Proposition 2.2.24.
We have the following: − The subcategory T U [ S et ] is closed in B by retracts. − T U -models have the gliding property relatively to local maps.Proof. For the first item, the proof is the same as for retract of objects in the range of U : if A isa retract of a T U -model A then for any f : K → A and covering family of K in J U | fp one has K A AK i A fη sf s A ∃ r So that A itself is J U | fp -flat by Yoneda lemma, hence is a T U -model.For the second item, let u : A → A be local with A a T U -object. Then for any f : K → A and covering family of K in J U | fp the composite uf : K → A extends through some n i ; but asthose n i are in Diag U , this extension induces a diagonalization K AK i A fn i u ∃ Hence A is a T U -model. Corollary 2.2.25. T U -models and local maps between them form altogether a (non full) multire-flective subcategory T U [ S et ] Loc ֒ → B In particular the full subcategory T U [ S et ] can be seen as a geometric envelope of the functor U .Remark . Observe that A is full in J U − Loc
Loc and T LocU as for a local map u : U ( A ) → U ( A ) 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 B arealso multireflective U ( A ) A U - Loc
Loc U B T U [ S ] Loc U eso,full Remark . More generally, one could ask, for a topology J , why it is worth to restrict us to acertain class of local maps between J -local objects rather that considering just general maps. Thereason is that etale maps would have to be iso, although cover generating J are not, so this wouldnot define an admissibility structure. 27rom what was said, it seems we can formalize the correspondence between Diers contexts andadmissibility structures. Definition 2.2.28.
For a fixed locally finitely presentable category B , we can either define a posetof Diers contexts on B and a poset of admissibility structures on B equiped with the respectiveorders: − ( U : A → B ) ≤ ( U : A → B ) if A BA U U − ( V , J ) ≤ ( V , J ) if V ⊆ V (so that Et V ⊆ Et V and Loc V ⊆ Loc V ) and J ≤ J . Hencein particular ( V , J ) is an admissibility structure.We denote by Diers B and Coste B those respective poset. Proposition 2.2.29.
We have an adjunction between Coste and Diers contexts on B Coste B ⊥ Diers B ( V , J ) T J [ Set ] Loc V ֒ → B ( Diag U ∩ −→B fp , J U ) ←− [ A U → B 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 givenadmissibility context itself induces the same admissibility by this process. However it is obviousthat such a thing can be true only if J has enough set-valued models, otherwise the induced stableinclusion may be void. Proposition 2.2.30. If ( V , J ) is an admissibility where J has enough points, then, the stableinclusion T J [ Set ] Loc ֒ → B induces an admissibility equivalent to ( V , J ) . In the previous section, we have seen that an admissibility structure generates a toposwisestable functor which can be seen as a situation of local adjunction. In the following we fix anadmissibility structure ( V , J ) for a finite limit theory T .We have a pseudofunctor, which is representable by S [ T ] GTop op Cat T [ − ] and is defined as follows: − for 0-cells, it returns the category T [ E ] of T -models − for 1-cell f : F → E , it returns the inverse image functor T [ E ] f ∗ −→ T [ F ]which is moreover lex and cocontinuous, for it is left adjoint to the direct image functor T [ F ] f ∗ −→ T [ E ] − on 2-cells α : f ⇒ g it returns the natural transformation also denoted as α T [ E ] T [ F ] f ∗ g ∗ α R T [ − ] GTop p T This bicategory can be considered as a category of all models of T regardless of their base topos.An object in this category is a modelled topos , that is, a pair ( E , E ) with E a Grothendieck toposand E in T [ E ]. However in the following, we choose to work with an algebraic convention in thesense that we want morphisms between modelled toposes to have the orientation of the morphismsof models rather than the orientation of the underlying geometric morphism. To this end, we usethe following, which is nothing but the direct fibration associated to T [ − ]: Definition 3.0.1.
The bicategory T - GTop of T -modelled toposes has − for 0-cells, modelled toposes ( E , E ) − for 1-cells, ( f, φ ) : ( F , F ) → ( E , E ) with f : E → F a geometric morphism and φ consistingof a pair ( φ ♭ , φ ♯ ) with f ∗ F f ♭ −→ E and F f ♯ −→ f ∗ E mates along the adjunction f ∗ ⊣ f ∗ − for 2-cells α : ( f, φ ) → ( g, ψ ), 2-cell α : f → g in GTop .From its construction, this bicategories is equiped with an obfibration p T : T − GTop → GTop op Now the local data associated to ( V , J ) also define an pseudofunctor GTop op Cat T J [ − ] Loc and again we can consider the associated direct 2-fibration; but for each topos E , we have a nonfull inclusion of category T J [ E ] Loc [ E ] ֒ → T [ E ]inducing the following non full sub-bicategory of T - GTop defined:
Definition 3.0.2.
The category T Loc J - GTop of T J -locally modelled toposes has − for 0-cells pairs ( E , E ) with E in T J [ E ] − for 1-cells pairs ( f, φ ) : ( F , F ) → ( E , E ) with f : E → F and φ is such that φ ♭ : f ∗ F → E isin Loc [ E ] − and the same 2-cells as T - GTop .In particular we have this inclusion is a strict morphism of opfibration T Loc J - GTop T - GTopGTop op ι J, Loc p J, Loc p T =However beware that p J, Loc does not inherit the fibration structure of p T for T J models arenot stable under direct images of geometric morphisms.In the following section, we construct the spectrum as the left adjoint of this inclusion ι J, Loc
In this section we present Cole construction of the spectrum as a sequence of universal con-struction exhibiting it as the classifier of the local form under a given model in a topos. Howeverwe try to provide a description as detailed as possible of the process, and also prove that it coin-cides with the description of the spectrum as the left adjoint to the inclusion between categories ofmodelled toposes. In this section, to emphasize the 2-dimensional aspects, we refer to the categoryof T -modelled toposes as the oplax slice at S [ T ], with, beware, the following arrow convention:29 efinition 3.1.1. For a Grothendieck topos E , we call the oplax slice the bicategory GTop // E whose 0-cells are pairs ( F , f ) consisting of a geometric morphisms f : F → E , 1-cells ( F , f ) → ( F , f ) are 2-cells in GTop of the form F F E f f f φ and 2-cells ( f, φ ) → ( g, γ ) are equalities of 2-cells F F E g f f fγσ = F F E f f fφ Remark . Beware that with this convention, 1-cells in our notion of oplax slice have the samedirection as their 2-cells part in
GTop , and the opposite direction of their underlying 1-cell. Wechoose this convention in order to allow us to see T - GTop as the oplax slice
GTop // S [ T ].Recall that the bicategory GTop of Grothendieck toposes has finite bilimits. In particular, itpossesses bipullbacks, bicomma, and bipower with 2. In particular the bipower of a topos E with2 is equiped with its universal 2-cell F F ∂ ∂ µ F such that any other 2-cell E F fgφ induces a universal arrow
E → F , unique up to unique invertible 2-cell, equiped with twoinvertible two cells α f , α g such that we have an equality of 2-cells E F fgφ = E F F fgt φ ∂ ∂ α f ≃ α g ≃ µ F Moreover, this construction is functorial in the sense that any 2-cell
E F tsσ induces a square in Geom (cid:2) E , S [ T ] (cid:3) ≃ T [ E ] ∂ t ∂ s∂ t ∂ s ∂ ∗ σµ ∗ t µ ∗ s∂ ∗ σ which codes for a commutative square in T [ E ], that is, a morphism µ ∗ t → µ ∗ s in T [ E ] . Con-versely, any square in T [ E ], seen as a morphism between objects of T [ E ] , induces uniquely such a2-cell. 30ollowing [7], we now describe how to use those universal constructions in order to exhibitthe spectrum as the classifier of local form under a given modelled topos. Let be ( T , J, V ) anadmissibility structure with ι J S [ T J ] ֒ → S [ T ] the embedding of the classifier of local objects intothe classifier of T . As the category of T -models T [ E ] ≃ Geom (cid:2) E , S [ T ] (cid:3) in any topos E inherits afactorization system ( Et E , L oc E ) from V , any two cell F S [ T ] FG φ coding for a morphism φ : F → G between T -models in F admits an admissible factorization F S [ T ] FGφ = F S [ T ] H φ FGn φ u φ In particular consider the admissible factorization of canonical 2-cell of the bipower of S [ T ] S [ T ] S [ T ] ∂ µ ∂ ∂ n µ u µ Proposition 3.1.3.
Let be φ : F → G a morphism in T [ F ] , with t φ : F → S [ T ] the associateduniversal map. Then choose n φ = n µ ∗ t φ and u φ = u µ ∗ t φ .Proof. As the factorization system is left generated, the category of etale maps between T -modelsis locally finitely presentable, being classified by the preseaf topos d V op . Hence they are stableunder inverse image along geometric morphisms. In particular, as n µ : ∂ → ∂ µ is an etale map in T [ S [ T ] ], its inverse image n φ ∗ t φ : ∂ t φ → ∂ µ t φ is an etale map in T [ F ], as well as its compositewith the isomorphism α F : F ≃ ∂ t φ .Now recall that local maps are also stable under inverse image. Hence , as u µ : ∂ µ → ∂ is a localmap in T [ S [ T ] ], its inverse image u φ ∗ t φ : ∂ µ t φ → ∂ t φ is a local map in T [ F ], as well as itscomposite with the isomorphism α G : ∂ t φ ≃ G . Moroever, by functoriality of whiskering, we have φ = α G u φ ∗ t φ n φ ∗ t φ α F , providing an admissible factorization of φ . Corollary 3.1.4.
The classifiers of etales and local maps between T -models are respectively theinverters of u µ and n µ : S [ Et ] S [ T ] S [ T ] ι Et ∂ µ ∂ u µ S [ Loc ] S [ T ] S [ T ] ι Loc ∂ ∂ µ n µ Similarly one can define the classifier of local maps between local objects as the following inverter S [ Loc J ] S [ T J ] S [ T J ] ι LocJ ∂ ∂ µ n µ Proof.
As for any orthogonal factorization system, etale maps are those whose local part is invert-ible and dually, local maps are those whose etale part is invertible. Then a morphism between T -models φ : F → G in a topos F is an etale map in T [ F ] if and only if the whiskering u µ ∗ t φ is invertible, and is a local map if and only if n φ ∗ t φ is invertible. Each of those conditions isequivalent to say that t φ factorizes through the corresponding inverter.31n the following we denote the canonical 2-cells of those three classifiers as S [ Et ] S [ T ] ∂ Et ∂ Et ν S [ Loc ] S [ T ] ∂ Loc ∂ Loc υ S [ Loc J ] S [ T J ] ∂ LocJ ∂ LocJ υ J Now any morphism of modelled topos ( F , F ) → w ( E , E ) with ( E , E ) a locally modelled toposdefines a 2-cell in GTop
E FS [ T J ] S [ T ] E f Fι J φ But considering the admissible factorization f ∗ F EH φφn φ u φ of the inverse image part φ ♭ : f ∗ F ⇒ E , where H φ is a local object in E we have on the left auniversal map classifying the etale part n φ together the 2-cell FE S [ Et ] S [ T ] S [ T J ] Fα Ff ≃ fH φ t nφ ∂ Et ∂ Et ι J α ιJHφ ≃ ν while we have on the right a universal map classifying u φ (seen as a local map between localobjects) together with the 2-cell E S [ Loc J ] S [ T J ] H φ Et uφ ∂ LocJ ∂ LocJ α Hφ ≃ α E ≃ υ J Before constructing the spectrum, we must first construct two different objects classifying onone side the factorization data, and on the other side the local data the spectrum classifies. Indeed,in the following we are going to exhibit the spectrum as a classifier of local forms under a given T -model, that are its etale maps toward a local object. Observe that from this very definition, itsuffices to have and both an object classifying etale maps under a fixed object, and another oneclassifying etale arrows toward arbitrary local objects.Consider the pullback of F along the universal codomain of the classifier of etale maps S [ F, Et ] FS [ Et ] S [ T ] π F π F y F∂ Et t : E → S [ F, Et ] is the name of an etale map in T [ E ] µ ∗ t : π F F t ≃ ∂ π F t → ∂ π F t with codomain F π F t , so any such morphism defines uniquely a morphism E FS [ T ] π F t∂ π F t Fµ ∗ t ⇐ We also consider the bicategory
GTop // S [ Et ] Loc , the (0,2)-full sub-bicategory of the oplaxslice
GTop // S [ Et ] with again 1-cell consisting of a local map as the underlying 2-cell in GTop . Proposition 3.1.5.
The assignment sending a T -modelled topos ( F , F ) to the arrow π F : S [ F, Et ] →S [ Et ] defines a pseudofunctor GTop // S [ T ] GTop // S [ Et ] Loc∂ Et ∗ Remark . Restricted to the pseudoslices, the functoriality of ∂ Et ∗ = S [ − , Et ] is obviousas it coincides with the base change functor along ∂ Et . However we are interested here withoplax slices, where the functoriality of this process is not trivially induced from the universalproperty of bipullback, but from the factorization data involved with etales maps, through amore complex sequence of universal properties. In fact, we are going to construct a morphismbetween the corresponding classifier of etales map from the “generic admissible factorization afterprecomposition” as below. Proof.
Consider a morphism of T -modelled toposes, that is, a morphism in the oplax slice E FS [ T ] fE Fφ ⇐ and its admissible factorization φ = u φ n φ in T [ E ]. Observe that for any etale arrow n : E → G under E in T [ E ], we have an admissible factorization F f EH nφ G φn nφ nu nφ We are going to consider the generic version of this process. Consider the composite 2-cell S [ E, Et ] E FS [ T ] π E ∂ Et π E fE Fφ ⇐ ν E ⇐ and its admissible factorization E FS [ E, Et ] S [ T ] f Fπ E H νEφ ∂ Et π E n ν E φ ⇓ u ν E φ ⇓ t n νEφ : S [ E, Et ] → S [ F, Et ] together withinvertible 2-cells α fπ E and α H νEφ such that n ν E φ decomposes as the following 2-cell E FS [ E, Et ] S [ F, Et ] S [ T ] fα fπE ≃ Ft nνEφ H νEφαHνEφ ≃ π E π F ∂ Et π F ν F Now observe that the composite π F t n νEφ classifies the same map t n νEφ but seen as an etale map,forgetting its domain; then we end with the following 2-cell S [ E, Et ] S [ F, Et ] S [ Et ] π E t nνEφ π F u νEφ ⇐ producing the desired morphism in the bicategory GTop // S [ Et ] Loc .For the sake of the precision, let us describe how this processes on the 2-cells. Let be some σ : ( f, φ ) ⇒ ( g, γ ) between ( F , F ) and ( E , E ). Then we have the 2-dimensional equation F f F gE F ∗ σφ γ Then observe that for any etale arrow n : E → G in T [ E ], the admissible factorizations of nφ and nγ are related as follows: consider the successive factorizations F f F g EH n nγ F ∗ σ H nγ G n nnγF ∗ σ F ∗ σ φ γn nγ nu nnγF ∗ σ u nγ then by uniqueness of the admissible factorization, one has n nφ = n n nγ F ∗ σ H nφ = H n nγ F ∗ σ u nφ = u nγ u n nγ F ∗ σ Applying this property to the generic etale map, this says that there is a generic transition 2-cell S [ E, Et ] S [ F, Et ] t nνEφ t nνEγ t ν E ∗ σ ⇓ such that the 2-cell u ν E φ factorizes as S [ E, Et ] S [ F, Et ] S [ Et ] π E t nνEφ t nνEγ π F t ν E ∗ σ ⇓ u νEγ ⇐ S [ Et ] the commutativediagram ∂ Et π F t ν E γ ∂ Et π F t ν E φ ∂ Et π E ∂ Et π F t ν E γ ∂ Et π F t ν E φ ∂ Et π E ∂ Et ∗ u νEγ ν ∗ π F t νEγ ν ∗ π F t νEφ ∂ Et π F ∗ t νE ∗ σ ∂ Et ∗ u νEφ ν ∗ π E ∂ Et ∗ u νEγ ∂ Et ∗ u νEφ ∂ Et π F ∗ t νE ∗ σ which is the generic form of the relations between the factorization of nφ and n γ above, but forthe generic etale map under E , knowing that the top triangle is actually, up to the canonical iso,the equation γF ∗ σ = φ , while the lower triangle express the equation between the local parts ofthe factorizations, and the vertical arrows are the etale parts of the factorization, with ν ∗ π E thegeneric etale map. Remark . Observe that in this process we obtained in particular a canonical invertible 2-cell S [ E, Et ] S [ F, Et ] E F
E t nνEφ α fπF ≃ Ff satisfying the following 2-dimensional equality S [ E, Et ] S [ F, Et ] E FS [ T ] E t nνEφ α fπF ≃ FfE Fφ ⇐ = S [ E, Et ] S [ F, Et ] E S [ Et ] FS [ T ] t nνEφ π E π E π F π F E u νEφ ⇐ ∂ Et F This invertible 2-cell will be involved in proving the functoriality of the spectral construction. Thissays that one returns the original arrow φ by whiskering with the domain functor ∂ Et ∗ . Remark . This 2-cell codes for the “generic etale map under F ” in the sense, which can alsobe seen as an etale map in T [ S [ F, Et ]]. Now observe that for a morphism ( f, φ ) : ( F , F ) → ( E , E )in GTop // S [ T ], exploiting the functoriality of the universal property of S [ Et ] returns a squarebetween those universal etale arrows ∂ Et F t n νEφ ∂ Et E∂ Et F t n νEφ ∂ Et E ν ∗ F t nνEφ ∂ ∗ u νEφ ν ∗ E∂ ∗ u νEφ which is the “generic admissible factorization of the composite of an etale map under E with φ ”; thetop row is indeed equal up to canonical iso to φ while ∂ Et F t n νEφ ≃ H ν E φ we have ∂ ∗ u ν E φ , beinglocal by stability of local maps along inverse image, is the local part of the generic factorization.The following proposition serves as an intermediate step in the construction of the spectrumfor the admissibility structure ( T , J, V ). In fact, it constitutes the spectral adjunction for theadmissibility structure ( T , T, V ) with the same factorization data V but without choice of localobjects, that is with the trivial topology T corresponding to the trivial geometric extension of35 into itself. To this purpose we introduce the (0,2)-full sub-bicategory GTop // S [ T ] Loc whoseobjects are T -modelled topos, and whose 1-cells are 2-cells E FS [ T ] fE Fυ ⇐ such that υ : F f → E is a local map in T [ E ], with no restrictions on 2-cells. Observe that thosemorphisms can also be visualized by the 2-cellIn particular we have a pseudofunctor GTop // S [ Et ] Loc
GTop // S [ T ] Loc
GTop // S [ T ] ∂ Et ! ι L oc sending any n : E → S [ Et ] to its domain ∂ Et n , and any morphism ( f, υ ) to the whiskering ∂ ∗ υ .We are going to prove that the arrow π F : S [ F, Et ] → F together with the isomorphism F π F ≃ ∂ Et π F is the unit of this biadjunction (the invertibility of the 2-dimensional part comesfrom it already serves as a unit for the restricted biadjunction between pseudo-slices). Theorem 3.1.9.
We have a biadjunction
GTop // S [ T ] Loc ⊥ GTop // S [ T ] ι Loc S [ − , Et ] where the pseudofunctor S [ − , Et ] sends ( F , F ) to the composite ∂ Et π F : S [ F, Et ] → S [ T ] .Proof. Let be a morphism ( f, φ ) : ( F , F ) → ( E , E ) in T op// S [ T ]. Then the etale part of theadmissible factorization of φ in T [ E ] produces a canonical arrow t n φ : E → S [ F, Et ] which togetherwith the local part of φ produces a morphism ( t n φ , u φ α H φ ) : ( S [ F, Et ] , π F ) → ( E , E ) such that the2-cell φ decomposes as the pasting E S [ F, Et ] FS [ Et ] S [ T ] S [ T ] S [ T ] f αf ≃ H φ t nφ E u φ α Hφ ⇐ π F y π F F∂ Et ∂ Et ν We left the process relatively to the 2-cell as an exercise for the reader. As those data uniquelydetermine ( f, φ ) and were produced out of universal properties, we have then an equivalence ofcategory
GTop // S [ T ] Loc (cid:2) ( S [ F, Et ] , π F ) , ( E , E ) (cid:3) ≃ GTop // S [ T ] (cid:2) ( F , F ) , ( E , E ) (cid:3) proving the desired adjunction. In particular the 2-cell S [ F, Et ] FS [ Et ] S [ T ] π F π F F∂ Et ν F where ν F = ν ∗ π F is the unit of the bi-adjunction, and is the generic etale map under F . It iseasy to see that the converse process of composing a 2-cell ( g, u ) : ( S [ F, Et ] , π F ) → ( E , E ) simplyhas to be pasted with the unit 2-cell to return a morphism ( F , F ) → ( E , E ), whose local part getsback u . 36he previous part was aimed at constructing spectrum for a purely factorial admissibilitystructure, for instance the one we get from an arbitrary admissibity structure by forgetting thespecification of geometric extension of T . Now consider the local data. A local form is an etalearrow toward a local object: then local forms can be classified by the pullback S [ Et, J ] S [ Et ] S [ T J ] S [ T ] ι J,Et ∂ Et,J y ∂ Et ι J Definition 3.1.10.
Let be ( T , V , J ) a geometry. Then define the spectrum of an T -modelled topos( F , F ) as the following sequence of pullbacks Spec ( F ) S [ F, Et ] FS [ Et, T J ] S [ Et ] S [ T ] S [ T J ] S [ T ] S [ T ] y π F π F y F∂ Et,J ι J,Et y ∂ Et ∂ Et ι J ν In the following we denote as η : Spec ( F ) → F the composite of the top side of this square, and e F : Spec ( F ) → S [ T J ] the composite of the left side, and h : F η ⇒ ι J e F the whiskering of ν withthe map induced from the top-left pullback.The spectrum of ( F , F ) is equipped with a canonical 2-cell Spec F FS [ T J ] S [ T ] e F η Fι J h In particular observe the following, which is just unravelling the universal property throughwhich the spectrum is constructed:
Proposition 3.1.11.
For any Grothendieck topos E , the category Geom [ E , Spec ( F )] has forobjects pairs ( f, x ) with f : E → F and x : F f → E x in Et [ E ] with E in T J [ E ] . Conversely, anysuch datum ( f, x ) in E defines uniquely a geometric morphism t ( f,x ) : E →
Spec ( F ) In particulara point of
Spec ( F ) is the data of a point p : S → F and a set-valued local form x : F p → E . Now some precision on the functoriality of this process. Let us consider a morphism of T -modelled toposes ( f, φ ) : ( F , F ) ⇒ ( E , E ). Then the following diagram Spec ( F ) S [ F, Et ] F Spec ( E ) S [ E, Et ] ES [ Et, T J ] S [ Et ] S [ T ] S [ T J ] S [ T ] S [ T ] ι F y y π F π F Fι E y y t nνEφ π E π E Ef∂ Et ∗ ι J ι ∗ J ∂ Et y ∂ Et ∂ Et ι J ν φ ⇐ u νEφ ⇐ But now we have in E a normal form under F with n φ : f ∗ F → H φ , which is sent to anormal form ι ∗ E π E ∗ n φ in Spec ( E ), inducing a unique arrow Spec ( φ ) : Spec ( E ) → Spec ( F ) which37oreover makes the top left square to commute, so that we have a composite invertible 2-cell Spec ( E ) Spec ( F ) S [ E, Et ] S [ F, Et ] E F ft nνEφ Spec ( φ ) ι F ι E π E π F α fπF ≃ ≃ And pasting this the upper invertible 2-cell with u ν E φ provides an invertible 2-cell π F ι F Spec ( φ ) ≃ π F t n νEφ ι E ⇒ π E ι E , and composing it again with the pullbacks intertible 2-cells returns a 2-cell ∂ Et ∗ ι J ( ∂ Et ∗ ι J ) ∗ π F ⇒ ∂ Et ∗ ι J ( ∂ Et ∗ ι J ) ∗ π E Spec ( φ ), which comes uniquely from a 2-cell( ∂ Et ∗ ι J ) ∗ π F e φ ⇒ ( ∂ Et ∗ ι J ) ∗ π E Spec ( φ )because ∂ Et ∗ ι J is an embedding and hence is a fully faithful morphism. This provides the desired2-cell Spec ( E ) Spec ( F ) S [ T J ] Spec ( φ ) e E e F e φ related to φ through the following diagram Spec ( F ) F Spec ( E ) ES [ T J ] S [ T ] e E Spec ( φ ) e F ι J η F Fη E fEν F e φ φν E where we have the equality of 2-cells ν E φ = ι J ∗ e φν F encapsulating the generic admissible factor-ization of the precomposition of the generic local form under E with φ .Now, as before, we consider the (0,2)-full sub-bicategory GTop // S [ T J ] Loc ֒ → GTop // S [ T J ]consisting of 1-cells with a local map as underlying 2-cell. It is equiped with an inclusion GTop // S [ T J ] Loc ι
Loc,J ֒ → GTop // S [ T ] Theorem 3.1.12 (Cole) . We have a bi-adjunction
GTop // S [ T J ] Loc ⊥ GTop // S [ T ] ι Loc,J
Spec
Proof.
At this point, this central result has become a corollary of the intermediate result theo-rem 3.1.9. For a morphism of modelled toposes ( f, φ ) : ( F , F ) → ( E , ι J E ) with ( E , E ) locallymodelled, the admissible factorization of φ ♭ : f ∗ F → E produces an etale map n φ : f ∗ F → H φ E Spec ( F ) FS [ T J ] S [ T ] f αf ≃ H φ t nφ E u φ α Hφ ⇐ e F η F Fι J h and the data of the universal map t n φ and the right part u φ of the factorization defines a morphismof locally modelled topos ( t n φ , u φ α H φ ) : ( Spec ( F ) , e F ) → ( E , E ) Proposition 3.1.13.
Let be ( F , F ) a T -modelled topos. Then for any geometric morphism f : G → F one has a bipullback
Spec ( F f ) G Spec ( F ) F f Spec (1 Ff ) η Ff η F y with f F f = e F Spec (1 F f ) .Proof. This is just a property of left-cancellation of pullbacks applied to the 2-dimensional equality:
Spec ( F f ) S [ F f, Et ] G Spec ( F ) S [ F, Et ] FS [ Et , T J ] S [ Et ] S [ T ] f Spec (1 Ff ) π F Fπ F ∂ Et ι j, Et π Ff yy yy = Spec ( F f ) S [ F f, Et ] GS [ Et , T J ] S [ Et ] S [ T ] ∂ Et ι j, Et π Ff π Ff F f yy In particular at any point p : S → F we have a bipullback
Spec ( F p ) S Spec ( F ) F p Spec (1 Fp ) η Fp η F y Remark . Here we should end with a remark justifying the construction above. We saw wecould consider purely factorization geometries without choice of local objects: for a factorizationsystem, we can classify etale maps under a given modelled topos - all of them are local formsas all objects are local - and the inclusion 2-functor
GTop // S [ T ] Loc ֒ → GTop // S [ T ] had a leftadjoint, which was the spectrum of the associated factorization geometry. Then we saw how thisprocess could be refined to take into account a choice of local objects - as long as they satisfiedadmissibility. Now for a choice of local object, that is, for a geometric extension T → T J , we canalso construct a classifier of arrows toward local objects as the comma[ F, T J ] FS [ T J ] S [ T ] ι J F e F µ and this does not involve factorization data at all. Then one could ask if this construction definesa spectrum in the sense of a left biadjoint to an inclusion of the bicategory of locally modelledtoposes into in the bicategory of modelled toposes. There are two ways to answer this question:39 either one considers that “no factorization data” means that any map is local and
GTop // S [ T J ] Loc = GTop // S [ T J ]; hence the only etale maps are isomorphisms. But then we do not have admis-sibility as the factorization of an arrow φ : F → E with E local returns a local objects H φ ≃ F only if F is already local, and the pseudofunctor [ − , T J ] cannot be a left bi-adjoint to theinclusion GTop // S [ T J ] ֒ → GTop // S [ T ]: indeed, for a morphism ( f, φ ) : ( F , F ) → ( E , ι J E ),the universal property of the comma category returns a universal arrow t φ and two invertible2-cells E [ F, T J ] FS [ T J ] S [ T ] ι J F e F fE t φ µ αf ≃ αE ≃ where ( t φ , α E ) : ( Spec ( F ) , e F ) → ( E , E ) is the corresponding morphism of locally modelledtopos and α E is always an iso. This defines an adjunction between homcategories GTop // S [ T J ] (cid:2) ( Spec ( F ) , e F ) , ( E , E ) (cid:3) GTop // S [ T ] (cid:2) ( F , F ) , ( E , ι J E ) (cid:3) ⊣ where the right adjoint sends ( g, ψ ) to the wiskering ( π g, ι J ∗ ψµ ∗ g ) and the left adjoint sends( f, φ ) on ( t φ , α E ) as above. But as an arbitrary ( g, ψ ) in GTop // S [ T J ] (cid:2) ( Spec ( F ) , e F ) , ( E , E ) (cid:3) has not necessarily an invertible ψ , the counit of this adjunction seldom is an isomorphism,and this adjunction does not induce an equivalence of categories: hence [ − , T J ] is not leftadjoint to the inclusion; − either one considers that “no factorization data” means that any map is etale; hence localmaps are all iso, and we trivially have admissibility. Then, from what was said above, sinceany morphism of locally modelled topos has now an invertible algebraic part, we actuallyhave a biadjunction GTop // S [ T J ] Iso ⊥ GTop // S [ T ] ι Loc,J [ − , T J ] But such a geometry is very rigid and of limited interest as any morphism of locally modelledtopos hence only consists in a restriction along a geometric morphism. In some sense this isthe most discrete geometry associated with T and J . In this section we present the site-theoretic aspects of the construction of the spectrum associ-ated to a geometry, following the recipe given in [8]. First we give the spectral site of set-valuedmodels, which gathers the finitely presented etale map under it; then we provide a pseudolimitdecomposition for the spectrum of arbitrary etale maps, and give some geometric properties ofthe spectra. In the second subsection, we recall the notion of fibered site and fibered topos asintroduced in [29], and introduce a variation of this notion to take account of a topology on thebasis. In the last subsection, we combine the two previous subsections to prove that the spectrumof an arbitrary modelled topos is a topos of continuous sections of a fibered topos. In this wholesection we fix a geometry ( T , V , J ) and denote as B = T [ S ] the locally finitely presented categoryof set-valued T -models and Et and Loc the associated factorization system in T [ S ]. In the previous section, the spectrum was constructed in an abstract manner through 2-dimensional universal properties. Now we would like to give a concrete site presentation of thespectrum. Following essentially [8], this section is devoted to the construction of the spectral site40or a set-valued model.Recall that for any T [ S ]-model B we denote as V B is the category of finitely presented arrowsunder B , which were obtained as pushouts under B of finitely presented etale maps in V . Definition 4.1.1.
For any B in B , V opB can be equipped with the Grothendieck topology J B defined from the duals of the families B CC inn i m i ! i ∈ I such that there exists some arrow b : K → C and a covering family ( l i : K → K i ) i ∈ I dual of acovering family in J such that for each i ∈ I we have K C BK i C i nm i n i bl i y Definition 4.1.2.
For an object B in B , Spec V ,J ( B ) = Sh ( V opB , J B )In the following we denote as a J B : d V op B → Spec ( B ) the associated sheafification functor left adjointto the inclusion ι B : Spec ( B ) ֒ → d V op B . Remark . Observe that the sheafification functor a J B extends into a functor between categoriesof T -models and T J -models. In fact we have a pseudocommutative square T [ d V op B ] T [ Spec ( B )] T J [ d V op B ] T J [ Spec ( B )] a ∗ JB a ∗ JB ≃ which is the pseudonaturality square of the natural transformation Geom [ d V op B , − ] Geom [ Spec ( B ) , − ] Geom [ ι B , − ] at the inclusion ι J : S [ T J ] ֒ → S [ T ].Now we turn to the functoriality of the construction. For a morphism f : B → B in T [ S ],the geometric morphism Spec ( f ) is computed from the pushout functor along f V B V B f ∗ sending a finitely presented etale arrow to its pushout B B C f ∗ C n f p f ∗ n But now, observe that the pushout functor sends finite colimits of V B to finite colimits in V B hence defines a lex functor V opB → V opB . Moreover, this functor is J B -continuous, by compositionof pushouts. Hence Diaconescu applies and returns an extension V op B V op B Spec ( B ) Spec ( B ) f ∗ lan aJB ょ ∗ f ∗ ≃ which is the inverse image part of Spec ( f ). 41 emark . V opB is a Lex site coding for “basic compact open inclusions”. Objects of thesheaf topos Spec ( B ) ֒ → [ V B , S ] should be seen as generalized opens of the spectral topology, whileobjects of Ind( V B ), which are arbitrary etale arrows under B , should be seen as saturated compactsof the spectral topology. In particular the embedding V op B ֒ → Spec ( B ) exibits V B as a basis of basic compact sets that are open - and V op B as a basis of open sets that are compacts .The following observation motivate the name for etale arrows: Proposition 4.1.5.
Finitely presented etale arrows n : B → C under B correspond to etalegeometric morphisms: Spec ( C ) ≃ Spec ( B ) / a J B ( ょ ∗ n ) Spec ( n ) −→ Spec ( B ) Proof.
For n : B → C in V B we have an equivalence of category V C ≃ n ↓ V B sending m : C → D to the triangle BC D n mmn and conversely any triangle l = mn in n ↓ V B to the underlying arrow m . In particular we have d V op C ≃ \ ( n ↓ V B ) op ≃ d V op B / ょ ∗ n But also by the expression of slices in a sheaf category (see [29][III, Proposition 5.4]), we knowthat the topology induced on n ↓ V B by J B is the same as J C - this is the corresponding topology J ′ corresponding to J C through the equivalence above - and we have Spec ( C ) ≃ Sh ( n ↓ V opB , J ′ ) ≃ Spec ( B ) / a J B ( ょ ∗ n ) Remark . Observe that we have a 2-pullback square in the bicategory of Grothendieck toposes
Spec ( B ) / a J B ょ ∗ n d V op B / ょ ∗ n Spec ( B ) d V op B Spec ( n ) ι B ょ ∗ n y exhibiting the etale geometric morphism Spec ( n ) as the 2-pullback of the etale geometric morphismassociated to ょ ∗ n . Remark . The further left adjoint of the inverse image will be induced from the postcompo-sition functor V opC → V opB sending m : C → D to the composite mn : B → D which is in V B . Thisfunctor defines a left adjoint V op B V op Cn ∗ n ∗ ⊣ The intuition that objects of V B are compact can be formalized thanks to the following property.Recall that a geometric morphism is said to be tidy if its direct image part preserves filtered colimits.From [26][Theorem 4.8] we know that tidy geometric morphisms are stable under 2-pullback. Proposition 4.1.8.
For n : B → C in V B , the geometric morphism Spec ( n ) : Spec ( C ) → Spec ( B ) is tidy.Proof. Recall we can express
Spec ( n ) as the pullback of the etale geometric morphism d V op B / ょ ∗ n → d V op B along ι B . But we know that ょ ∗ n is a finitely presented object in the presheaf topos d V op B , sothat the associated internal hom functor ( − ) ょ ∗ n preserves filtered colimits: hence its associatedetale geometric morphism is tidy, and hence its pullback Spec ( n ) also is.42 emark . Arbitrary etale arrows are not in the topos
Spec , but rather from the side ofpoints and saturated compacts. Hence they do not correspond to etale geometric morphisms ingeneral. In fact observe that an arbitrary etale map l : B → C is an object of Ind( V B ), for thefactorization system ( Et , Loc ) was left generated from V ; but we have Ind( V B ) ≃ Pro( V op B ) op ,which is the pro-completion of V op B , whose objects are those functors V op B → S which are cofilteredlimits of representable: this mimics the fact that arbitrary etale maps are construted as cofilteredintersection of basic open compact sets. For this reason, [3] says pro-etale for what we call arbitraryetale, reserving “etale” for the basic ones. This can be formalized into the following result: Theorem 4.1.10.
Let be l : B → C an arbitrary etale arrow under B . Then Spec ( C ) decomposesas a cofiltered pseudolimit Spec ( C ) ≃ lim ←− ( n,a ) ∈ R ょ l Spec (cod( n )) Proof.
From theorem 1.3.7, we know that V C ≃ colim −−−→ ( n,a ) ∈ R ょ l V cod( n ) Now recall that for each ( n, a ) in R ょ l , the opposite category of the etale generator V cod( n ) can beequipped with a topology J cod( n ) , and the opposite category of the pseudocolimit (colim ( n,a ) ∈ R ょ l V cod( n ) ) op can be equipped with the coarsest topology making the canonical inclusion q op ( n,a ) : V op cod( n ) → (colim ( n,a ) ∈ R ょ l V cod( n ) ) op continuous, which is h [ ( n,a ) ∈ R ょ l q ( n,a ) ( J cod( n ) ) i From [29] and [14], and also the general version of theorem 4.2.6 on cofiltered pseudolimits ofGrothendieck toposes, we know that the corresponding sheaf topos is the pseudolimit Sh (cid:0) (colim ( n,a ) ∈ R ょ l V cod( n ) ) op , h [ ( n,a ) ∈ R ょ l q ( n,a ) ( J cod( n ) ) i (cid:1) ≃ lim ←− ( n,a ) ∈ R ょ l Spec (cod( n ))and moreover, this topology is exactly the image of the induced topology on the pseudocolimit alongthe equivalence of category above with V C . Now we can also glue the image of the J cod( n ) -coveringfamilies along the pushout functors a ∗ to generate a topology on V op C h [ ( n,a ) ∈ R ょ l a ∗ ( J cod( n ) ) i We must prove that any covering family a ∗ ( J cod( n ) ) is covering in J C , and conversely that any J C covering family is covering in the topology induced from the colimit.For ( n, a ) in R ょ l , m an object of V cod( n ) and a covering family ( l i ) i ∈ I induced as K D cod( n ) K i D il i k i b m m i y in J cod( n ) ( m ), consider the composition of pushouts as in the diagram below K D cod( n ) Ba ∗ D CK i D i a ∗ D il i k i b m y m ∗ a an la ∗ m i m ∗ l i yy y m i y i ∈ I the arrow a ∗ l i is also the pushout ( m ∗ a b ) k i , and this exhibits thepushout family ( a ∗ l i ) i ∈ I as a pushout of a family in J , and hence as a covering family of J C .Conversely, for a covering family CK DK i D im m i l i bk i y we know from the essential surjectivity of the equivalence result in theorem 1.3.7 that m is inducedas some a ∗ m ′ for some ( n, a ) ∈ R ょ l cod( n ) CD D D im m i l i bm ′ a y Now observe that from the situation belowcod( n ) CD ′ DK mm ′ a y b there exists from theorem 1.3.5 some ( n , a ) in R ょ l and a factorization of b through the interme-diate pushout as below cod( n ) cod( n ) CD ′ a ′∗ D ′ DK mm ′ ba ′ y a a ′∗ m ′ y c and now we can pushout the family ( k i ) i ∈ I of J along c to get a family ( c ∗ k i ) i ∈ I in J cod( n ) of theobject a ′∗ m ′ in V cod( n ) cod( n ) cod( n ) CD ′ a ′∗ D ′ Dc ∗ K i D i K K i m m i l i c ∗ k i m ′ m ′ i a y bk i m ′ a ′ c y y y y Moreover, the objects (( n, a ) , m ′ ) and (( n , a ) , a ′∗ m ′ ) are identified in the pseudocolimit for theyare related through an opcartesian morphism, so that ( c ∗ k i ) i ∈ I is both covering for the class of(( n, a ) , m ′ ) in the induced topology on the pseudocolimit colim ( n,a ) ∈ R ょ l V cod( n ) , and is sent to thecovering family ( l i ) i ∈ I of J C through the pushout functor a ∗ . Hence the J C -cover ( l i ) i ∈ I . The44atter says that ( l i ) i ∈ I is in the induced topology a ∗ ( J cod( n ) ), and hence in the jointly generatedtopology. This proves that the equivalence of category of theorem 1.3.7 induces an equivalence ofsites (cid:0) ( colim −−−→ ( n,a ) ∈ R ょ l V cod( n ) ) op , h [ ( n,a ) ∈ R ょ l q ( n,a ) ( J cod( n ) i (cid:1) ≃ (cid:0) ( V op C , J C (cid:1) so that they induce the same sheaf topos, which proves the desired limit decomposition of Spec ( C ). Remark . Observe that we also have directly from theorem 1.3.7 d V op C ≃ lim ←− ( n,a ) ∈ R ょ l d V op B / ょ ∗ n Since each
Spec ( Cod ( n )) for ( n, m ) ∈ R ょ l expresses as an etale geometric morphism obtained asa pullback as in theorem 4.1.6, and pullback commutes with limits, we havelim ←− ( n,m ) ∈ R ょ l Spec ( B ) / a J ょ ∗ n lim ←− ( n,m ) ∈ R ょ l d V op B / ょ ∗ n Spec ( B ) d V op Bι B y so that we have a pullback in GTop Spec ( C ) d V op C Spec ( B ) d V op Bι C Spec ( n ) ι B y This means that the natural inclusion
Spec ( − ) ֒ → d V op ( − ) is cartesian at etale maps l ∈ Et .In particular we have the following, for local forms are etale arrows (that are seldom finitelypresented): Proposition 4.1.12.
Points of
Spec ( B ) correspond to local forms x : B → A . More generally,geometric morphisms f : E →
Spec ( F ) correspond to etale arrows x : ! ∗E B → E in Et [ E ] with E in T J [ E ] .Proof. First observe that a point of the spectrum, that is a J B -continuous lex functor x in Lex J B [ V opB , S ]; this is in particular an ind-object in c V B , hence an object of B ↓ B as V B ֒ → ( B ↓ B ) fp . We can write x as an arrow x : B → A ; now the condition of continuity says that fora covering ( m opi : n i → n ) i ∈ I in V opB one has ` i ∈ I x ( n i ) x ( n ) h x ( m i ) i i ∈ I But one also has as above, by Yoneda, x ( n ) ≃ Ind( V B )[ ょ n , x ] = { f : C → A | f n = x } and the surjectivity property above expresses the existence of the dashed arrow in the followingdiagram for some i ∈ I : BCC i A n i n xm i f ∃ And this expresses that x is a relatively local object under B , hence a local form of B . In particular x ∗ e B = Cod ( B → A ) = A is a local object. The same argument applies if we replace S by anarbitrary topos. 45t the level of points, etale maps n : B → C produce discrete opfibrations pt ( Spec ( C )) ≃ pt ( Spec ( B ) / a J B ( ょ ∗ n )) → pt ( Spec ( B ))Moreover, for an arbitrary etale map l : B → C , as the functor of points pt ≃ Geom [ S , − ]preserves pseudolimits, we have a pseudolimit of category pt ( Spec ( C )) ≃ lim ←− ( n,a ) ∈ R ょ l pt ( Spec (cod( n )) Remark . Observe that arbitrary etales maps under B correspond to points to the presheaftopos d V op B as Lex [ V op B , S ] ≃ Ind( V B ).As well as the term “etale” was justified by the fact that finitely presented etale morphisms in V B where sent to etale geometric morphism by Spec , the name of “local” for objects is justifiedas follows. Recall that a geometric morphism f : F → E is said to be local if its inverse imagepart f ∗ is full and faithful and moreover the direct image part f ∗ has a further right adjoint f ! . Inparticular, a Grothendieck topos E is said to be local if the global section functor Γ! E → S has afurther right adjoint - the full-and-faithfulness condition being automatic in this context.The prototypical example of local geometric morphism is the universal domain map ∂ : E → E of a topos. Now for any point p : S → E , we can consider the
Grothendieck Verdier localization at p , which is defined as the pseudopullback E p E S E ∂ pp ∗ ∂ y Its universal property is that for any Grotendieck topos F , we have an equivalence with thecocomma category Geom [ F , E p ] ≃ p ! F ↓ Geom [ F , E ]In particular, when F is S , we have an equivalence of category pt ( E p ) ≃ p ↓ pt ( E )From [22][Theorem 3.7] we know that if E has ( C , J ) as a lex site of definition, then E p can beexpressed as a cofiltered pseudolimit of etales geometric morphism E p ≃ lim ←− ( C,a ) ∈ R p ∗ E / a J ょ ∗ C where R p ∗ is the cofiltered category of elements of the J -flat functor p ∗ : C → S . Proposition 4.1.14.
Let A be a local object in T [ S ] : then Spec ( A ) is a local topos.Proof. Suppose that A is in T J [ S ]. Then in particular we know that A is J -local for the generalizedcovers associated to J in T [ S ] in the sense of theorem 2.1.1. Then in V op A , the terminal object 1 A is J A -local as it lifts through all its covers. Then the functor ょ A : V op A −→S is lex J A -continuousand defines a points p A . But now consider the Grothendieck-Verdier localization at this point Spec ( A ) p A Spec ( A ) S Spec ( A ) ∂ p A p ∗ A ∂ y which is calculated as Spec ( A ) p A ≃ lim ←− ( n,a ) ∈ R p ∗ A Spec ( A ) / a J A ょ ∗ n Observe then that R p ∗ A has then an initial object (1 A , A ), corresponding to the identity triangleof 1 A in V A . Then the limit above reduces to Spec ( A ) p A ≃ Spec ( A ) / a J A ょ ∗ A , but 1 A was theinitial object of V A , and ょ ∗ turns it into the terminal object of d V op A , which is preserved by a J A .Therefore Spec ( A ) p A ≃ Spec ( A ), exhibiting Spec ( A ) as a local topos.46e also have this corollary from theorem 4.1.10: Corollary 4.1.15.
For x : B → A x a local form under B , the geometric morphism Spec ( x ) : Spec ( A x ) → Spec ( B ) is the Grothendieck-Verdier localization of Spec ( B ) at the point p x : S →
Spec ( B ) . Now we turn to the structural sheaf.
Definition 4.1.16. e B is the structural sheaf of local forms of B , that is the sheafification of thepresheaf returning the codomain of morphism of the site V B , that is e B = a J B codThe structural sheaf can also be described as follows: recall that any T -model B is in Ind( B fp ) ≃ Lex [ C T , S ]. Moreover, we can consider the conerve of the codomain functor T [ S ] op d V op B B T [ S ] (cid:2) B, cod (cid:3) cod ∗ which can be composed along the embedding C T ֒ → T [ S ] op V op B T [ S ] op C T d V op B cod ょ ∗ cod ∗ χ to produce a lex functor C T d V op B { x, φ } T [ S ] (cid:2) K φ , cod (cid:3) cod ∗ which we can now compose with the lex localization a J B : d V op B → Spec ( B ) to get a lex functor C T Spec ( B ) { x, φ } a J B T [ S ] (cid:2) K φ , cod (cid:3) e B Proposition 4.1.17.
The structural sheaf e B is in T J [ Spec ( B )] . In particular for any point x : S →
Spec ( B ) the stalk x ∗ e B is in T J [ S ] .Proof. We have to prove that for any J -cover ( θ i : { x i , φ i } → { x, φ } ) i ∈ I we have an epimorphismin the category of sheaves Spec ( B ) = Sh ( V op B , J B ) a i ∈ I a J B T [ S ] (cid:2) K φ i , cod (cid:3) a J B T [ S ] (cid:2) K φ , cod (cid:3) h a JB T [ S ] h f θi , cod i i i ∈ I Now recall that epimorphisms in categories of sheaves have to be tested locally. Actually what weare going to prove is directly that even before sheafification, h T [ S ] (cid:2) f θ i , cod (cid:3) i i ∈ I allready satisfiesthe localness condition; then, for sheafification is a left adjoint, it preserves coproducts, that is a i ∈ I a J B T [ S ] (cid:2) K φ i , cod (cid:3) ≃ a J B a i ∈ I T [ S ] (cid:2) K φ i , cod (cid:3) and moreover, the localness condition is preserved after sheafification, and makes a J B h T [ S ] (cid:2) f θ i , cod (cid:3) i i ∈ I ≃h a J B T [ S ] (cid:2) f θ i , cod (cid:3) i i ∈ I an epimorphism in Spec ( B ).Let us prove the localness condition for h T [ S ] (cid:2) f θ i , cod (cid:3) i i ∈ I , which amounts to it to be a localsurjection - see for instance [25]. Let be b : K φ → cod( n ); then one can push the J -cover ( f θ i : K φ → K φ i ) i ∈ I along b so we get a cover of n in V op B K φ cod( n ) BK φ i b ∗ K φ i nb ∗ f θi nb ∗ f θi bf θi y i ∈ I , we have T [ S ] (cid:2) K φ , b ∗ f θ i (cid:3) ( b ) = b ∗ f θ i b = f θ i ∗ b f θ i = T [ S ] (cid:2) , f θ i , K φ i (cid:3) ( f θ i ∗ b ),which exactly says that the restriction of b along each member of the cover has an antecedent along h T [ S ] (cid:2) f θ j , b ∗ K φ i (cid:3) i j ∈ I : hence the natural transformation h T [ S ] (cid:2) f θ j , cod (cid:3) i j ∈ I is a local surjectionrelatively to the Grothendieck topology J B , hence its sheafification is an epimorphism in Spec ( B ). Remark . Beware that the structural sheaf e B = a J B cod needs not to return local objectsas values; in particular, whenever J B is subcanonical, e B = cod, but the codomains of basic etalearrow have no reason to be local objects. This is because local objects are models of a geometricextension of T , being a model of which is a local notion that do not hold globally.Now recall that S is terminal amongst Grothendieck toposes, and for any Grothendieck topos E , the terminal geometric morphism ! E : E → S has for direct image part Γ = E (1 , − ). Now as T is a finite limit theory, it is stable under direct image, so that Γ induces a functor T [ E ] Γ −→ T [ S ]In particular for any locally T J -modelled topos ( E , E ), we can apply ! E ∗ to wE to get a set-valued T -model Γ E , and for a morphism of locally T J modelled topos ( f, φ ) : ( E , E ) → ( F , F ), we havea morphism Γ φ ♯ : Γ E → Γ F , as Γ f ∗ F = Γ F for direct images commute with global sections.Moreover, for a 2-cell α : ( f, φ ) → ( g, ψ ), the equality ψ ♭ F ∗ α ♭ = φ ♭ corresponds to an equality φ ♯ = α ♯ ∗ F ψ ♯ with α ♯ : g ∗ ⇒ f ∗ ; but Γ sends F ∗ α ♯ into an equality, so that α is collapsed intothe equality of the morphism Γ φ ♯ = Γ ψ ♯ in T [ S ]. This defines a 2-functor T Loc J - GTop Γ −→ T [ S ]Hence the adjunction of theorem 3.1.12 reduces in particular to the following: Theorem 4.1.19.
We have an adjunction T [ S ] T Loc J - GTop
Spec Γ ⊣ Proof.
Let be B in T [ S ], ( E , E ) a T J -locally modelled topos and φ : B → Γ( E , E ). Observe that T [ S ] admits a full and faithful embedding T ֒ → T - GTop sending B on ( S , B ) and φ can be seen asa morphism of modelled topos (! E , φ ) : ( ∗ , B ) → ( E , E ), so from the Spec ⊣ w adjunction we havea unique morphism of locally modelled topos ( g, ψ ) : ( Spec ( B ) , e B ) → ( E , E ). In particular observethat the unit ( h ( ∗ ,B ) , η ( ∗ ,B ) ) of ( ∗ , B ) necessarily has ! Spec ( B ) as underlying geometric morphism,and therefore the direct image part η ♯ ( ∗ ,B ) : B → Γ e B of η ( ∗ ,B ) is the unit of the Spec ⊣ Γadjunction.This process can be explicitly generalized for B -objects in arbitrary Grothendieck toposes. Butthe process will be better understood relatively to the concept of fibered sites and fibered toposes .The next section is devoted to some prerequisites on this notion, but also contains a new variationof it and some results allowing to adapt it to our situation. We first recall, and adapt slightly, [29][VI, part 7] results on the notion of fibered site and fiberedtopos . In this source, they were introduced over a base category without topology; we shall proposein this section an adaptation for the case of a topology on the base category, as it shall be used inthe next section.
Definition 4.2.1. A fibered lex site on a small category C is an indexed category V : C op → C at such that in each c ∈ C , V c is lex and equiped with a Grothendieck topology J c such that for each s : c → c the corresponding transition functor V s : V c → V c is a morphism of lex site.For a fibered site on small, lex category C , one can consider the Grothendieck construction Z V p V −→ C
48t the indexed category V , which is the oplax colimit of V , equiped with the canonical oplax cocone V c R VV c ι c ι c V s φ s Lemma 4.2.2.
For a fibered lex site V : C op → C at over a lex category C , the oplax colimit R V islex, as well as the fibration p V .Proof. The finite limit of a finite diagram F : I → R V is constructed as follows: first take thelimit lim i ∈ I p V ( F ( i )) in C ; then, we have a pseudococone diagram in C V p V ( F ( i )) V lim i ∈ I p V ( F ( i )) V p V ( F ( j )) V pV ( F ( d )) V pi V pj ≃ producing a finite diagram ( V p i ( F ( i ))) i ∈ I in V lim i ∈ I p V ( F ( i )) , where one can take the limit lim i ∈ I V p i ( F ( i )).Then we have lim F ≃ (lim i ∈ I p V ( F ( i )) , lim i ∈ I V p i ( F ( i )))Now we can equip R V with a coarsest topology making the inclusions ι c lex continuous: definethe topology J V = h [ c ∈C ι c ( J c ) i as generated by the inclusion of all fiber topologies, then trivially each ι c : ( V c , J c ) ֒ → ( R V, J V ) isa morphism of lex sites.Now one can ask for the structure obtained if one considers the sheaf topos associated to eachfiber. This is the purpose of the following notion Definition 4.2.3. A fibered topos on a category C is an indexed category E ( − ) : C op → C at suchthat for any c ∈ C the fiber E c is a Grothendieck topos, and for each s : c → c the transitionfunctor E s : E c → E c is the inverse image part of a geometric functor f s : E c → E c .Then we can also consider the Grothendieck construction at a fibered topos and define thefibration Z E ( − ) p E −→ C Now, to a fibered site V , we can canonically associate a fibered topos whose fiber at c is the sheaftopos Sh ( V c , J c ) and the transition at s : c → c is the inverse image of the geometric morphism Sh ( V s ) : Sh ( V c , J c ) → Sh ( V c , J c ) induced by V s . Then we can consider the Grothendieckconstruction associated to the fibered topos Z Sh ( V ( − ) , J ( − ) ) p V −→ C For an a Grothendieck fibration p : M → C , we denote as Γ( p ) the category whose objects aresections MC C x p E ( − ) : C → C at defines also a Grothendieck fibration on C op thanks to the adjunctions f ∗ s ⊣ f s ∗ , where the fiber at c is still E c but the transition morphismat s : c → c is now the direct image functor f s ∗ : E c → E c . We denote as p ′E : R E ′ ( − ) → C op theassociated fibration. In the following we call this fibration the direct fibration of the fibered topos E ( − ) .Then in particular a section X : C → R E ′ ( − ) of the direct fibration of a fibered topos returnsat each object c an object X c of the topos E c and at an arrow s : c → c a morphism ( s, X s ) :( c , X c ) → ( c , X c ) whith X s : X c → f s ∗ X c with f s ∗ the direct image of the transition geometricmorphism E s .The following says that the sheaf topos over the indexed site is the category of sections of thedirect fibration associated to the fibered topos constructed by sheafification of the fibers. Proposition 4.2.4.
Let V : C → C at a fibered lex site; then we have an equivalence of categories Sh ( Z V, J V ) ≃ Γ( p V ′ )Moreover, sheafification turns the oplax cocone made of the inclusions ( ι c ) c ∈C into an oplaxcone of geometric morphism Sh ( V c , J c ) Sh ( R V, J V ) Sh ( V c , J c ) Sh ( i c ) Sh ( i c ) Sh ( V s ) φ s ∗ where φ s ∗ is the mate of the transformation φ ∗ s : Sh ( ι c ) Sh ( V s ) ⇒ Sh ( ι c ) induced from the cocone φ s . Then we also have the following: Proposition 4.2.5.
We have in the bicategory of Grothendieck topoi that Sh ( Z V, J V ) ≃ oplaxlim c ∈C Sh ( V c , J c ) Proof.
It would be actually expectable for oplaxcolimits of lex sites to be turned into oplaxlimitsof toposes, as well as finite colimits of sites are turned into finite limits and filtered colimits intocofiltered limits. To see this, observe that an oplax cone Sh ( V c , J c ) E Sh ( V c , J c ) f c f c Sh ( V s ) ψ s ∗ is the same as an oplax cocone of lex-continuous functors( V c , J c ) E ( V c , J c ) f ∗ c f ∗ c V s ψ s ∗ which factorizes uniquely in C at through the oplaxcolimit f ∗ : R V → E , and this functor is both lexand J V continuous for all its restrictions at fibers are lex continuous, so that it defines a geometricmorphism f : E → Sh ( R V, J V ). 50n particular, it can be shown that the opcartesian morphisms in the oplax colimit R V havea left calculus of fraction in the sense of Gabriel-Zisman, as explained in [29][Proposition 6.4], sothat we can consider the localization of the oplax colimit R V at the opcartesian morphisms; weknow that this localization is the pseudocolimit of the pseudofunctor V in C at , that is,colim V ≃ Z V [Σ − V ]where Σ V denotes the class of opcartesian morphisms. Moreover, the topology J V is transferred tothe localization - we still denote the induced topology as J V . This provides the following expressionof pseudolimits: Proposition 4.2.6.
We have in the bicategory of Grothendieck topoi that Sh (colim V, J V ) ≃ lim c ∈C Sh ( V c , J c ) Remark . In [29][Theorem 8.2.3] the underlying category is just supposed to be cofiltered forthe equivalence above to hold; in our case this condition is automatically satisfied as we supposed C to be lex. Beware that the results above are not necessarily true for an arbitrary small category C . In fact, it is not known whether the bicategory of Grothendieck toposes has arbitrary smallpseudolimits.As this is the content of [29][Sections 6 and 8, in particular 8.2.3] (and has also its bilimitversion for bifiltered diagrams at [14][Theorem 2.4]) we do not prove it again: we now focusrather on the following adaptation in the case where the underlying category is endowed with aGrothendieck topology, asking for a way to make the fibration in a fibered site a comorphism of site.First observe that any fibered lex site has a terminal section 1 ( − ) : C → R V associating to each c the terminal object 1 V c of V c and to each s : c → c . In particular this is a cartesian section asthe transitions morphisms V s for each s : c → c are lex so that V s (1 V c ) = 1 V c , so the value ofthis section at s : c → c is the cartesian lift s : ( c , V s (1 V c ) → ( c ; 1 V c ). Now for a topology J on C , and V : C op → C at a fibered site, observe that any covering family ( c i → c ) i ∈ I can be lifted toa family ( s i : ( c i , V ci ) → ( c, V c )) i ∈ I : we call such families horizontal . Then consider the topology J V,J = h J V ∪ V ( − ) ( J ) i where 1 ( − ) ( J ) consists of all families of the form ( s i : ( c i , V ci ) → ( c, V c )) i ∈ I .Actually, while we used the terminal element to canonically lift J -covers in C to covers in R V ;but once the topology is generated from those data, we get actually horizontal covers by lifting J -covers at any object in a fiber. To see this, use the following general lemma expressing thatcartesian lifts of an arrow form altogether a cartesian transformation: Lemma 4.2.8.
Let V : C op → C at be a lex fibration. Then for any s : c → c the following squareis a pullback ( c , V s ( a )) ( c , a )( c , V c ) ( c , V c ) ( s, Vs ( a )(1 c , ! Vs ( a ) ) (1 c , ! a )( s, Vs ( a ) y Proof.
In any other square ( c , a ′ ) ( c , a )( c , V c ) ( c , V c ) ( s, Vs ( a )( u, ! a ′ ) (1 c , ! a )( t,f ) the vertical component of the left map is forced to be the terminal map ! a ′ as V u preserves theterminal element. But then such a square is the same as a situation testing the cartesianness of51he lift ( s, V s ( a ) ) which always produces a unique map as the dashed arrow below( c , V s ( a )) ( c , a )( c , a ) c c c u st ( s, Vs ( a ) )( t,f ) which provides in particular the desired factorization.As a consequence, from we now that Grothendieck topologies are closed under pullback ofcovering families, it appears the following: Lemma 4.2.9.
Let be V be a fibered site on a lex category C and C a Grothendieck topology on C . Then for any J -cover ( s i : c i → c ) i ∈ I and any a ∈ V c , the family (( s i , V si ( a ) ) : ( c i , V s i ( a )) → ( c, a )) i ∈ I is a covering family in J V,J . As defined above, the topology J V,J is the simplest that allows to induce a geometric morphismtoward Sh ( C, J ). In fact:
Proposition 4.2.10.
The topology J V,J is the coarsest topology such that we have simultaneouslythe two following conditions: − for each c in C , the inclusion ι c : ( V c , J c ) → ( R V, J
V,J ) − have a comorphism of site p V : ( R V, J
V,J ) → ( C , J ) . Moreover, as this topology refine the topology J V , there is a corresponding inclusion of topos Sh ( Z V, J
V,J ) ֒ → Sh ( Z V, J V )exhibiting objects of Sh ( R V, J
V,J ) as a specific kind of sections. The intuition is that some notionof continuity relatively to the base topology is involved. We introduce the following notion to thisend:
Definition 4.2.11.
Let be E ( − ) a fibered topos on a lex site ( C , J ). Then by a continuous sectionof E ( − ) we mean a section of the associated direct fibration R E ′ ( − ) C op C opX p ′E such that for any J -covering family ( c i → c ) i ∈ I the lifting ( X s i : X c → f s i ∗ X c i ) i ∈ I exhibits X c alimit in the fiber E c of the diagram X c = lim i ∈ I (cid:16) Y i ∈ I f s i ∗ X c i ⇒ Y i,j ∈ I f s iij ∗ X c ij (cid:17) where the double arrow is induced from the transitions ( f s i ∗ ( X s iij ) : f s i ∗ X c i → f s iij ∗ X c ij ) i,j ∈ I . Wedenote as Γ J ( p E ′ ) the category of continuous sections and natural transformations between them. Theorem 4.2.12.
Let ( C , J ) be a small lex site, V : C op → C at a fibered lex site on C with p V : R Sh ( V ( − ) , J ( − ) ) → C the associated fibered topos and p V ′ : R Sh ( V ( − ) , J ( − ) ) ′ → C op thecorresponding direct fibration. Then one has an equivalence of categories Sh ( Z V, J
V,J ) = Γ J ( p V ′ )52 roof. For one direction, observe that any sheaf X in Sh ( R V, J
V,J ) can be composed with theduals of the fiber inclusions V opc R V op Set V opc ι opc ι opc V ops Xφ ops and as the topology of each fiber is part of the topology J V,J the restriction Xι opc is a sheaf for thetopology J c , hence is an object of the sheaf topos Sh ( V c , J c ); moreover, for each s : c → c , thediagram below provides us by whiskering with a transformation X ∗ φ ops : Xι opc → Xι opc V ops But precomposition with V s is what the direct image Sh ( V s ) ∗ consists in. So we can associate to X the section b X : C op → R E with b X c = Xι opc = Sh ( ι c ) ∗ X and b X s = X ∗ φ s . Now we have tocheck that X is a continuous section. But we know that X is a sheaf for the horizontal topology,and also relatively to any horizontal cover (( s i , V si ( a ) ) : ( c i , V s i ( a )) → ( c, a )) i ∈ I , which hence issent on a limit diagram b X c ( a ) = lim i ∈ I (cid:16) Y i ∈ I b X c i ( V s i ( a )) ⇒ Y i,j ∈ I b X c ij ( V s iij V s i ( a )) (cid:17) where V s iij V s i ( a ) = V s jij V s j ( a )for each i, j ∈ I : but from the fact that limits are pointwise in categories of sheaves, this is sufficientto ensure that b X is a continuous section.For the converse, suppose we have a continuous section b X : C op → R E ′ ( − ) . Then each X c is asheaf on ( V c , J c ); by the property of the oplaxcolimit, the data of the cocone V c Set op V c V s X opc X opc X ops induces a unique functor V c R V Set op V c V s X opc X opc ι c ι c X op φ s X ops whose restriction at each fiber coincides with X c and whose whiskering with some φ ops is X s . Hence X is a sheaf for the topology h S c ∈C J c i , with values X ( c, a ) = X c ( a ) for a ∈ V c , and X ( s, u ) isobtained as the composite X c ( a ) X c ( V s ( a )) X c ( a ) X ( s,u )( X s ) a X c ( u ) where ( X s ) a is the component at a of the natural transformation X c → Sh ( V s ) ∗ X c and X c ( u )is the restriction map of X c at u : a → V s ( a ). But now, being continuous as a section, forany J -cover ( s i : c i → c ) i ∈ I the transitions ( X s i : X c → Sh ( V s ) ∗ X c i ) i ∈ I form a limit diagram in53 h ( V c , J c ), and as limits are pointwise in categories of sheaves, this means in particular that in1 V c , the induced horizontal cover (( s i , s i ) : ( c i , V ci ) → ( c, V c ) where 1 V ci = V s i (1 V c )) i ∈ I is sent toa limit diagram X ( c, V c ) = lim i ∈ I (cid:16) Y i ∈ I X ( c i , V ci ) ⇒ Y i,j ∈ I X ( c ij , V cij ) (cid:17) which exactly means that X is a sheaf for the topology 1 V ( − ) ( J ).Now it is clear that those two processes are mutually inverse from the fact that R V is theoplaxcolimit, X and b X are mutually determined as functors. For a Grothendieck topos F with a lex site of presentation S h ( C F , J F ) and F a B -object in F , define the site ( V opF , J F ) as: − V F has as objects the pairs ( c, l ) with c ∈ C F and l a morphism in V F ( c ), that is some l : F ( c ) → B c,l gotten as a pushout of map in V . As morphisms, it has the pairs ( s, h ) :( c, l ) → ( c ′ , l ′ ) with s : c ∈ c ′ and h : B c,l → B c ′ ,l ′ such that F ( c ) B c,l F ( c ′ ) B c ′ ,l ′ lF ( s ) hl ′ − J F is the Grothendieck topology conjointly generated by J F and each of the J F ( c ) , moreexactly by the families of the form(( c, c ) ( s i ,F ( s i )) −→ ( c i , c i )) i ∈ I with ( c s i → c i ) i ∈ I ∈ J op F ( c )that are covers in the definition site transferred through ι : ( C F , J F ) → V F sending c on( c, c ), and families of the form(( c, l ) (1 c ,h i ) −→ ( c, l i )) i ∈ I with F ( c ) B c,l B c,l i ll i h i ! i ∈ I ∈ J opF ( c ) ( l )The spectrum of the T B -modelled topos ( F , F ) is defined as Spec V ,J ( F , F ) = Sh ( V opF , J F )Moreover the e F is defined as the sheafification of the preasheaf of B -objects defined on V F asthe codomain functor F : ( c, l ) B c,l In particular this construction coincides with the specific one above when considering a set valuedmodel, that is a modelled topos of the form ( ∗ , B ) with B seen as the constant sheaf on the point.The spectral site defines an indexed category C op V opF ( − ) → L exc
7→ V opF ( c ) c s → c
7→ V opF ( c ) ( s ∗ ) op → V opF ( c ) where as the transition morphism at an arrow s : c → c in C F , one has a pushout functor V F ( c ) F ( s ) ∗ → V F ( c ) V opF ( c ) V opF V opF ( c ) F ( s ) op ∗ ι opc ι opc φ s whose component φ sn at an object n : F ( c ) → C is given by the opposite of the pushout map n ∗ F ( s ) op as below F ( c ) F ( c ) C F ( s ) ∗ C n F ( s ) p F ( s ) ∗ nn ∗ F ( s ) The data of the fibers inclusions ι c : V op F ( c ) ֒ → V op F together with those transformations definean oplax cocone exhibiting V opF as the oplax colimit of the indexed category V opF ( − ) : C op → C at Then the construction above exhibits the spectral site of F the oplax-colimit in Lex V F = oplaxcolim c ∈C F V F ( c ) But now, at each c in C , the fiber V F ( c ) is itself equiped with a topology J F ( c ) . Moreover wehave the following: Proposition 4.3.2.
The transition morphisms F ( s ) ∗ are cover preservingProof. For a family ( m ′ i : n ′ → n ′ i ) in V F ( c ′ ) obtained as a pushout along some f of a coveringfamily ( k i : K → K i ) of finite presentation under n as visualised below F ( c ) F ( c ′ ) F ( s ) ∗ B c ′ ,n ′ B c ′ ,n ′ KF ( s ) ∗ B c ′ ,l ′ i B c ′ ,l ′ i K iF ( f ) ∗ n ′ i F ( f ) ∗ n ′ F ( s ) n ′ n ′ i q F ( s ) ∗ ( m i ) m i fk i q where the front and back square both are pushouts, so that, by left cancellation of pushouts, thebottom square also is. Then the composite of the two bottom pushouts still is a pushout: F ( s ) ∗ ( m ′ i ) = ( l ∗ F ( s )) ∗ ( k i )and, as a pushout of finitely presented covering family, it is a covering family of F ( s ) ∗ ( n ′ ) in V F ( c ) .Hence we have a fibered lex site V op F ( − ) : C op → Lex ; denote as Z V op F ( − ) C p F the associated fibered site. Now recall that each of the sites ( V op F ( c ) ) , J F ( c ) ) are the spectra site of F ( c ), that is, Spec ( F ( c )) = Sh ( V op F ( c ) ) , J F ( c ) ). Then the associated fibered topos has the spectrumof the F ( c ): as fiber, that is, we have a fibered topos Z Spec ( F ( − )) C p F Spec ( F ( s )) ∗ : Spec ( F ( c )) → Spec ( F ( c )) for s : c → c . We can consider its associated direct fibration Z Spec ( F ( − )) ′ C p F ′ Recognize now in the definition of the topology J F that it is generated from the horizontal andvertical families as done in the previous section. Applying theorem 4.2.12 we have the following: Theorem 4.3.3.
The spectrum of F is the topos of continuous sections of the direct fibrationassociated to the fibered spectral topos: Spec ( F ) ≃ Γ J ( p F ′ ) Proposition 4.3.4.
We have a cloven fibration of sites defined by the projection V opF → C F , whichis cover reflecting, and its right adjoint given by the inclusion of C F which is cover preserving.Hence there is a geometric surjection Spec ( F ) ։ F Proof.
We address the fibrational aspect (where cocartesiannes is an easy consequence of the useof the pushout) of this in the next remarks; first just observe that the projection V opF → C F is coverreflecting because of the very existence of the adjoint inclusion as for any ( c, l ) in V F any J F -cover( c i → c ) in C F is lifted to a cover (( c i , F ( c i ) ) → ( c, F ( c ) )) in V F .On the other side, taking the etale part of the composite of an etale map with the restrictionmap should define some kind of adjoint of those transition functors: F ( c ) F ( c ′ ) B c,l B lF ( s ) l F ( s ) n lF ( s ) u lF ( s ) However the etale part n lF ( s ) in such factorization needs not be a finitely presented map even if l was. This can be fixed at the level of the indcompletion of the underlying site in which live allsuch factorizations. Actually we have the following adjunction for each s : c → c ′ Ind( V F ( c ′ ) ) ⊥ Ind( V F ( c ) ) F ( s ) ∗ ( − ) n ( − ) ◦ F ( s ) Where the unit and counit are defined as F ( c ) F ( c ′ ) F ( s ) ∗ B c ′ ,l ′ B c ′ ,l ′ B F ( s ) ∗ l ′ F ( f ) ∗ l ′ F ( s ) l ′ n F ( s ) ∗ l ′ q η l ′ u F ( s ) ∗ l ′ F ( c ) F ( c ′ ) F ( s ) ∗ B lF ( s ) B lF ( s ) B c,l l F ( s ) ∗ n lF ( s ) F ( s ) n lF ( s ) ǫ l u lF ( s ) where η ′ l is induced from the etale/locale factorization being the terminal one amongst those withan etale map on the left, while the counit is induced by property of pushout. Beware that thisadjunction will be actually flipped upside down between the actual underlying sites of the spectrum.Observe that the right adjoint n ( − ) ◦ F ( s ) is finitary as its left adjoints send finitely presentedobject into finitely presented objects. It is in fact the morphism of locally finitely presentablecategories in induced by F ( s ) op ∗ seen as a left exact functor V opF ( c ′ ) → V opF ( c ) Proposition 4.3.5.
We have a cloven bifibration defined by the projection
Ind( V opF ) → C F and itsright adjoint given by the inclusion of C F . roof. We have to prove the cartesianness and cocartesiannes of the projection. First, for a given( c, l ) and an arrow c → c ′ , the cleavage comes from the factorization by taking the etale partof lF ( s ) as pullback object s ∗ ( c, l ) = ( c ′ , n lF ( s ) ) and the local part as the lifting. This choice iscocartesian: indeed, for any situation as the following left diagram: F ( c ) F ( c ′ ) F ( c ′′ ) B c,l B lF ( s ) B c ′′ ,l ′′ c c ′ c ′′ l F ( s ) n lF ( s ) F ( t ) l ′′ F ( x ) u lF ( s ) hxs t F ( c ) F ( c ′ ) F ( c ′′ ) F ( s ) ∗ B c ′ ,l ′ B c ′ ,l ′ B c ′′ ,l ′′ c cc ′′ F ( t ) F ( s ) ∗ l ′ q F ( s ) l ′ F ( x ) l ′′ ∃ h st x we get two factorizations of lF ( s ) F ( t ): one with an etale part on the left hl ′′ and one with a localpart on the right u lF ( s ) n lF ( s ) F ( t ) so by the property of admissibility there exists a composite arrowbetween them B c ′′ ,l ′′ → B lF ( s ) F ( t ) → B lF ( s ) providing the desired lifting. On the other hand, cartesianness it obvious from the use of pushoutas depicted in the right diagram.Concerning the structural sheaf, observe that one has the following restrictions at the lever ofthe structural presheaf V opF BV opF ( c ) Fi c F ( c ) as the codomain functor has to be applied fiberwise in the indexed site. Hence i ∗ c F = F ( c ). Butnow inverse image commute with sheafification, so that one has i ∗ c e F = g F ( c )That is, for any ( c, n ) ∈ V F one has e F ( c, n ) = g F ( c )( n )while at a point ( p, x ) of Spec ( F ), that is a local form ξ : x ∗ F → A ξ with x : S → F , one has e F | ( x,ξ ) ≃ A x In general Stone duality is considered from the point of view of concrete dualities. Howeverit is possible to provide a spectral account of it, or at least to reconstruct the Stone dual of adistributive lattice as the spectra of a certain geometry. But in this process the stone dual is alsoendowed with a structural sheaf which is not considered in classical Stone duality, and the spec-trum will be adjoint to the global section functor rather than the “compact open” functor, which,in Stone duality, returns the basis of compact open set of Stone spaces to which it is hence restricted.As for Stone duality, this construction allows can be done in two manners, equipping the Stonedual with either the
Zariski or the coZariski topology: this depends on the way we define theadmissibility structure. 57e recall here the admissibility structure for Zariski. Our ambient locally finitely presentablecategory is DLat, the category of bounded distributive lattices. Recall that distributive latticesare not 1-regular, that is, for any given ideal there are several congruences whose class in 1 is thisideal. For etale maps one can choose : those are morphisms A ։ A/θ with θ minimal amongst congruences whose class in 1 is [1] θ . One can easily prove this class is closed bycomposition and colimits, contains iso and is left-cancellative. For a lattice D finitely presented1-minimal quotients are of the form D → D/θ ( a, Then it can be shown that one has a factorization system (
M inQuo , − Cons ) on DL at , where1 − Cons is the class of maps that reflect the top element, that is those f : D → D ′ such that f − (1) = { } .Then define the category L oc DL at − cons having: − as objects local distributive lattices , where { } is prime filter − as morphisms 1-conservative morphisms f .Then L oc DL at − cons ֒ → DL at is a multireflection. But we can also axiomatize the category of locallattices as follows: define J on DL at opfp generated by ( f i : D ։ D/θ i ) such that T i ∈ I θ i = diag D .Now observe that a distributive lattice D is J -local if an only if { } is a prime ideal, that is, D has a minimal point L → a = 1 on 0. Local lattices are the points of the topos S h ( DL at op fp , J ).The associated spectrum for D is( Spec ( D ) = ( F P rimeD , τ
ZariskiD ) , e D )with e D defined on the basis as e D ( U coZara ) = D/θ ( a, for any a ∈ D . Then we have an adjunction DL at op ⊥ DL at ∗ − S paces Spec
Zar Γ Then one recovers the Stone spaces as the underlying spaces of affine DL at -spaces.The spectral site of a distributive lattice D is ( Zar opD , J ( D )) where Zar D consists of finitelypresented 1-minimal quotients of D ; in particular for a filter F of D , a factorization as below D ։ D/θ ( a, , and a factorization D D/θ F D/θ ( a, q a q F expresses the fact that a lies in F .Now, at a distributive lattice D , the induced topology J ( D ) consists of finite families ( D ։ D/θ ( a i , ) i ∈ I with W a i = 1. Being made of epi, Zar D is a poset and Zar D ≃ D op and we have D ֒ → τ Zar . J ( D ) coincides with the coherent topology on D . The spectrum is spatial and isequiped with the Zariski topology which is the frame of filters F D .Opens of Zariski topology form the frame τ Zar = Sh ( Zar opD , J ( D )) = Sh ( I D ): Zariski openscorrespond to ideals of D and D ֒ → I D is a base of compact open of Zariski topology. On the otherside, D ֒ → ( F D ) op , but a filter F of D defines a filtered diagram whose colimit is the 1-minimalquotient at F S ։ S/θ minF = colimf ilt a ∈ F S/θ ( a, saturated compact of Zariski topology. A prime filter x corresponds to the 1-quotient D ։ D/θ x , which is the saturated compact ↑ x , the focal component in x .58owever Zariski geometry is not the only way to retrieve Stone duality. One could have eitherdefined the factorization system (0 − minQuo, − cons ), whith 0 − minQuo as the minimal quotientwith a fixed ideal, and 0 − cons the morphisms f such that f − (0) = { } , and could have takenas local objects those distributive lattices with { } prime.The CoZariski site would have been ( coZar opD , J ( D )) with coZar D made of the minimal quo-tients D ։ D/θ ( a, and J ( D ) defined by ( a i ) i ∈ I such that V i ∈ I a i = 0.Then D ≃ CoZar D , so that D op ֒ → τ coZar = Sh ( coZar opD , J ( D )) ≃ ( F D ) op . Then filters arethe closed subsets of coZariski topology. On their side ideals I D define filtered colimits of finitelypresented minimal quotients maps in coZar D , hence correspond to saturated compacts.We recover Hochster duality from Isbell duality at a lattice seen as a category. Dτ Zar ≃ I D τ op coZar ≃ ( F D ) op [ D op ,
2] [ D, ⊣⊣ Jipsen-Moshier duality is an example of an geometry without specification of local objects, sothat any object is actually local.In a space X with specialization order ⊑ , a compact open filter is an upset F for ⊑ which isboth open and compact. For X denote KOF ( X ) its set of compact open filters. A point x is basiccompact open if ↑ is a compact open filter. Hofmann-Mislove-Stralka spaces are sober spaces X such that KOF ( X ) is a basis closed under finite intersection.Denote HMS the category of HMS spaces with continuous maps f : X → Y such that f − restrictsto KOF ( Y ) → KOF ( X ). Any compact open filter of a HMS space has a focal point. Moreover, ina HMS space, any point is a directed join of basic compact open points. The specialization ordermakes ( X, ⊑ ) a complete lattice, and there are both an initial and terminal points in such a X .Then one can recover the spectra of Jispen and Moshier duality for ∧ -semilattices with unit: ∧ − SL at op ≃ HMS Defines
Spec ( S ) = ( F S , ↓ S ). For X HMS,
KOF ( X ) is a ∧ -semilattice. Moreover, for a semillaticeone has S ≃ KOF ( Spec ( S )), while for a HMS space, one has X = Spec ( KOF ( X )).If S is a ∧ -semilattice, F S ≃ ( I primeS ) op is a complete lattice. Moreover any filter of a ∧ -semilattice is trivially prime. This says that Spec ( S ) = ∧ − Slat [ S, − M inQuo, − Cons ) also is a factorization system on ∧ − SL at , and no topology. Finitelypresented etale maps under a ∧ -semilattice S are principal 1-minimal quotients S ։ S/θ ( a, F one has a minimalquotient S ։ S/θ minF = colimf ilt a ∈ F S/θ ( a, θ minF is the congruence in F given as θ minF = T { θ | [1] θ = F } . This defines a point of Spec ( S ), and any saturated compact actually has a focal point. Observe that this is a situation ofthe purely factorization geometries as in theorem 3.1.9,which is induced from the stable inclusionof the right class of a factorization system 1 − Cons ֒ → ∧ −
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