2-filteredness and the point of every Galois topos

2-FIL TEREDNESS AND THE POINT OF E VER Y GALOIS TOPOS EDUARDO J. DUBUC Abstra ct. A locally connected top os is a Galois top os if the Galois ob jects generate the t op os. W e show that the full sub category of Galois ob jects in any connected lo cally connected top os is an in versely 2-fi ltered 2-category , and as an app lication of t he construction of 2-filtered bi- limits of top oi, we show that every Galois top os has a p oint. introduction. Galois topoi (definition 1.5) arise in Grothendiec k’s Ga- lois theory of lo cally connected top oi. They are an sp ecial kind of atomic top oi. It is w ell known that atomic top oi ma y b e p oint less [6], ho w ever, in this pap er w e show that any Galo is top os has p oin ts. W e sh o w h ow the full sub catego ry of Galois ob jects (definition 1.2) in any connected lo cally connected top os E h as an structure of 2-filtered 2-category (in the sense of [3]). Then w e sho w that the assignmen t, to eac h Galois ob ject A , of the category D A of connected lo cally constant ob jects trivialized by A (definition 3.1) , determines a 2-functor in to the category of catego ries. F urthermore, this 2-system b ecomes a p ointed 2-system of p oin ted sites (considering the top ology in whic h eac h single arro w is a co v er). By the results on 2-filtered bi-limits of top oi [4], it follo w s that, if E is a Galois top os, th en it is the b i-limit of this system, and thus, it has a p oin t. context. Throughout this pap er S = S ets denotes the top os of sets. All top oi E are assu m ed to b e Gr othendiec k to p oi (ov er S ), the s tr ucture map will b e denoted by γ : E → S in all cases. 1. Galo is to poi and the 2-fil tered 2-ca tegor y of Galo is ob jects W e recall now the definition of Galois ob j ect in a top os. The original definition of Galois ob ject giv en in [5 ] was r elativ e to a su rjectiv e p oin t of the top os: 1.1. Definition. L et E γ − → S b e a top os furnishe d with a surje ctive p oint, that is, a ge ometric morphism S p − → E whose inverse image functor r efle cts isomorph isms. Then, an obje ct A is a Galois obje ct if: i) Ther e exists a ∈ p ∗ A su c h that the map Aut ( A ) a ∗ − → p ∗ A , define d by a ∗ ( h ) = p ∗ ( h )( a ) is a b ije ction (the same holds then for any other b ∈ p ∗ A ). ii) A is c onne cte d and A → 1 is epimorphic. Notice that in the con text of the classica l Galois theory (Artin’s interpre- tation) this definition coincides with the defin ition of n orm al extension. It is easy to c h ec k that in the presence of a (su rjectiv e) p oin t the follo wing unp oin ted defin ition is equ iv alen t. 1 2 EDUARDO J. DUBUC 1.2. Definition. An obje ct A i n a top os γ : E → S is a Galois obje ct if: i) The c anonic al map A × γ ∗ Aut ( A ) − → A × A , describ e d by ( a, h ) 7→ ( a, h ( a )) , is an isomorphism. ii) A is c onne cte d and A → 1 is epimorphic. In p articular, A is a c onne cte d lo c al ly c onstant obje ct. It fol lows: iii) “ Z ⊂ A ⇒ Z = ∅ or Z = A ”. 1.3. Prop osition. L et A b e a Galois obje ct in a lo c al ly c onne cte d top os E γ − → S , and let X b e any obje ct such that ther e exi sts an epimorphism A e − → X . Then, the c anonic al map A × γ ∗ [ A, X ] → A × X , describ e d by ( a, f ) 7→ ( a, f ( a )) , is an i somorph i sm. In p articular, X is a c onne cte d lo c a l ly c onstant obje ct split by the c over A → 1 . Pr o of. C on s ider the follo wing comm u tativ e d iagram: A × γ ∗ Aut ( A ) ∼ = / / A × γ ∗ e ∗   A × A A × e   A × γ ∗ [ A, X ] / / A × X This sho w s that the m ap (b ottom r o w) is an epimorphism. T o see that it is also a monomorphism, let Z s / / t / / A × γ ∗ [ A, X ] (with Z 6 = ∅ an d connected) b e a pair of maps which b ecome equal into A × X . T he maps s and t are giv en b y pairs ( u, f ) and ( v , g ), with Z u / / v / / A and A f / / g / / X , suc h that u = v and f ◦ u = g ◦ v . Then , E q ual iz er ( f , g ) 6 = ∅ . It follo ws from iii) in d efi nition 1.2 that f = g . Since connected ob jects generate the top os, th is finish es the pro of.  Giv en an y t wo Galois ob jects A , B , in a connected lo cally connected top os E , an y connected comp onent of the pr o duct A × B is a connected lo cally constan t ob ject. It follo ws from the existence of Galois closure (see for example [2] A.1.4) that there is a Galois ob j ect C and morphisms C → A , C → B . The f u ll sub category A of Galois ob jects fails to b e (in v ersely) filtered b ecause, cle arly , differen t morph ism s A u / / v / / B b et w een Galois ob jects cannot b e equalized from a Galois ob ject C → A unless they are already equal. Ho we v er, we ha ve : 1.4. Prop osition. The c ate gory A of Galois obje cts in a c onne cte d lo c al ly c onne cte d top o s b e c omes a (inversely) 2-filter e d 2-c ate gory (in the sense of [3] ) by adding a formal 2-c el l A u / / v ⇓ θ vu / / B b etwe en any two morphisms, satisfying the fol lowing e qu ations: θ uu = id, θ w v ◦ θ vu = θ w u , ( thus θ uv = θ − 1 vu ) , θ vu θ sr = θ sv ru Pr o of. the pro of is v ery easy , we let the in terested reader look at the defini- tion of 2-filtered 2-catego ry giv en in [3] and v erify the assertion.  2-FIL T EREDNESS AND THE POINT OF EVER Y GALOIS TOPOS 3 After Grothend iec k “Categories Galoisiennes” of [5] and Mo erdiejk “Ga- lois T op os” of [7 ], we state the f ollo wing defin ition: 1.5. Definition. A Galois T op os is a c onne cte d lo c a l ly c onne cte d top os gen- er ate d by its Galois obje cts, or, e quivalently, such that any c onne cte d obje ct is c over e d by a Galois obje ct (notic e that we do not r e quir e the top os to b e p ointe d). Since Galois ob jects are conn ected lo cally constan t ob j ects, it f ollo ws that Galois top oi are generated by lo cally constan t ob jects. O n the other hand , the existence of Galois cl osure (see for example [2] A.1.4) shows that any suc h top os is a Galois top os. T h u s, a c onne cte d top o s is a Galois top os if and only if it is gener ate d by its c onne cte d lo c al ly c onstant obje cts. . It follo ws: 1.6. Proposition. Any Galois top os is a c onne cte d atomic top os; that is, is a c onne cte d lo c al ly c onne cte d b o ole an top os. 2. Galo is Topoi as fil tered bi-limits of top oi with points. Consider no w a connected lo cally connected top os E , and let C b e a f ull sub catego r y of connected generators. Let A ∈ C b e a Galois ob ject. W e denote by C A ⊂ A th e full su b category whose ob jects are the X ∈ C b elo w A (remark that this is not the comma category ( A, C )). If there is a morph ism A → B b et wee n Galois ob jects, clearly C B ⊂ C A , and if E is a Galois top os, b y definition th e category C is the filtered un ion: . . . C B   / / C A   / / . . .   / / C The top os E is the top os of sh eav es for the canonical top ology on C , and eac h C A is itself a site with th is top ology . It follo ws from the theory of filtered in verse bi-limits of top oi ([1] Exp ose VI) that, if E A is the top os of sheav es on C A , then the top os E is an inv erse bi-limit of top oi: · · · E B ← − E A ← − · · · ← − E The represen table fu nctor C A [ A, − ] − → S is a p oin t of the site, thus th e top oi E A are all p ointed top oi. Ho w ever, a p oint for the site C is equiv alen t to a sim u ltaneous c hoice of p oin ts for eac h A comm uting w ith all the inclusions C B ⊂ C A . That is, an elemen t of the in verse limit of sets: · · · P oint s ( C B ) ← − P oints ( C A ) ← − · · · ← − P oint s ( C ) whic h , a p riori, ma y b e empty . 3. Galo is Topoi as p ointed 2-fil tered bi-limits o f p ointed to poi. W e shall consider next a d ifferen t category asso ciated to an y Galois ob ject. 3.1. Definition. L et E γ ∗ − → S b e a c on ne cte d lo c al ly c onne cte d top o s, let C b e the sub c ate gory of c onne cte d obje cts, and let A ∈ C b e any Galois obje ct. The c ate gory D A is define d as the bi- pul lb ack of c ate gories: D A / /   C A × ( − )   S γ ∗ / / E / A 4 EDUARDO J. DUBUC The obje cts and arr ows of D A c an b e describ e d as fol lows (wher e π 1 denotes the first pr oje ction): O b : tr ipl es ( X, S, σ ) , X ∈ C , ∅ 6 = S ∈ S , ( π 1 , σ ) : A × γ ∗ S ∼ = − → A × X Ar r : ( X, S, σ ) → ( Y , T , ξ ) : X f − → Y , S η − → T | ξ ( a, η ( s )) = f ( σ ( a, s )) . Giv en an y ( X , S, σ ) ∈ D A , since S 6 = ∅ it f ollo ws that [ A, X ] 6 = ∅ . Thus, X ∈ C A . F urthermore, since X is connected and lo cally constant , an y m ap A → X is an epimorphism. In fact, for the same reason, any map X → Y is an epimorphism. It follo w s that so it is A × γ ∗ ( η ). Since A is connected, it follo ws that η is also an ep imorphism, thus a surj ectiv e fu nction of sets. 3.2. Remark. Given any arr ow ( X, S, σ ) f , η − → ( Y , T , ξ ) , f is an epimor- phism and η a surje ctive function. 3.3. Prop osition. The func tor D A ∼ − → C A , ( X, S, σ ) 7→ X , is an e quiva- lenc e of c ate gories. D A has a site structur e such that any single arr ow is a c over. The induc e d morphism E A ∼ − → P A is an e quivalenc e (wher e P A denotes the top os of she aves on D A ). Pr o of. J u st by d efinition of connected ob ject it immediately follo ws that giv en ( X, S, σ ), ( Y , T , ξ ), and an arrow X f − → Y , there exists a un ique S η − → T whic h determines an arrow ( f , η ) in D A . Thus, th e fun ctor is fu ll and faithful. That it is essen tially surjectiv e follo ws by p rop osition 1.3. The second assertion is clear (consider r emark 3.2).  3.4. Prop osition. a) The functor D A p ∗ A − → S , p ∗ A ( X, S, σ ) = S , determines a p oint of the site. This p oint is natur al ly isomor phic to the r epr esentable fu nctor [ A, − ] under the e quivalenc e D A ∼ − → C A . b) A morphism b etwe en Galois obje cts A u − → B determines a morphism of sites D B u ∗ − → D A c ommuting with the p oints p ∗ A ◦ u ∗ = p ∗ B . c) Given any two morphisms b etwe en Galois obje cts A u / / v / / B , ther e is a c anonic a l natur al tr a nsformation D A u ∗ / / v ∗ ⇓ θ vu / / D B satisfying the e quations in definition 1.4 d) The fol lowing diagr am c ommutes: C B   / / C B D A ∼ = O O u ∗ / / v ∗ ⇓ θ vu / / D B ∼ = O O Pr o of. a) I t follo ws from remark 3.2 and the fact that A is conn ected. F ur ther- more, we know there exists an epimorp hism A → X . The natural b ijection [ A, X ] ∼ = S follo ws in the same wa y as item c) b elo w (recall prop osition 1.3). 2-FIL T EREDNESS AND THE POINT OF EVER Y GALOIS TOPOS 5 b) follo ws by th e unive r sal prop ert y of bi-pullb ac ks. Giv en A u − → B , an explicit constru ction of u ∗ is the follo wing: ( X, S, σ ) = u ∗ ( Y , T , ξ ), X = Y , S = T , and σ is the map u n iquely determined by the equation σ ( s, a ) = ξ ( s, u ( a )). c) Consider the description in b) and the follo wing d iagram: u ∗ ( Y , T , ξ ) : θ vu   A × γ ∗ S ∼ = / / A × γ ∗ ( η )   A × X id   v ∗ ( Y , T , ξ ) : A × γ ∗ S ∼ = / / A × X By definition of connected ob ject, there exists a u nique S η − → S making the square comm utativ e. Define θ vu = ( id X , η ). C learly , the equations h old by uniqueness. Finally , d) is clear by definition of u ∗ , v ∗ and θ vu  Ever y Galois to pos E h as a point: It follo w s from prop osition 3.4 b) and c) that the assignment of the site D A to a Galois ob ject A d etermines a 2-filtered 2-system of categ ories wh ic h has a bi-colimit that w e denote D (see [3]). . . . D B u / / v ⇓ θ vu / / D A / / / / . . . / / D It f ollo ws from the resu lts in [4] that the site stru ctur es in the categories D A determine a site stru cture on D in such a wa y that we h a ve an in verse bi-limit of the top oi of shea ves: . . . P B P A u o o v ⇓ θ vu o o . . . o o o o P o o where the top os P is the top os of shea ves on the site D . F rom p rop osition 3.4 a) it follo ws that all th e top oi P A are p oin ted, and from b) it follo ws that in this case these p oin ts ind uce a p oint S → P of the bi-limit top os P . Fin ally , from prop ositions 3.4 d) and 3.3, it follo ws that the top oi E and P are equiv alen t top oi, E ∼ − → P . T hus, E has a p oin t S → E determined by any inv erse equiv alence and the p oint of P . Referen ces [1] Artin M, Grothendiec k A, V erdier J., S GA 4 , (1963-64) , Springer Lecture Notes in Mathematics 270 ( 1972). [2] Dubuc, E. J., On th e rep resentation th eory of Galois and Atomi c T opoi , Journal of Pure and Ap plied Algebra 186 (2004) 233 - 275. [3] Dubuc, E. J., Street, R ., A construction of 2-filtered bicolimits of categories , Cahiers de T opologie et Geometrie Differentielle, (2005). [4] Dubuc, E. J., Y uhjtman, S., A constru ct ion of 2-filtered b i-limits of T opoi , to app ear. [5] Grothendiec k A., SGA1 ( 1960-61) , Springer Lecture Notes in Mathematics 224 (1971). [6] Makk ai, M., F ull contin uous embeddings of top oses , T rans. AMS 269 (1982) 167-196. [7] Moerdijk I., Pro discrete groups and Galois top oses Proc. Kon. Nederl. Aka d . v an W etens. S eries A, 92-2 (1989).