A construction of 2-cofiltered bilimits of topoi

A CONSTRU CTION OF 2-COFI L TERED BILIMITS OF TOPOI EDUARDO J. DUBUC, SERGIO YUHJTMAN introduction W e sho w the existence of bilimits of 2-cofiltered diagrams of top oi, gen- eralizing the construction of cofiltered b ilimits dev eloped in [2]. F or an y giv en suc h diagram, we sho w that it can b e repr esen ted by a 2-cofiltered diagram of small sites with fin ite limits, and we constr u ct a sm all site for the inv erse limit top os. Th is is d on e b y taking the 2-filtered bicolimit of the underlying categories and inv erse image functors. W e use th e constru ction of this bicolimit develo p ed in [4], wh ere it is prov ed th at if the cate gories in the diagram ha ve finite limits and the transition fun ctors are exact, then the b icolimit category has finite limits and th e pseud o cone functors are ex- act. An applicati on of our result here is the f act that every Galois top os has p oin ts [3]. 1. Back ground, term inology and no t a t ion In this sectio n we recall some 2-c ategory and top os theory that we sh all explicitly need, and in this w a y fix notation and terminology . W e also include some in-edit pro ofs w hen it seems necessary . W e distinguish b et ween smal l and lar ge sets. Categories are supp osed to ha ve s mall hom-sets. A categ ory with large hom-sets is ca lled i l le gitimate . Bicolimits By a 2-c ate gory w e mean a C at enric hed category , and 2-functors are C at functors, w h ere C at is the category of sm all categories. Given a 2-cat egory , as usual, w e d enote horizon tal comp osition b y juxtap osition, and ve rtical comp osition b y a ′′ ◦ ′′ . W e consider ju xtap osition more bin ding than ′′ ◦ ′′ (th u s xy ◦ z means ( xy ) ◦ z ). If A , B are 2-catego ries ( A small), w e will denote by [[ A , B ]] th e 2- catego ry whic h has a s ob jects t he 2-functors, as arro ws the pseudonatur a l tr ansformations , and as 2-c ells the mo dific ations (see [5] I,2.4.). Give n F, G, H : A − → B , there is a functor: (1.1) [[ A , B ]]( G, H ) × [ [ A , B ]]( F , G ) − → [[ A , B ]]( F , H ) T o ha v e a handy reference we will explicitly describ e these data in the par- ticular cases w e use. A pseudo c one o f a diagram giv en b y a 2-fun ctor A F − → B to an ob ject X ∈ B is a pseudon atur al tr ansformation F h − → X from F to th e 2-functor whic h is constan t at X . It consists of a family of arro ws ( h A : F A → X ) A ∈A , and a family of inv ertible 2-cells ( h u : h A → h B ◦ F u ) ( A u − → B ) ∈A . A mor- phism g ϕ = ⇒ h of pseudo cones (with same v ertex) is a mo d ification, as suc h , it consists of a family of 2-cells ( g A ϕ A = ⇒ h A ) A ∈A . These data is sub ject to the follo wing: 1 2 EDUARDO J. DUBUC, SERG IO YUHJTMAN 1.2 ( Pseudo c one and morph ism of pseudo c o ne e quations) . p c0. h id A = id h A , for e ach obje ct A p c1. h v F u ◦ h u = h vu , for e ach p a ir of arr ows A u − → B v − → C p c2. h B F γ ◦ h v = h u , for e ach 2-c el l A u / / γ ⇓ v / / B p cM. h u ◦ ϕ A = ϕ B F u ◦ g u , for e ach arr ow A u − → B W e state and pro ve now a lemma whic h, although exp ected, needs nev er- theless a pro of, and for which w e do not ha ve a reference in the literature. As t he rea der will realize, th e statemen t concerns g eneral pseudonatural transformations, but we treat here the particular case of pseudo cones. 1.3. Lemma. L et A F − → B b e a 2-functor and F g − → X a pseudo c one. L et F A h A − → X b e a family of mo rphisms to gether with invertible 2 -c el ls g A ϕ A = ⇒ h A . Then, c onjugating by ϕ determines a pseudo c one structur e f or h , unique such that ϕ b e c o mes an isomorphism of pseudo c ones. Pr o of. If ϕ is to b ecome a pseudo cone morp hism, the equation p cM. ϕ B F u ◦ g u = h u ◦ ϕ A m u st hold. Thus, h u = ϕ B F u ◦ g u ◦ ϕ − 1 A determines and defi nes h . Th e pseud o cone equations 1.2 for h follo w from the resp ectiv e equations for g : p c0. h id A = ϕ A ◦ g id A ◦ ϕ − 1 A = ϕ A ◦ id g A ◦ ϕ − 1 A = id h A p c1. A u − → B v − → C : h v F u ◦ h u = ( ϕ C F v ◦ g v ◦ ϕ − 1 B ) F u ◦ ϕ B F u ◦ g u ◦ ϕ − 1 A = ϕ C F ( v u ) ◦ g v F u ◦ ϕ − 1 B F u ◦ ϕ B F u ◦ g u ◦ ϕ − 1 A = ϕ C F ( v u ) ◦ g v F u ◦ g u ◦ ϕ − 1 A = ϕ C F ( v u ) ◦ g vu ◦ ϕ − 1 A = h vu p c2. F o r A u / / ⇑ γ v / / B w e m ust see h B F γ ◦ h v = h u . Th is is the same as h B F γ ◦ ϕ B F v ◦ g v ◦ ϕ − 1 A = ϕ B F u ◦ g u ◦ ϕ − 1 A . Canceling ϕ − 1 A and comp os- ing with ( ϕ B F u ) − 1 yields (1) ( ϕ B F u ) − 1 ◦ h B F γ ◦ ϕ B F v ◦ g v = g u . F rom the compatibilit y b et ween ve rtical and horizon tal comp osition it follo ws ( ϕ B F u ) − 1 ◦ h B F γ ◦ ϕ B F v = ( ϕ − 1 B ◦ h B ◦ ϕ B )( F u ◦ F γ ◦ F v ) = g B F γ . Thus, after r ep lacing, (1) b ecomes g B F γ ◦ g v = g u .  Giv en a small 2-diagram A F − → B , the category of pseudo cones and its morphisms is, by definition, pc B ( F , X ) = [[ A , B ]]( F, X ). Give n a p seudo- cone F f − → Z and a 2-cell Z s / / ξ ⇓ t / / X , it is clear and straight forw ard ho w to define a morphism of pseudo cones F sf / / ξ f ⇓ tf / / X wh ich is the co mp osite F f − → Z s / / ξ ⇓ t / / X . This is a particular case of 1.1, thus comp osing with f d etermines a functor (denoted ρ f ) B ( Z , X ) ρ f − → pc B ( F , X ). A CONSTRUCTION OF 2-COFIL TERED BILIMITS OF TOPOI 3 1.4. Definition. A pseudo c one F λ − → L is a bicolimit of F if f or every obje ct X ∈ B , the functor B ( L, X ) ρ λ − → pc B ( F , X ) is an e quivalenc e of c ate gories. This amo unts to the fol lowing: bl) Given any pseudo c one F h − → X , ther e exists an arr ow L ℓ − → X and an invertible morphism of pseudo c ones h θ = ⇒ ℓλ . F urthermor e, given any other L t − → X and h ϕ = ⇒ tλ , ther e e xists a unique 2 -c el l ℓ ξ = ⇒ t su ch that ϕ = ( ξ λ ) ◦ θ (if ϕ is invertible, then so it is ξ ). 1.5. Definition. When the functor B ( L, X ) ρ λ − → pc B ( F , X ) is an i somor- phism of c ate gories, the bic olimit is said to b e a p seudo colimit . It is kn own that the 2-category C at of s mall catego ries has all small pseu- do colimits, then a “fortiori” all small bicolimits (see for example [7]). Giv en a 2-fun ctor A F − → C at we denote by L im − − → F the v ertex of a bicol imit cone. In [4] a sp ecial construction of the pseud o colimit of a 2-filtered diagram of categories (not necessarily s mall) is made, and u sing this construction it is p ro ved a result (t heorem 1 .6 b elow) whic h is the k ey to our construction of small 2-filtered bilimits of top oi. Notice that ev en if the categories of the system are large, condition b l) in d efinition 1.4 mak es sense and it defin es the bicolimit of large categories. W e denote by C A T f l the il le gitimate (in the sense that its hom-sets are large) 2-cate gory of finitely complete c atego ries a nd exact (that is, finite limit pr eserving) fun ctors. 1.6. Theorem ([4] T heorem 2.5) . C A T f l ⊂ C AT is close d under 2-filter e d pseudo c olimits. Namely, given any 2-filter e d diagr am A F − → C AT f l , the pseudo c olimit pseudo c one F A λ A − → L im − − → F taken in C AT is a pseudo c olimit c one in C AT f l . If the index 2-c ate gory A as wel l as al l the c ate g ories F A ar e smal l, then L im − − → F is a smal l c ate gory.  T op oi By a site we mean a category furnished with a (Grothendieck) top ology , and a small s et of ob jects capable of co v ering an y ob ject (ca lled top olo gic al gener ators in [1]). T o simplify we wil l c onsider only sites with finite limits. A mor phism of sites w ith fin ite l imits D f − → C is a c ontinous (that is, co v er pr eserving) and exact fu nctor in the other direction C f ∗ − → D . A 2-cell D f / / γ ⇓ g / / C is a natural transformation C g ∗ / / γ ⇓ f ∗ / / D 1 . Under the p resence of top ological generators it can b e easily seen there is only a small set of natural transformations b et we en any t w o con tinous functors. W e denote b y S it the resu lting 2-category o f sites with finite limits. W e denote by S it ∗ the 2-ca tegory whose ob jects are the sites, but taking as arro ws and 2-cells the functors f ∗ and natural transformations resp ective ly . Thus S it is obtained b y formally in v erting the arro ws and the 2- cells of S it ∗ . W e ha ve b y defin ition S it ( D , C ) = S it ∗ ( C , D ) op . 1 Notice that 2- cells are also tak en in t he opp osite direction. This is Grothendieck original con vention, later changed by some auth ors. 4 EDUARDO J. DUBUC, SERG IO YUHJTMAN A top os (also “Grothendiec k top os”) is a category equiv alen t to the cate- gory of shea ves on a site. T op oi are considered as sites furnishin g them with the canonical top ology . T his d etermines a full su b category T op ∗ ⊂ S it ∗ , T op ∗ ( F , E ) = S it ∗ ( F , E ). A morph ism of top oi (also “geometric m orphism”) E f − → F is a pair of adjoint fun ctors f ∗ ⊣ f ∗ (called inv er s e and direct image r esp ectiv ely) E f ∗ / / F f ∗ o o together with an adjun ction isomorphism [ f ∗ C, D ] ∼ = − → [ C , f ∗ D ]. F urtherm ore, f ∗ is r equ ired to p reserv e fi nite limits. Let T op b e the 2-cate gory of top os with geometric morph ism s. 2-arrows are pairs of n at- ural transformations ( f ∗ ⇒ g ∗ , g ∗ ⇒ f ∗ ) compatible with the adjunction (one of the natural transformations co mpletely determines the o ther). T he in verse image f ∗ of a morp hism is an arro w in T op ∗ ⊂ S it ∗ . This deter- mines a forgetful 2-functor (iden tit y on th e ob jects) T op − → S it whic h establish an equiv alence of categories T op ( E , F ) ∼ = S it ( E , F ). Notice that T op ( E , F ) ∼ = T op ∗ ( F , E ) op , n ot an equalit y . W e recall a basic resu lt in the theory of morphism s of Grothend iec k top oi [1] exp ose IV, 4 .9.4. (see f or example [6] Chapter VI I, section 7). 1.7. Lemma. L et C b e a site with finite limits, and C ǫ ∗ − → e C the c anonic al morphism of sites to the top os of she aves e C . Then for any top os F , c om- p osing with ǫ ∗ determines a f unctor T op ∗ ( e C , F ) ∼ = − → S it ∗ ( C , F ) which i s an e q u ivalenc e of c ate g ories. Thus, T op ( F , e C ) ∼ = − → S it ( F , C ) . By the comparison lemma [1] Ex. I I I 4.1 w e can state it in the follo wing form, to b e used in the pr o of of lemma 2.3. 1.8. Lemma. L et E b e any top os and C any smal l set of gener ators close d under finite limits (c onsider e d as a site with the c anonic al top olo gy). Then, f or any top os F , the inclusion C ⊂ E induc e a r estriction functor T op ∗ ( E , F ) ρ − → S it ∗ ( C , F ) which is an e quivalenc e of c ate g ories. 2. 2-cofil tered bilimits of topoi Our w ork with sites is auxiliary to p ro ve our results for top oi, and for this all w e need are s ites with finite limits. The 2-categ ory S it has all s m all 2-cofiltered p seudolimits, which are obtained by fur nishing the 2-filtered pseudo colimit in C AT f l (1.6) of th e un derlying categ ories with the coarsest top ology m aking the co ne injecti ons site morphisms. E x p licitly: 2.1. Theorem. L et A b e a smal l 2-filter e d 2-c ate gory, and A op F − → S it ( A F − → S it ∗ ) a 2-functor. Then, the c ate gory L im − − → F is furnishe d with a top olo gy such that the pseudo c one functors F A λ ∗ A − → L im − − → F b e c ome c ontinuous and ind uc e a n isomo rphism of c ate gories S it ∗ [ L im − − → F , X ] ρ λ − → P C S it ∗ [ F , X ] . The c orr esp onding site is then a pseudo c olimit of F in the 2-c ate gory S it ∗ . If e ach F A is a smal l c ate g ory, then so it is L im − − → F . A CONSTRUCTION OF 2-COFIL TERED BILIMITS OF TOPOI 5 Pr o of. Let F A λ A − → L im − − → F b e th e colimit pseudo cone in C AT f l . W e giv e L im − − → F the top ology generate d b y the families λ A c α − → λ A c , wh ere c α − → c is a co v ering in some F A , A ∈ A . With th is topology , the fun ctors λ A b e- come con tin u ous, thus they corresp ond to s ite morphisms. This d etermines the u pp er horizonta l arro w in the follo w ing diagram (where the vertical arro ws are full su b categories and the low er horizon tal arro w is an isomor- phism): S it [ L im − − → F , X ] / /   pc S it [ F , X ]   C at f l [ L im − − → F , X ] ∼ = / / pc C at f l [ F , X ] T o show that the up p er horizonta l arr o w is an isomorph ism we hav e to c hec k that giv en a pseudo cone h ∈ pc S it [ F , X ], the unique fu nctor f ∈ C at f l [ L im − − → F , X ], corresp onding to h und er the lo w er arr o w, is con tin- uous. But this is clear since from the equation f λ = h it follo ws that it preserve s th e generat ing co ve rs, and th us all co v er s as w ell. Finally , b y the construction of L im − − → F in [4] w e know th at ev ery ob ject in L im − − → F is of the form λ A c for some A ∈ A , c ∈ F A . It follo ws then that the colle ction of ob jects of the form λ A c , with c v aryin g on the set of to p ological g enerators of eac h F A , is a set of top ological generators for L im − − → F .  In the next prop osition we show that an y 2-diagram of top oi restricts to a 2-diagram of small site s with finite limits b y means o f a 2-natural (th us a fortiori p seudonatural) transform ation. 2.2. Proposition. Give n a 2-functor A op E − → T op ther e exists a 2-func tor A op C − → S it suc h that: i) F or any A ∈ A , C A is a small fu l l gener ating sub c ate gory of E A close d under finite limits, c onsider e d as a site with the c anonic al top olo gy. ii) The arr ows and the 2 - c el ls in the C diagr am ar e the r estrictions of those in the E diagr am: F or any 2 c el l A u / / γ ⇓ v / / B in A , the fol lowing diagr am c ommutes (wher e we omit notation for the action of the 2 functors on arr ows and 2 -c el ls): E A u ∗ / / v ∗ γ ⇓ / / E B C A u ∗ / / v ∗ γ ⇓ / / ?  i A O O C B ?  i B O O Pr o of. It is wel l kn o wn that any small set C of ge nerators in a top os ca n b e enlarged so as to d etermine a (non canonical ) small full sub category C ⊃ C closed under finite limits: Cho ose a limit co ne for eac h fi nite diagram, and rep eat this in a denumarable pr o cess. O n the other hand , for the v alidit y of condition ii) it is enough that for eac h transition f unctor E A u ∗ − → E B 6 EDUARDO J. DUBUC, SERG IO YUHJTMAN and ob ject c ∈ C A , we ha ve u ∗ ( c ) ∈ C B (with this, natural transformations restrict automatically). Let’s start with an y set of generators R A ⊂ E A for all A ∈ A . W e w ill naiv ely add ob jects to th ese sets to remedy the failure of eac h condition alternativ ely . In this wa y w e ac hiev e sim ultaneously the t w o conditions: Define C 0 A = R A ⊃ R A . Defin e R n +1 A = S X u − → A u ∗ ( C n X ). R n +1 A is small b ecause A is s mall. C n X ⊂ R n +1 A due to id A . Supp ose no w c ∈ R n +1 A , c = u ∗ ( d ) with d ∈ C n X , and let A v − → B in A . W e h av e v ∗ ( c ) = v ∗ u ∗ ( d ) = ( v u ) ∗ ( d ), thus v ∗ ( c ) ∈ R n +1 B . Defin e C n +1 A = R n +1 A ⊃ R n +1 A . T hen, it is straigh tforward to c hec k that C A = S n ∈ N C n A satisfy th e t wo conditions.  A generalization of lemma 1. 8 to pseudo cones holds. 2.3. Lemma. Given any 2-diagr am of top oi A op E − → T op , a r e striction A op C − → S it as b efor e, and any top os F , the inclusions C A ⊂ E A induc e a r estriction fu nctor pc T op ∗ ( E , F ) ρ − → p c S it ∗ ( C , F ) which is an e quivalenc e of c ate gories. Pr o of. The restrictio n functor ρ is just a particular ca se of 1.1, so it is w ell defined. W e will c heck that it is essen tially sur jectiv e and fully-faithful. The follo wing diagram illustr ates the situation: C A   i A / / u ∗   g ∗ A   E A u ∗   h ∗ A + + V V V V V V V V V V V V ∼ = ϕ A ⇓ h u F C B   i B / / ≡ g ∗ B H H E B h ∗ B 3 3 h h h h h h h h h h h h ∼ = ϕ B essential ly surje ctive : Let g ∈ pc S it ∗ ( C , F ). F or eac h A ∈ A , tak e by lemma 1.8 E A h ∗ A − − → F , ϕ A , h ∗ A i A ϕ A ≃ g ∗ A . By lemma 1.3, h ∗ i inherits a pseu - do cone str ucture such that ϕ b ecomes a pseudo cone isomorphism. F or eac h arro w A u − → B we h a ve ( h ∗ i ) A ( h ∗ i ) u ⇒ ( h ∗ i ) B u ∗ . Since ρ A is fully-faithfu l, there exists a unique h ∗ A h u ⇒ h ∗ B u ∗ extending ( h ∗ i ) u . In th is wa y we obtain data h ∗ = ( h ∗ A , h u ) that restricts to a pseud o cone. Again from the fully- faithfulness of eac h ρ A it is s tr aigh tforwa rd to c h ec k that it satisfies the pseudo cone equ ations 1.2. ful ly-faithful: Let h ∗ , l ∗ ∈ pc T op ∗ ( E , F ) b e t wo pseudo cones, and let e η b e a morp hism b et ween the pseud o cones h ∗ i and l ∗ i . W e ha v e natural transformations h ∗ A i A f η A + 3 l ∗ A i A . Since the inclusions i A are dense, we can extend f η A uniquely to h ∗ A η A + 3 l ∗ A suc h that e η = η i . As b efore, fr om the fully-faithfulness of eac h ρ A it is straigh tforw ard to c hec k that η = ( η A ) satisfies th e morphism of pseudo cone equation 1.2.  A CONSTRUCTION OF 2-COFIL TERED BILIMITS OF TOPOI 7 2.4. Theorem. L et A op b e a smal l 2-filter e d 2-c ate gory, and A op E − → T op b e a 2-fu nctor. L et A op C − → S it b e a r estriction to smal l sites as in 2.2. Then, the top os of she aves ^ L im − − → C on the site L im − − → C of 2.1 is a bilimit of E in T op , or, e qui valently, a bic olimit in T op ∗ . Pr o of. Let λ ∗ b e the p seudo colimit p seudo cone C A λ ∗ A − − → L im − − → C in the 2-cat egory S it ∗ (2.1). Consider the comp osite pseudo cone C A λ ∗ A − − → L im − − → C ε − → ^ L im − − → C and let l ∗ b e a pseudo cone from E to ^ L im − − → C suc h that l ∗ i ≃ ǫ ∗ λ ∗ giv en b y lemma 2.3. W e h av e the follo wing d iagrams com- m u ting u p to an isomorphism: F ^ L im − − → C o o ∼ = L im − − → C ε ∗ o o E l ∗ O O C λ ∗ O O i o o T op ∗ ( ^ L im − − → C , F ) ρ l   ρ ε / / ∼ = S it ∗ ( L im − − → C , F ) ρ λ   pc T op ∗ ( E , F ) ρ / / pc S it ∗ ( C , F ) In the diagram on the right the arro ws ρ ε , ρ λ and ρ are equiv alences of cat- egories (1.7, 2.1 and 2.3 resp ectiv ely), so it follo w s that ρ l is an equiv alence. This finish es the p ro of.  This th eorem sho ws the existence of s mall 2-cofiltered bilimits in the 2-cate gory of topoi and geometric morphisms . But, it sh o ws more, namely , that giv en any small 2 -filtered diagram of top oi, without loss of generalit y , w e can constru ct a small site with fi nite limits for the b ilimit top os out of a 2-cofiltered sub-diagram of small sites with finite limits. Ho wev er, this dep end s on the axiom of choic e (needed f or Prop osition 2.2). W e notic e for the int erested reader that if w e allo w large sites (as in Th eorem 2.1), we can tak e the top oi themselves as sites, and the pro of of theo rem 2.4 with C = E do es n ot use Prop osition 2.2. Thus, without the use o f c h oice we ha v e: 2.5. Theorem. L et A op b e a smal l 2-filter e d 2-c ate gory, and A op E − → T op b e a 2-fu nc tor. Then, the top os of sh e aves ^ L im − − → E on the site L im − − → E of 2.1 is a bilimit of E in T op , or, e qui v alently, a bic olimit in T op ∗ . Referen ces [1] Artin M, Grothendieck A, V erd ier J., SGA 4 , (1963-64) , Lecture Notes in Mathe- matics 269 S pringer, (1972). [2] Artin M, Grothendieck A, V erdier J., SGA 4 , (1963-64) , Springer Lecture Notes in Mathematics 270 (1972). [3] Dubu c, E. J., 2-Filteredness and the point of eve ry Galois topos , Proceed in gs of CT2007 , Applied Categorical S tructures, V olume 18, Issue 2, Springer V erlag (2010). [4] Dubu c, E. J., Street, R., A construct ion of 2-fi ltered bicolimits of categories , Cahiers de T op ologie et Geometrie Differentielle, (2005). [5] Gray J. W., F ormal Category Theory: Adjointness for 2 -Catego ries , Springer Lecture Notes in Mathematics 391 (1974). [6] Mac Lane S., Mo erdijk I ., Shea ves in Geometry and Logic , Springer V erlag, (1992). [7] Street R., Limits indexed by category-v alued 2 -functors J. Pure Appl. A lg. 8 (1976).