Totally distributive toposes

T otally distributiv e top oses Rory B.B. Lucysh yn-W righ t ∗ Y ork Univ ersit y , 4700 Keele St., T oron to, ON, Canada M3J 1P3 Abstract A lo ca lly small category E is total ly distributive (as defined by Rosebru gh-W o od ) if there exists a string of adjoint functors t ⊣ c ⊣ y , w here y : E → b E is the Y oneda em b edding. Sa ying that E is lex total ly distributive if, moreo ve r, the left adjoin t t pre- serv es finite limits, w e show that the lex totally d istributiv e categories w ith a small set of generators are exactly the inje ctive Gr othendie ck top oses , studied by Johns to n e and Jo ya l. W e c haracterize the totally distr ib utiv e catego r ies w ith a small set of generators as exactly the essential sub to p oses of pr eshea f top oses, studied by Kelly-La w v ere and Kennett-Riehl-Ro y-Zaks. 1 In tr oduction The aim of th is pap er is to establish certain connections b et ween the work of Mar- molejo, Rosebrugh, and W o o d [14, 13] on total ly distributive c ate gories and t wo other b o dies of wo r k on distinct topics: Firstly , that of Johnstone and Jo ya l [4, 7] on in- je ctive top oses and c ontinuous c ate gories , an d secondly , that of Kelly-La w vere [8] and Kennett-Riehl-Ro y-Zaks [9] on essential lo c alizations and essential subtop oses . One of our observ ations, 1.5.9 (2), when tak en together with a theorem of Kelly-La wv ere which w e recall in 1.5.6, yields a concrete com binatorial description of all totally distrib utiv e catego r ies with a small s et of generators. W e adopt th e foundational con ven tions of [6] (and [4, 7]), since our on ly use of the stronger found at ional assu mptions of [17 , 16 , 18 , 14 , 13] is made in finally dedu cing our main results (1.5.9) as strengthened v ariants of prop ositions which precede them. W e let CAT r epresen t th e m eta-2-catego r y of categories, fu ncto rs, and natural trans- formations (see [6], 1.1.1), and we let CA T b e its fu ll su b-(meta )-2-category consisting of locally small categories. ∗ Partial financial assista nce by the O n tario Graduate Scholarship prog ram is gratefully acknowledged. † 2010 Mathematics Sub ject Classification: 1 8A35, 18B25 , 18B30, 18B3 5 ‡ Keywords: to ta lly distributive categor y; top os; injective topos ; essential subtop os; essential lo c a lization; contin uous categor y 1 1.1 Completely distributiv e lattices, tot ally d istributiv e categories. A p oset E is a c onstructively c ompletely distributive lattic e [2 ], or c c d lattic e , if there exist adjunctions E ↓ ↓ ↓ ⊤ 7 7 ↓ ⊤ ' ' Dn ( E ) ∨ o o where Dn ( E ) is the p oset of d own-closed subsets of E , ord ered b y inclus ion, and ↓ : E → Dn ( E ) is the embedd ing giv en by v 7→ ↓ v := { u ∈ E | u 6 v } . The existence of the left adjoin t ∨ of ↓ is equiv alent to th e co co m plete ness of E , i.e. the cond ition that E b e a complete lattice, and if such a map ∨ exists, it n ec essarily sends eac h do wn -c losed subset to its join in E . In the presence of the axiom of choic e, a p oset is a ccd lattice iff it is a completely d istributiv e lattice in the us u al sense [2]. Rosebrugh and W o o d [14] ha ve defined an analogue of th is n ot ion for arbitrary catego r ies rather than just p osets 1 . A lo ca lly small category E is total ly distributive if there exist adjun cti ons E t ⊤ 9 9 y ⊤ % % b E c o o where b E is the presheaf category [ E op , Set ] and y is the Y oneda embedd ing, giv en by v 7→ b v := E ( − , v ). W e sa y that a totally distributiv e category E is lex total ly distributive if the asso ciate d functor t : E → b E preserves finite limits. The existence of the left ad j oi n t c of y is the requirement that E b e total [17], or total ly c o c omp lete . This left adj oi n t c of y is c haracterized by the prop ert y that cE ∼ = colim b u → E u = colim(( E ↓ E ) → E ) ∼ = Z u ∈E E u · u (1) naturally in E ∈ b E , so that totalit y is equiv alen t to the existence of a colimit in E of the (p ossibly large) canonical diagram of eac h presh ea f E on E . Note that an y totally distribu tiv e categ ory E is in particular lex total , meaning that E is total and th e asso ciated functor c : b E → E pr ese rv es finite limits. W oo d [18] attributes to W alters the theorem that those lex total categories with a small set of generators are exactly the Grothendiec k top oses; the pap er [16] of Street includes a pro of of this resu lt. 1.2 Con tinu ous dcp os, con t in uous categories. A p oset X is a c ontinuous dcp o if there exist adjunctions X ↓ ↓ ⊤ 6 6 ↓ ⊤ ( ( Idl ( X ) ∨ o o 1 Marmolejo, Ro sebrugh, and W oo d [13] have also studied an a ppa ren tly distinct a nalogue — the notion of c ompletely distributive c ate gory . 2 where Idl ( X ) is the p oset of id ea ls of X (i.e. upw ard-directed d o wn-closed subsets of X ), ordered by inclusion, and ↓ : X → Idl ( X ) is the emb ed ding giv en by y 7→ ↓ y := { x ∈ X | x 6 y } . T h e existence of the left ad j oi n t ∨ of ↓ is equiv alen t to the existence of all directed j oins in X , i.e. the condition that X b e a dcp o , or dir e cte d c omplete p art ial or der , and if such a map ∨ exists, it n ec essarily sends eac h ideal to its join in E . Johnstone and J o y al [7] ha ve defined a generalization of this notion to arb itrary catego r ies, rather than jus t p osets, as follo ws. W e sa y that a lo ca lly small category X is c ontinuous if there exist adjun ct ions X w ⊤ 7 7 m ⊤ ' ' Ind X colim o o , where Ind X is the ind-c ompletion of X , w hose ob jects are all small filtered diagrams in X , and m is th e canonical full em b eddin g sending eac h ob ject x ∈ X to the diagram 1 → X , indexed by the terminal category 1, with constant v alue x . The existence of the left adjoint colim of m : X → Ind X is equiv alen t to the requirement that X be equ ipp ed with colimits f o r all s mall filtered d iag r ams, and colim necessarily send s eac h D ∈ Ind X to a colimit of D in X . 1.3 Stone dualit y for con tinuou s dcp os. It was sho wn by Hoffmann [3] and La wson [10] that the category of cont inuous d cpos an d directed-meet-preserving maps is dually equiv alen t to the category of completely distributiv e lattice s and maps preserving finite meets and arbitrary joins . The category of cont inuous d cpos is isomorphic to the full sub cate gory of top olo gical spaces consisting of con tin uous dcp os endow ed with the Sc ott top olo gy , and the giv en du al equiv alence of th is category of s p ac es with the category of completely distribu tiv e lattice s is a restriction of the dual equiv alence b et w een sob er spaces and spatial frames (see, e.g., [5]), asso ciating to a sp ace its fr ame of op en sets. Subsequ ent work of Banasc hewski [1] entail s that this dual equiv alence restricts further to a d ual equ iv alence b et w een c ontinuous lattic es (i.e. those con tin uous d cpos whic h are also complete lattices) and stably sup er c ontinuous lattic es , also kno wn as lex c c d lattic es [13] or lex c ompletely distributive lattic es , whic h are those ccd lattices for whic h the left adj oint ↓ ↓ ↓ preserve s finite meets. Scott [15 ] had shown earlier that the con tinuous lattic es, when endow ed with their Scott top ologie s, are exactly the inje ctive T 0 spaces. 1.4 Con tinu ous categories and injective top oses. Scott’s isomorphism b e- t wee n injectiv e T 0 spaces and cont inuous lattices [15] has a top os-theoretic analogue, giv en by Johnstone-Jo yal [7], whic h we no w recall. First let us record the f ol lowing earlier result of John stone [4]: Theorem 1.4.1. (John stone [4]). A Gr othendie c k top os E is inje ctive (with r esp e ct to ge ometric inclusions) if and only if E i s a r etr act, by ge ometric morphisms, of a pr eshe af top os b C with C a smal l finitely-c omplete c ate gory. 3 W e call s uc h Grothendieck top oses i nje ctive top oses . A quasi-inje ctive top os [7] is defined as a Grothendiec k topos which is a r et r ac t, b y geometric morphisms, of an arbitrary p resheaf top os b C (with C a small category). A con tinuous category X is ind- smal l if th er e exists a small ind-dense sub category A of X , by which w e mean a small, full, d ense sub category A of X for whic h eac h comma category ( A ↓ x ), w ith x ∈ X , is filtered 2 . Theorem 1.4.2. (Johnstone-Joy al [7]). 1. Ther e is an e quivalenc e of 2-c ate g o ries b etwe en the 2-c ate gory of quasi-inje ctive top oses, with ge ometric morphisms, and the 2-c ate g ory of ind-smal l c ontinuous c ate gories, with morphisms al l filter e d-c olimit-pr eserving functors. This e quiva- lenc e sends a quasi-inje ctive top os E to its c ate gory of p oints pt ( E ) . 2. This e quivalenc e r estricts to an e quivalenc e b etwe en the fu l l sub-2-c ate gories of inje ctive top oses and c o c omp lete ind-smal l c ontinuous c ate gories. 1.5 T otally distributiv e toposes. Ha v in g seen that Scott’s isomorphism b e- t wee n injectiv e T 0 spaces and con tinuous lattice s has a top os-theoretic analog u e r ela ting injectiv e top oses an d co co mplete ind -small con tin uous categories, w e are led to seek a top os-theo retic analogue of the dual equiv ale nce b etw een contin uous lattice s and lex completely distributive lattices. W e prov e the follo wing, wh ere b y a smal l dense g e n- er ator f or a category E we mean a small dense full sub catego r y G of E . Recall that ev ery Gr othend iec k top os has a small dense generator. Theorem 1.5.3. The lex total ly distributive c ate gories with a smal l dense gener ator ar e exactly the inje ctive top oses. Henc e, the 2-c ate gory of c o c omplete ind-smal l c ontinuous c ate gories (1.4.2) is e quivalent to the the 2-c ate gory of lex total ly distributive c ate gories with a smal l dense ge ner ator (with ge ometric morphisms). One ma y also ask whether there is a similar analogue of the b roader dual equiv alence b et w een contin uous dcp os and completely distrib utiv e lattic es, and in th is regard we pro vid e a partial result, as follo w s: Prop osition 1.5.4. Every quasi-inje ctive top os is total ly distributive. In pr o ving these theorems, w e come up on a further r esult of indep endent in terest. An essential subtop os of a top os F is a top os E for whic h there is a geometric inclusion i : E → F w h ose in v erse image functor i ∗ : F → E has a left adjoin t. Theorem 1.5.5. Those total ly distributive c ate g or i e s having a smal l dense gener ator ar e exactly the essential subtop oses of pr eshe af top oses b C = [ C op , Set ] (with C a smal l c ate gory). Remark 1.5.6. It w as sho wn by Kelly-La w v ere [8 ] that the essen tial subtop oses of a presheaf top os b C corresp ond bijectiv ely to idemp otent ide als of arro w s in the sm al l catego r y C . 2 The term ind-smal l w a s int ro duced not in [7] but later in [6], where it is defined in terms of a differen t criterion, which, by 2.17 of [7] and C4.2.18 of [6], is equiv alent to the given condition, employ ed in [7 ]. Chapter C4 of [6] includes an alterna te exp o sition of muc h of the conten t of [7]. 4 Example 1.5.7. The cases in whic h b C is the top os b ∆ of simplicial sets , the top os b I of cubic al sets , or the top os b G of r eflexive globular sets are of in terest in homotop y theory and h ig her category theory . It is sho wn in [9] that the essen tial subtop oses of these top oses are classified by the dimens ions n ∈ N . In general, the essent ial s u btop oses of a top os F (or rather, their asso ciated equiv alen t full replete sub categories of F ) form a complete lattice [8]. Remark 1.5.8. As noted in 1.1, it was pro ved in [16 ] that any lex total category E with a small s et of generators is a Grothendiec k top os. Using this result, wh ose pr oof in [16] app ears to make use of the foun d ati onal assu mption that th ere is a category of sets S ′ suc h th a t b oth E and th e catego r y Set of small sets are categories internal to S ′ , we obtain the follo win g corollaries to Theorems 1.5.3 and 1.5.5: Theorem 1.5.9. 1. Those lex total ly distributive c ate gories having a smal l set of gener ato rs ar e exactly the inje ctive top oses. 2. Those total ly distributive c ate gories having a smal l set of gener ators ar e exactly the essential subtop oses of pr eshe af top oses b C = [ C op , Set ] (with C smal l). 2 Preliminaries on totally distributiv e categor ies It is shown in [14], by means of a result of [17], th at eve r y presheaf category b C on a small category C is totally distribu tiv e. In order to clearly establish this in the absence of the foun d ati onal assump tio n s of [14], we giv e a self-con tained elemen tary pr oof, by means of the follo wing lemma (cf. C oroll ary 14 of [17]). W e p r o v e also that if C is finitely complete, then b C is lex totally d istr ibutiv e. Lemma 2.1. L et C b e a smal l c ate gory. Then ther e is an adjunction b C y b C ⊤ $ $ b b C c y C o o , wher e y C : C → b C and y b C : b C → b b C ar e the Y one da emb e ddings. Pr o of. Eac h C ∈ b b C is a co end C ∼ = R C ∈ b C C ( C ) · b C , and these isomorphisms are natural in C . Using this and the Y oneda L emm a , w e obtain isomorphisms ( b y C ( C ))( c ) = C ( b c ) ∼ = Z C ∈ b C C ( C ) × b C ( b c ) ∼ = Z C ∈ b C C ( C ) × C ( c ) natural in C ∈ b b C and c ∈ C . Hence w e ha ve an isomorphism b y C ( C ) ∼ = Z C ∈ b C C ( C ) · C natural in C ∈ b b C , s o with reference to (1), b y C ⊣ y b C . 5 Prop osition 2.2. L et C b e a smal l c ate g o ry. Then b C is total ly distributive. Mor e over, if C has finite limits, then b C is lex total ly distributive . Pr o of. W e ha ve an adj unction as in Lemm a 2.1, and the left adjoin t b y C : b b C → b C has a further left adjoint ∃ y C : [ C op , Set ] → [ b C op , Set ], w h ic h is give n by left Kan-extension along y op C : C op → b C op . Hence b C is totally d istr ibutiv e. If C h as finite limits, then y C : C → b C is a cartesian fu n cto r b et w een cartesian categories, and it follo ws that the asso cia ted fu nctor ∃ y C is also cartesian. The follo win g lemma, based on Lemma 3.5 of Marmolejo-Rosebrugh-W o od [13], pro vid es a means of dedu cing that a categ ory is totally distributiv e. W e hav e aug- men ted the lemma sligh tly in order to handle lex tota lly distrib u tiv e categories as w ell. Lemma 2.3. L et D and E b e lo c al ly smal l c ate gories. Supp ose we ar e given adjunctions D q ⊤ 9 9 s ⊤ % % E r o o with q , s ful ly faithful and E total ly distributive. Then D is total ly distributive. Mor e over, if E is lex total ly distributive and q pr eserves finite limits, then D is lex total ly distributive. Pr o of. There is a 2-fun cto r d ( − ) := CA T (( − ) op , Set ) : CA T coop → CAT , where CA T coop is the (meta)- 2-category gotten by reversing b oth th e 1-cells and 2-cells in CA T . T h is 2-functor sends the adju nctio ns q ⊣ r ⊣ s : D → E in CA T to adjunctions b q ⊣ b r ⊣ b s , so we h a v e a diagram D q ⊤ 6 6 s ⊤ ( ( y ′   E r o o t ⊣   y ⊣   b D g g b q ⊤ w w b s ⊤ b E / / b r c O O where y ′ is the Y oneda em b edd in g. Observe that y ′ ∼ = b s · y · s , since w e hav e ( b s · y · s )( d ) = b s ( E ( − , sd )) = E ( s op − , sd ) ∼ = D ( − , d ) = y ′ ( d ) naturally in d ∈ D , as s is fu lly faithful. Therefore, letting c ′ := r · c · b r and t ′ := b q · t · q w e fi nd that D t ′ ⊤ 9 9 y ′ ⊤ % % b D c ′ o o so D is totally distrib u tiv e. If t and q are cartesian, then since b q is also cartesian, t ′ = b q · t · q is cartesian and hence D is lex totally d istributiv e. 6 3 A construction of Johnstone-Jo y al Let X b e an in d-small con tinuous catego r y , and let A b e a small ind-dense sub category of X . W e n o w recall fr om [7 ] an explicit mann er of constructing a qu asi-i njectiv e top os F s u c h that X is equ iv alen t to the category of p oin ts of F . Firstly , there is an asso ciated fun ctor W : X op × X → Se t , given by W ( x, y ) := Ind X ( m x, w y ) , x, y ∈ X . The elemen ts of W ( x, y ) are called wavy arr ows from x to y in X . John s to n e and Jo yal [7] sh o w that this functor W , wh en view ed as a profunctor W : X ◦ → X , un derlies an idemp otent pr ofunctor c omonad on X , and that the restriction V : A op × A → Set of W is again an idemp oten t pr ofunctor comonad on A . In the latter case, sin ce A is sm a ll, this means pr ec isely th at V : A ◦ → A is an idemp oten t comonad on A in th e bicatego r y Prof of small categories, profunctors, and morp hisms of profun cto r s. F urth er , V is left-flat , meaning that for eac h y ∈ A , V ( − , y ) : A op → Set is a flat presheaf. Recall that f or small catego r ies C , D , eac h profun ct or M : C ◦ → D (b y whic h we mean a functor M : C op × D → Set ) gives rise to a cocont in u o us f unctor f M : [ C , Set ] → [ D , Set ]. Indeed, f M is the left Kan extension along the Y oneda em b eddin g C op → [ C , Set ] of the transp ose C op → [ D , Set ] of M . This passage defines an equiv alence of the bicategory Prof with another bicategory , in fact a 2-cate gory , whose ob jects are again all small categories, bu t wh ose 1-cells C → D are all co con tin uous f unctors [ C , Set ] → [ D , Set ], and wh ose 2-cells are all natural transformations. Hence our idemp otent comonad V : A ◦ → A in Prof determines an idemp otent comonad e V : [ A , Set ] → [ A , Se t ]. Moreo v er, since V ( − , y ) : A op → Set is fl at f o r eac h y ∈ A , it follo w s th at e V pr ese r v es fi nite limits and so is said to b e a c artesian c omona d . F urther, sin ce e V is also coconti n u ous, e V is the in verse-imag e p art of a geometric morphism: Definition 3.1. Giv en an ind -small cont inuous category X with a small in d-dense sub category A , the asso c i a te d ge ometric endomorphism is d efined to b e th e geomet- ric morphism m A , X : [ A , Set ] → [ A , Set ] whose inv erse-image part is the asso ciate d idemp otent c omonad m ∗ A , X = e V . Prop osition 3.2. (John sto n e-Jo y al [7]). L et X b e an ind-smal l c ontinuous c ate gory, and let A b e a smal l ind-dense sub c ate gory of X . L et [ A , Set ] → F → [ A , Set ] b e a fac- torization of the asso ciate d ge ometric endomor phism m A , X into a ge ometric surje ction fol lowe d by a ge ometric inclusion. Then F is a quasi-inje ctive top os whose c ate gory of p oints of is e quivalent to X . F urther, i f X i s c o c ompl e te, then we may take A to b e finitely c o c omplete, and it fol lows that F is an inje ctive top os. 4 T otally distributiv e top oses from contin uous categories W e no w sh o w th at the top oses corresp onding to con tinuous catego ries under the equiv- alence of Theorem 1.4.2 are totall y distributiv e, so that ev ery qu asi- in ject iv e top os is totally distributive. 7 Lemma 4.1. L et i : C → D b e a ful ly faithful functor with a right adjoint r , and supp ose that the induc e d c omonad i · r on D has a right adjoint n . Then r has a right adjoint s := n · i , so that i ⊣ r ⊣ s . Pr o of. C ( r ( d ) , c ) ∼ = D ( i · r ( d ) , i ( c ) ) ∼ = D ( d, n · i ( c )) = D ( d, s ( c )) , naturally in d ∈ D , c ∈ C . Lemma 4.2. L et X b e an ind-smal l c ontinuous c ate gory, let A b e a smal l ind-dense sub c ate gory of X , and let i : F ֒ → [ A , Set ] b e the c or efle ctive emb e dding induc e d by the asso ciate d idemp otent c omonad m ∗ A , X on [ A , Set ] (so that F i s the c ate gory of fixe d p oints of m ∗ A , X ). Then 1. i pr eserves finite limits; 2. The right adjoint r : [ A , Set ] → F to i has a further right adjoint s , so that F i ⊤ 5 5 s ⊤ ) ) [ A , Set ] ; r o o 3. F is a quasi-inje ctive top os whose c ate gory of p oints is e quivalent to X ; 4. If X is c o c omplete, we may take A to b e finitely c o c omplete, and F is then an inje ctive top os. Pr o of. Since F is isomorph ic to the category of coalgebras of the cartesian comonad m ∗ A , X , F is an elemen tary topos, and the forgetful functor i : F ֒ → [ A , Set ] is the in verse-imag e part of a geometric surj ec tion p : [ A , Set ] ։ F ; see, e.g., [6 ], A4.2 .2. F urther, the idemp oten t comonad i · r = m ∗ A , X has a right adjoin t m A , X ∗ , so we deduce by Lemma 4.1 that r has a right adjoin t s , so that i ⊣ r ⊣ s . I n p articular, r is left adjoin t and cartesian, so we obtain a geometric morph ism q : F → [ A , Set ] with q ∗ = r and q ∗ = s . Since i ⊣ r ⊣ s and i is fully faithful, it follo ws that s = q ∗ is also fu lly faithful, so q : F → [ A , Set ] is a geometric inclusion. F urth er, the comp osite [ A , Set ] p − → F q − → [ A , Set ] is m A , X , or, more p recise ly , h as inv erse-image part ( q · p ) ∗ = p ∗ · q ∗ = i · r = m ∗ A , X . Hence 3 and 4 follo w from Prop osition 3.2. Definition 4.3. F or an ind-small con tinuous category X and a small ind-dense s u b- catego r y A of X , we call the top os F of Lemma 4.2 the asso ciate d top os . Lemma 4.4. L et X b e an ind-smal l c ontinuous c ate gory, so that X has some smal l ind-dense sub c ate gory A . Then the the asso ciate d top os F is total ly distributive. If X is also c o c omplete, then we may take A to b e finitely c o c omp lete, and it fol lows that F is lex total ly distributive. Pr o of. By Lemm a 4.2, we ha ve adjun ctio n s F i ⊤ 5 5 s ⊤ ) ) [ A , Set ] r o o 8 with i, s fully faithfu l and i cartesian. By P roposition 2.2, [ A , Set ] is totally distribu- tiv e, so w e d educe b y Lemma 2.3 that F is tota lly distrib utiv e. If X is also co complete, then we can tak e A to b e fin it ely co complete, so A op is finitely complete and hence, by 2.2, d A op = [ A , Set ] is lex totally distrib u tiv e, so we d educe b y 2.3 that F is lex totally distributive . Theorem 4.5. Eve ry q u asi-inje ctive top os is total ly distributive, and every inje ctive top os is lex total ly distributive. Pr o of. Giv en a quasi-injectiv e top os E , Th eo r em 1.4.2 entai ls that the cate gory of p oin ts X := pt ( E ) of E is an ind-small con tin u ou s category . T aking any small ind -dense sub category A of X , the asso ciated top os F is a q u asi-i njectiv e top os w hose category of p oin ts is equiv alen t to X , so by Theorem 1.4.2 w e dedu ce that E is equiv alen t to F . But the latter top os is totall y distrib utiv e by Lemma 4.4, and total distr ib utivit y is clearly inv ariant u n der equiv alences, so E is totally distribu tiv e. Th e second statemen t ma y b e dedu ce d analogously . 5 T otally distributiv e categories as essen tial lo calizations Prop osition 5.1. L et E b e a total ly distributive c ate gory with a smal l dense gener ator i : G ֒ → E . We then c onclude the fol lowing: 1. Ther e ar e adjunctions E t ′ ⊤ 9 9 y ′ ⊤ % % b G c ′ o o with y ′ and t ′ ful ly faithful, wher e y ′ is the c omp osite E y − → b E b i − → b G . 2. E is an essential subtop os of b G and, in p articular, a Gr othendie c k top os. 3. If E is lex total ly distributive, then t ′ : E → b G pr eserves finite limits. Pr o of. W e let c ′ := c · ∀ i = ( b G ∀ i − → b E c − → E ) , t ′ := b i · t = ( E t − → b E b i − → b G ) , where ∀ i : b G → b E is the functor give n b y right Kan extension along i op : G op ֒ → E op . Since b i ⊣ ∀ i and t ⊣ c , we hav e that t ′ = b i · t ⊣ c · ∀ i = c ′ . Since i : G ֒ → E is fu lly faithful, the counit of the adjunction b i ⊣ ∀ i is an isomorphism (e.g., b y [11], X.3.3), so ∀ i is fully faithful. Observe that the diagram E y   y / / b E b E b i / / b G ∀ i O O 9 comm utes up to isomorphism, since the d ensit y of G in E giv es us exactly that u ∼ = R g ∈G E ( g , u ) · g n a turally in u ∈ E , so ( y v ) u = E ( u, v ) ∼ = E ( Z g ∈G E ( g , u ) · g , v ) ∼ = Z g ∈G [ E ( g , u ) , E ( g , v )] = (( ∀ i · b i · y ) v ) u naturally in u, v ∈ E . W e fin d that c ′ = c · ∀ i ⊣ b i · y = y ′ , since by using the adjoin tness c ⊣ y , the comm utativit y of the ab o ve diagram, and the fact that ∀ i is fully faithful, w e deduce that E ( c · ∀ i ( G ) , v ) ∼ = b E ( ∀ i ( G ) , y v ) ∼ = b E ( ∀ i ( G ) , ∀ i · b i · y ( v )) ∼ = b G ( G, b i · y ( v )) naturally in G ∈ b G , v ∈ E . Since G is a dense generator f o r E w e hav e that y ′ is fully faithful, and s in ce t ′ ⊣ c ′ ⊣ y ′ it follo ws that t ′ is fully faithful as wel l. If E is lex totally distributive, then t preserves finite limits, so since b i preserves all limits, t ′ = b i · t preserves finite limits. Theorem 5.2. L et E b e a lex total ly distributive c ate gory with a smal l dense gener ator. Then E is an inje ctive Gr othendie ck top os. Pr o of. By 5.1 we know that E is a Grothend iec k top os, and it f o llo ws from Giraud’s Theorem that there exists a finitely c omplete small den s e full sub category G of E . (Indeed, this follo ws readily from 4.1 and 4.2 in the App end ix of [12], for example). W e ha ve adjun ct ions t ′ ⊣ c ′ ⊣ y ′ as in P roposition 5.1, with y ′ fully faithful and t ′ cartesian. Hence we obtain geometric morphisms s : E → b G and r : b G → E with s ∗ = y ′ , s ∗ = c ′ , r ∗ = c ′ , r ∗ = t ′ , sin ce c ′ is right adjoint and hen ce cartesian. F urther, since y ′ is fully faithful and c ′ ⊣ y ′ , we ha ve that ( r · s ) ∗ = r ∗ · s ∗ = c ′ · y ′ ∼ = 1 E , so E is a (pseud o-) r etrac t of the presheaf top os b G by geometric morphisms, and the result follo ws by 1.4.1. Hence Theorem 1.5.3 is pr o v ed. T o prov e Theorem 1.5.5, it r ema in s only to show the follo wing: Prop osition 5.3. L et E b e an essential sub top os of a pr eshe af top os b C (with C smal l). Then E is total ly distributive and has a smal l dense gener ator. Pr o of. There is a geometric inclusion s : E → b C wh ose in ve r se-ima ge functor s ∗ : b C → E has a left adj oi nt s ! . Hence w e ha ve s ! ⊣ s ∗ ⊣ s ∗ with s ! and s ∗ fully faithful, so E is totally distributive, by Lemma 2.3. References [1] B. Banasc hewski, On the top olo gies of inje ctive sp ac es , Con tin u ous lattices and their applications (Bremen, 1982) , Lecture Notes in Pur e and Appl. Math., vol. 101, Dekk er, 1985, pp. 1–8. 10 [2] B. F a wcet t and R. J. W o od , Constructive c omplete distributivity. I , Math. Pro c. Cam br idge Philos. So c. 107 (1990), no. 1, 81–89. [3] R.-E. Hoffmann, Continuous p osets, prime sp e ctr a of c ompletely distributive c om- plete lattic es, and hausdorff c omp actific ations , Lecture Notes in Math. 871 (1981), 159–2 08. [4] P . T. Johnstone, Inje ctive top oses , Lecture Notes in Math. 871 (1981), 284–297 . [5] , Stone sp ac es , Cambridge Universit y Pr ess, 1982. [6] , Sketches of an elephant: a top os the ory c omp endium. Vols. 1,2 , Oxford Univ ersity Press, 2002. [7] P . T. Johnstone and A. J oy al, Continuous c ate gories and exp onentiable top oses , J. Pure Appl. Algebra 25 (1982), n o . 3, 255–296 . [8] G.M. Kelly and F.W. La w v ere, O n the c omplete lattic e of essential lo c alizations , Bull. So c. Math. Belg. S´ er. A 41 (1989), no. 2, 289–319. [9] C. Kennett, E. Riehl, M. Roy , and M. Zaks, L evels in the top oses of simplicial sets and cubic al sets , J. Pu re Ap pl. Algebra 215 (2011), no. 5, 949–9 61. [10] J . D. La w son, The duality of c ontinuous p osets , Houston J. Math. 5 (1979), 357– 386. [11] S . Mac Lane, Cate gories for the working mathematician , second ed., Springer- V erlag, 1998. [12] S . Mac Lane and I. Moerd ijk, She aves in ge ometry and lo gic , S pringer-V erlag, 1994. [13] F. Marmolejo, R. Rosebrugh, and R. J. W o od, Completely and total ly distributive c ate gories I , J. Pur e Appl. Algebra, to app ear, 2010. [14] R . Rosebrugh and R. J. W o od, An adjoint char acterization of the c ate gory of sets , Pro c. Amer. Math. So c. 122 (1994), no. 2, 409–4 13. [15] D. Scott, Continuous lattic es , Lecture Notes in Math. 274 (1972), 97–136. [16] R . Street, Notions of top os , Bull. Austral. Math. So c. 23 (1981) , no. 2, 199–20 8. [17] R . Street and R. W alters, Y one da structur es on 2-c ate gories , J. Algebra 50 (1978), no. 2, 350–379 . [18] R . J. W o od, Some r emarks on total c ate gories , J. Algebra 75 (1982), no. 2, 538– 545. 11