Grothendieck quasitoposes

GR OTHENDIE CK QUASITOPOSES RICHARD GARNER AND STEPHEN LA CK Abstract. A full re flective subcategor y E of a presheaf ca tegory [ C op , Set ] is the ca tegory o f she aves for a to p o lo gy j on C if and only if the reflection from [ C op , Set ] into E pr eserves finite limits. Such an E is then called a Gr othendieck topo s. More gener a lly , one can consider t wo top olo gies, j ⊆ k , and the category of sheav es for j whic h are also separa ted for k . The c ategories E of this form for some C , j , and k are the Gr othendieck quasitop oses of the title, previously studied by Bo rceux and P edicchio, and include ma n y examples of ca tegories of spaces. They also include the category o f concrete sheav es for a co nc r ete site. W e show that a full r eflective sub c ategory E of [ C op , Set ] ar ises in this w ay for some j a nd k if and o nly if the r eflection pres erves monomo rphisms as w ell a s pull- backs ov er elemen ts of E . More genera lly , for any quas itop o s S , we define a subqua sitop os o f S to b e a full reflective subca teg ory of S for which the r eflection pre serves monomorphisms as well as pullbacks ov er ob jects in the sub categor y , a nd we c harac ter ize such sub q uasitop oses in terms of universal closur e op era tors. 1. Intr oduction A Grothendiec k t o p os is a catego ry of the form Sh ( C , j ) for a sm all category C and a (Gro t hendiec k) top olo gy j on C . These categories ha v e b een of fundamental imp ortance in geometry , logic, and other ar- eas. Suc h categories w ere ch aracterized by Giraud as the co complete categories with a generator, satisfyin g v ario us exactne ss conditions ex- pressing compatibility b etw een limits and colimits. The category Sh( C , j ) is a full sub category of the presheaf categor y [ C op , Set ], and the inclusion has a finite-limit-preserving left adjoin t. This in fact leads to another c haracterization of Grothendiec k top oses. A full sub category is said to b e r efle ctive if the inclusion has a left adjoin t, and is said to b e a lo c alization if mor eov e r this left adjo in t preserv es finite limits. A category is a Grothendiec k top os if and only if it is a lo calization of some presheaf category [ C op , Set ] o n a small category C . (W e shall henceforth only consider preshea v es on small categories.) Date : Novem ber 2011. 1 2 RICHARD GARN ER AND STEPHEN LACK Elemen tary t o p oses, in tro duced b y Lawv ere and Tierney , generalize Grothendiec k top oses; the non- elemen tary conditio ns of co complete- ness and a generator in the Giraud c haracterization are replaced by the requiremen t that certain functors, whic h the G iraud conditions guar- an tee are con tin uous, m ust in fact b e represen table. Y et another c har- acterization of the G rothendiec k top oses is as the elemen tary top oses whic h are lo cally presen table, in the sense of [ 10 ]. W e cite the ency- clopaedic [ 12 , 13 ] as a general reference for top os-theoretic matters. A quasitop os [ 14 ] is a generalization of the notio n of elemen t a ry top os. The main difference is that a quasitop os need not b e balanced: this means that in a quasitop o s a morphism ma y b e b oth an epimor- phism and a mono mo r phism without b eing inv ertible. Rather than a classifier for all sub ob jects, there is only a classifier f o r str ong subo b- jects (see Section 2 b elow). The definition, then, of a quasitop os is a category E with finite limits a nd colimits, for whic h E and each slice category E /E o f E is cartesian closed, and whic h has a classifier for strong sub o b jects. A simple example of a quasitop os whic h is not a top os is a Heyting alg ebra, seen as a category by taking the ob jects to b e the elemen ts of the Heyting algebra, with a unique arr ow from x to y just when x ≤ y . Other examples include the category of conv ergence spaces in the sense of Cho quet, o r v arious categories of differen tiable spaces, studied b y Chen. See [ 12 ] once again for generalities ab out quasitop oses, a nd [ 1 ] for t he examples in v olving differentiable spaces. The notion of Grothendiec k quasitop os w as in tro duced in [ 2 ]. Once again, there are v arious p ossible characterizations: ( i ) the lo cally presen table quasitop oses; ( ii ) the lo cally prese n table categories whic h are lo cally cartesian closed and in whic h ev ery strong equiv alence relation is the kernel pair of its co equalizer; ( iii ) the categories of t he fo rm Sep( k ) ∩ Sh( j ) for top o logies j a nd k on a small category C , with j ⊆ k . In ( ii ), an equiv alence relation in a category E is a pair d, c : R ⇒ A inducing an equiv alenc e relation E ( X , R ) on eac h hom-set E ( X , A ); it is said to b e str on g if the induced map R → A × A is a strong monomorphism. In ( iii ), we write Sh( j ) for the shea v es for j , and Sep( k ) for the category of separated ob jects for k ; these are defined lik e shea v es, exce pt that in the sheaf condition we a sk only for the uniqueness , not the existenc e, of the gluing. A category C equipped with t o p ologies j and k with j ⊆ k is called a bisite in [ 13 ], and a presheaf on C whic h is a j - sheaf and k -separated is then said to b e ( j, k )- bise p ar ate d . GROTHENDIECK QUASITOPOSES 3 A sp ecial case is where C has a terminal ob ject and the represen table functor C (1 , − ) is fa it hful, and k is the to p ology generated b y the co v ering families consisting, for eac h C ∈ C , o f the totalit y of maps 1 → C . If j is an y sub canonical top ology contained in k , then ( C , j ) is a concrete site in the sense of [ 1 ] (see also [ 8 , 9 ]) for whic h the concrete shea v es are exactly the ( j, k )-biseparated preshea v es. In the case of G rothendiec k to p oses, a full reflectiv e sub category of a presheaf category [ C op , Set ] has t he form Sh( j ) for some (necessarily unique) top ology j if and only if the r eflection preserv es finite limits. The lac k of a corresp onding result for G rothendiec k quasitop oses is a noticeable gap in the existing theory , and it is precis ely this gap which w e aim t o fill. It is w ell-kno wn that the reflection fr om [ C op , Set ] to Sep( k ) ∩ Sh( j ) preserv es finite pro ducts and monomor phisms. In Example 3.9 b elow, w e sho w that t his do es not suffice to c haracterize suc h reflections, using the reflection o f directed graphs in to preorders as a coun terex ample. W e provide a remedy for this in Theorem 6.1 , where we sho w t ha t a reflection L : [ C op , Set ] → E has this form for top ologies j and k if and o nly if L preserv es finite pro ducts and monomorphisms a nd is also semi-left-exa c t , in the sense of [ 6 ]. Alternatively , suc h L can b e c haracterized as those whic h preserv e monomorphisms and hav e stabl e units , again in the sense of [ 6 ]. The stable unit condition can most easily b e stated b y sa ying that L preserv es all pullbacks in [ C op , Set ] o v er ob jects in the sub category E . F or eac h ob ject X ∈ E , the slice category E /X is a f ull sub category of [ C op , Set ] /X , with a reflection L X : [ C op , Set ] /X → E /X giv en on ob j ects b y the action of L on a morphism into X ; the condition that L preserv e all pullback s o v er ob jects of E is equiv alently the condition tha t each L X preserv e finite pro ducts. Since a subtop os of a top os S is b y definition a full reflec tiv e sub cate- gory of S f or whic h the reflection preserv es finite limits, the Grothen- diec k top oses are precisely the subtop oses of presheaf top oses. Sub- top oses o f an arbitrary to p os can b e c haracterized in terms of Law v ere- Tierney to p ologies; more imp ortantly for our purp oses, they can b e c haracterized in terms of univ ersal closure op erators. By analogy with this case, w e define a sub quasitop os of a quasitopos S t o b e a full reflectiv e sub category of S for whic h the reflection preserv es monomorphisms and has stable units. Th us a Grothendiec k quasitop os is precisely a sub quasitop os of a presheaf top os. W e also giv e a characterization of subquasitop o ses of an arbitrary quasitop os S , using univ ersal closure op erators. 4 RICHARD GARN ER AND STEPHEN LACK W e b egin, in the follow ing section, by recalling a few basic notions that will b e used in t he rest of the pap er; then in Section 3 we study v arious weak enings o f finite-limit-preserv atio n for a reflection, and the relationships b et w een these. In Section 4 we study conditions under whic h reflectiv e sub categor ies o f quasitopo ses a r e quasitop oses. In Sec- tion 5 w e characterize sub quasitop oses of a general quasitop o s, b efore turning, in Section 6 , to sub quasitop oses of presheaf to p oses and their relationship with Grothendiec k quasitop oses. Ac kno wledgemen t s. W e are gra teful to the anon ymous referee for sev eral helpful commen ts on a preliminary ve rsion of the pap er, in particular for suggesting the f orm ulation of Theorem 5.2 , whic h is more precise than the tr eat ment giv en in an earlier v ersion o f the pa p er. This researc h w as supp orted under t he Australian Researc h Coun- cil’s Disc overy Pr oje cts funding sc heme, pro ject n um b ers DP11010236 0 (Garner) and D P1094883 (La ck). 2. Preliminaries W e recall a few ba sic notions that will b e used in the rest of the pap er. A monomorphism m : X → Y is said to b e str ong if for all commu- tativ e diagrams X ′ e / /   Y ′   X m / / Y with e an epimorphism, there is a unique map Y ′ → X making the tw o triangles comm ute. Str ong e p imorphisms are defined dually . A strong epimorphism whic h is also a monomorphism is in v ertible , a nd dually a strong monomorphism whic h is also an epimorphism is in v ertible. A we ak sub obje ct cla ssifier is a morphism t : 1 → Ω with the prop ert y that f or a n y strong monomorphism m : X → Y there is a unique map f : Y → Ω for which the diag ram X m / /   Y f   1 t / / Ω is a pullback. A category with finite limits is said to b e r e gular if ev ery morphism factorizes as a strong epimorphism follow ed b y a monomorphism, and GROTHENDIECK QUASITOPOSES 5 if moreo v er an y pullbac k o f a strong epimorphism is again a strong epimorphism. It then follows that the strong epimorphisms are pre- cisely t he r e gular epimorphism s ; that is, the morphisms whic h are the co equalizer of some pair of maps. Our regular catego r ies will alwa ys b e assumed to hav e finite limits. A full sub category is r efle ctive whe n t he inclusion ha s a left adjoint; this left adjoint is called the r efle ction . Throughout the pap er, S will b e a category with finite limits; later on w e shall make further a ssumptions on S , suc h as b eing r egula r; when w e finally to come to our c haracterization of Gro t hendiec k qua- sitop oses, S will b e a presheaf top os. Lik ewise , throughout the pap er, E will b e a full reflectiv e subcat- egory of S . W e shall write L for the reflection S → E and also sometimes for the induced endofunctor of S , and w e write ℓ : 1 → L for the unit of the reflection. It is con v enien t to assume that the in- clusion E → S is replete, in the sense that any ob ject isomorphic to one in the imag e is itself in the image. It is also conv enien t to assume that ℓA : A → LA is the iden tit y whenev er A ∈ E . Neither a ssumption affects the results of the pap er. W e shall sa y that the reflection has monom orphic units if each com- p onen t ℓX : X → LX of the unit is a monomorphism, with an a na lo- gous meaning for str ongly epimorph ic units . When L preserv es finite limits it is said to b e a lo c aliza tion . An ob ject A of a category C is said to b e ortho gonal to a morphism f : X → Y if each a : X → A f actorizes uniquely thr o ugh f . If instead eac h a : X → A factorizes in at most one wa y through f , the ob j ect A is said to b e sep ar ate d with r esp e ct to f , or f -sep ar ate d . If F is a class of morphisms, we sa y that A is F -o rthogonal or F -separated if it is f -orthogonal or f -separated fo r eac h f ∈ F . 3. Limit-preser ving conditions f or reflections In this section w e study v arious conditions on a reflection L : S → E w eak er tha n b eing a lo calization. First observ e that a n y reflectiv e sub category is closed under limits, so the terminal ob ject of S lies in E , a nd so L alw a ys preserv es the terminal ob ject. Th us preserv ation of finite limits is equiv alent to preserv ation of pullbac ks; our conditions all sa y that c ertain pullbac ks ar e preserv ed. Pr eserva tion of finite pr o ducts. Since L preserv es the terminal ob ject, preserv ation of finite pro ducts amoun ts to preserv ation of binary pro d- ucts, or to preserv atio n of pullbac ks o v er the terminal o b ject. 6 RICHARD GARN ER AND STEPHEN LACK By a we ll-kno wn result due to Brian Da y [ 7 ], if S is cartesian closed, then L preserv es finite pro ducts if and o nly if E is an exponential ideal in S ; it then follo ws in particular that E is cartesian closed. Stable units. F or eac h o b ject B ∈ E , the reflection L : S → E induces a reflection L B : S /B → E / B onto the full sub category E /B of S / B . The origina l reflection L is said to ha v e stable units when eac h L B preserv es finite pro ducts, or equiv alen tly when L prese rv es all pullbac ks o v er ob jects o f E . Since the terminal ob ject lies in E , this implies in particular that L preserv es finite pro ducts. If S is lo cally cartesian closed then, by the Day reflection theorem [ 7 ] again, L has stable units just when each E /B is an exp onen tial ideal in S /B ; it then follow s that E is lo cally cartesian closed. The name stable units w as originally in tro duced in [ 6 ] for an appar- en tly we ak er condition, namely that L preserv e eac h pullbac k of the form P q / / p   A u   X ℓX / / LX but it w as observ ed in [ 4 , Section 3 .7] that these tw o conditions are in fact equiv alen t. Notice a lso that since LℓX is in v ertible, t o say that L preserv es the pullbac k is equiv alen tly to say that L in v erts q . F r ob enius. W e say that L satisfies the F r ob enius c ondition when it pre- serv es pro ducts of the form X × A , with A ∈ E . The condition is often give n in the more g eneral con text of an a d- junction, not necess arily a reflection, b etw een categories with finite pro ducts. In this case, the condition is that the canonical map ϕ : L ( X × A ) → LX × A, defined using the comparison L ( X × I A ) → LX × LI A and the counit LI A → A , should b e inv ertible. As is w ell-kno wn, if E and S are b o th cartesian closed, then this condition is equiv alen t to the righ t adjoin t I : E → S preserving in ter- nal homs. As is p erhaps less w ell-kno wn, in our setting of a reflection it is enough to assume that S is cartesian closed, and then the condition ensures that the in ternal ho ms restrict to E : see Prop osition 4.2 b elo w. Th us if S is cartesian closed, then L satisfie s the F rob enius conidition if and only if E is closed in S under internal homs. GROTHENDIECK QUASITOPOSES 7 In f act, for a n y monadic adjunction satisfying the F rob enius condi- tion, internal homs may b e lifted a long the right a dj o in t. More gen- erally still, t here is a v ersion of the F robenius conditio n defined f or monoidal categories in whic h the tensor pro duct is not required to b e the pro duct, and once again the in ternal homs can b e lifted along the righ t adjoint: see [ 3 , Prop osition 3.5 a nd Theorem 3.6]. Semi-left-exact. W e say , following [ 6 ], t ha t L is semi-le ft-exact if it preserv es eac h pullback P q / / p   A u   X ℓX / / LX with A ∈ E . This is clearly implied b y the stable units conditio n. By [ 6 , Theorem 4.3] it in fa ct implies, and so is equiv alen t to, t he apparen tly stronger condition that L preserv e eac h pullbac k of t he form P q / / p   A u   X v / / B with A and B in E . But this latter condition is in turn equiv alent to the condition that eac h L B : S /B → E / B b e F rob enius. Th us w e see that semi-left-exactness is in fact a “lo calized” v ersion of the F rob enius condition. In particular, w e ma y tak e B = 1, a nd see that semi-left-exactness implies the F rob enius condition. Once again, if S is lo cally cartesian closed, so tha t each S /B is cartesian closed, then L is semi-left-exact just when eac h E /B is closed in S /B under in ternal homs. This implies that each E /B is cartesian closed, and so t hat E is lo cally cartesian closed; see Lemma 4.3 below. Pr eserva tion o f monomorphisms. Preserv atio n o f monomorphisms will also b e an imp ortant conditio n in what follo ws. Once again it can b e seen as preserv ation of certain pullbacks . Notice also that L : S → E satisfies this condition if and only if eac h L B : S / B → E /B do es so. R elations hips b etwe en the c onditions. W e summarize in the diagram stable units + 3   semi-left-exact   mono-preserving   finite-pro duct-preserving + 3 F rob enius mono-preserving 8 RICHARD GARN ER AND STEPHEN LACK the relationships found so fa r b et w een these conditions. Eac h condition in the to p ro w amounts to requiring the condition b elo w it to hold for all L B : S /B → E /B with B ∈ E . In Theorem 3.5 b elo w, w e shall see that if S is regular and L pre- serv es monomorphisms, then the stable units condition is equiv alent to the conjunction of the semi-left-exactness and finite-pro duct-preserving conditions. In o r der to prov e this, w e start b y considering separately the case where the comp onen ts of the unit ℓ : 1 → L are monomor- phisms and that where they are strong epimorphisms. Theorem 3.1. If S is finitely c om plete and the r efle ction L : S → E has monomo rp h ic units, then the fol lowing ar e e quivalent: (i) L pr eserves finite limits; (ii) L ha s stable units; (iii) L is semi - l e ft-exact and pr eserves finite pr o ducts. Pr o of. The implication ( i ) ⇒ ( ii ) is trivial, while ( ii ) ⇒ ( iii ) w as observ ed ab ov e. Th us it remains to ve rify the implication ( iii ) ⇒ ( i ). Supp ose then that L is semi-left-exact and preserv es finite pro ducts. W e m ust sho w that it preserv es equalizers. Giv en f , g : Y ⇒ Z in S , form the equalizer e : X → Y of f and g , and the equalizer d : A → LY of Lf and Lg ; of course A ∈ E since E is closed in S under limits. There is a unique map k : X → A making the diagram X e / / k   Y f / / g / / ℓY   Z ℓZ   A d / / LY Lf / / Lg / / LZ comm ute. It fo llo ws easily from t he fact that ℓZ is a monomorphism that the square on the left is a pullbac k. Since A ∈ E , it follo ws b y semi-left-exactness that L inv erts k , whic h is equiv alen tly to sa y that L preserv es the equalizer of f and g .  Theorem 3.2. If S is r e gular, and the r efle ction L : S → E has str ongly ep i m orphic units, then the fo l lowing ar e e quivalent: (i) L pr eserves finite pr o ducts an d monomorphi s ms; (ii) L ha s stable units and pr eserves monomorp h isms. Pr o of. Since ha ving stable units alw ays implies the preserv at io n of finite pro ducts, it suffices to sho w that ( i ) implies ( ii ). Supp ose then that L GROTHENDIECK QUASITOPOSES 9 preserv es finite pro ducts and monomorphisms. Let P q / / p   Y u   X ℓX / / LX b e a pullbac k. Since ℓ X is a strong epimorphism, so is its pullback q ; but the left adjoint L preserv e s strong epimorphisms and so Lq is also a strong epimorphism in E . Since L preserv es finite pro ducts and monomorphisms, it preserv es join tly mo no morphic pairs; th us Lp and Lq are jo in tly monomorphic. It follow s that the canonical comparison from LP to the pullback of Lu a nd LℓX is a mono mo r phism, but this comparison is just Lq . Thus Lq is a strong epimorphism a nd a monomorphism, and so inv ertible. This pro v es that L has stable units.  Ha ving understo o d separately the case of a reflection with mono mor- phic units and that of one with stro ng ly epimorphic units, w e no w com bine these to deal with the general situation. The first step in this direction is w ell-kno wn; see [ 6 ] for example. Prop osition 3.3. Supp ose that S is r e gular. I f R is the closur e of E in S under sub obje cts, then the r efle ction of S into E fa c torizes as E L w w 7 7 ⊥ R L ′ w w 6 6 ⊥ S wher e L ′ has s tr ongly epimorphic units an d L has monom o rphic units. Pr o of. A straigh tforw ard argumen t shows that an o b ject X ∈ S lies in R if and only if the unit ℓX : X → LX is a monomorphism. Then the restriction L : R → E o f L is clearly a reflection of R into E . Since S is regular w e may factor ize ℓ : 1 → L as a strong epimor- phism ℓ ′ : 1 → L ′ follo w ed by a monomorphism κ : L ′ → L . Since L ′ X is a sub ob ject of LX , it lies in R . W e claim that ℓ ′ X : X → L ′ X is a reflection of X in to R . Given an o b ject Y ∈ R , the unit ℓY : Y → LY is a monomorphism, and no w if f : X → Y is any morphism, then in 10 RICHARD GARN ER AND STEPHEN LACK the diagra m X ℓ ′ X / / f   L ′ X κX   LX Lf   Y ℓY / / LY ℓ ′ X is a strong epimorphism and ℓY a monomorphism, so there is a unique induced g : L ′ X → Y with g .ℓ ′ X = f and ℓY .g = Lf .κX . This giv es the existence of a factorization of f thro ugh ℓ ′ X ; the uniqueness is automatic since ℓ ′ X is a (strong) epimorphism.  Corollary 3.4. Supp ose that S is r e gular. If L : S → E is semi- left-exact and pr eserve s fini te pr o ducts an d monomorphi sms, then its r estriction L : R → E to R p r eserves fin i te limits, while L ′ : S → R has stable units and pr ese rv e s monomorphi s ms. Pr o of. Since L is semi-left-exact and preserv es finite pro ducts and mono- morphisms, the same is true of its restriction L . Thus L preserv es finite limits by Theorem 3.1 and the fa ct that L has monomorphic units. As for L ′ , since it has strongly epimorphic units it will suffice , b y Theorem 3.2 , to sho w that it preserv es finite pro ducts a nd mo no mo r - phisms. First observ e that if m : X → Y is a monomorphism in S , then w e ha v e a comm utativ e diagram X ℓ ′ X / / m   L ′ X κX / / L ′ m   LX Lm   Y ℓ ′ Y / / L ′ Y κY / / LY in whic h Lm and κX are monomorphisms, and th us also L ′ m . This pro v es that L ′ preserv es mo no morphisms. On the o t her hand, for any ob jects X, Y ∈ S , w e hav e a comm uta- tiv e diagram X × Y ℓ ′ ( X × Y ) / / ℓ ′ X × ℓ ′ Y ' ' P P P P P P P P P P P P L ′ ( X × Y ) π ′   κ ( X × Y ) / / L ( X × Y ) π   L ′ X × L ′ Y κX × κY / / LX × LY GROTHENDIECK QUASITOPOSES 11 in whic h π and π ′ are the canonical comparison maps. Now ℓ ′ X × ℓ ′ Y is a strong epimorphism, since in a regular category these are closed under pro ducts, and κX × κY is a monomorphism, since in an y category thes e are closed under pro ducts. Since L preserv es pro ducts, π is inv ertible, and it no w follow s that π ′ is also in v ertible. Th us L ′ preserv es finite pro ducts.  Theorem 3.5. L et S b e r e gular, and L : S → E an arbitr ary r efle c- tion onto a ful l sub c ate gory, wi th unit ℓ : 1 → L . Then the fol lowing ar e e quivalent: (i) L is semi-le f t-exact, and pr ese rv es finite pr o ducts and monom or- phisms; (ii) L ha s stable units and pr eserves monomorp h isms. Pr o of. The non- trivial part is tha t ( i ) implies ( ii ). Supp ose then t ha t L satisfies ( i ), and factorize L as LL ′ as in Prop osition 3.3 . By Corol- lary 3.4 , w e kno w that L preserv es finite limits, while L ′ preserv es pullbac ks o v er ob jects of R a nd mo no morphisms, thu s the comp osite LL ′ preserv es pullbac ks o v er ob jects of R and mono mo r phisms, and so in particular has stable units and preserv es monomorphisms.  Remark 3.6. In fact w e hav e sho wn that a reflection L satisfying the equiv alen t conditions in the theorem preserv es all pullbacks ov er an ob ject of R ; that is, o v er a sub ob ject of an ob ject in the sub category E . Ha ving describ ed the p o sitiv e relationships b etw een our v arious con- ditions, w e now show the exten t t o whic h they are indep enden t. W e shall give three examples; in each case S is a presheaf category . Example 3.7. Let 2 b e the full sub category of Set × Set consisting of t he ob jects (0 , 0) and (1 , 1). This is reflectiv e, with the reflection sending a pair ( X , Y ) to (0 , 0) if X and Y a re b ot h empty , and (1 , 1 ) otherwise. It’s easy to see tha t this is semi-left-exact and preserv es monomorphisms, but fails to pr eserv e the pro duct (0 , 1) × (1 , 0) = (0 , 0) since L (0 , 1) = L (0 , 1) = 1 but L (0 , 0) = 0. Example 3.8. Consider Set as t he full reflectiv e subcategory of R Gph consisting o f the discrete reflexiv e graphs. This time the reflector sends a graph G t o its set of connected comp onen ts π 0 G . This is w ell-kno wn to prese rv e finite pro ducts. F urthermore, it preserv es pull- bac ks o v er a discrete reflexiv e graph X , since Set /X ≃ Set X and R Gph /X ≃ RG ph X and the induced π 0 /X : RGph /X → Set /X is just π X 0 : RGph X → Set X , whic h preserv es finite pro ducts since π 0 do es so. Thus π 0 has stable units, and so is semi-left-exact (and, a s we ha v e already seen, preserv es finite pro ducts). But π 0 do es not preserv e 12 RICHARD GARN ER AND STEPHEN LACK monomorphisms: an y set X gives rise b oth to a discrete reflexiv e graph and t he “complete” reflexiv e gra ph K X with exactly o ne directed edge b et w een each pair of ve rtices. The inclusion X → K X is a mono mo r - phism, but π 0 X is j ust X , while π 0 K X = 1. Th us π 0 do es not preserv e this monomorphism if X has more than one ve rtex. Example 3.9. Let RGph b e the category of reflexiv e graphs, and Preord the full reflectiv e sub category o f preorders. Since the r eflec- tion sends a graph G to a preorder on the set of vertice s o f G , it clearly preserv es monomorphisms. An easy calculation sho ws tha t for a ny reflexiv e graph G a nd preorder X , the in ternal hom [ G, X ] in R Gph again lies in Preord , correspo nding to the se t of graph homomo r phisms equipped with the p oint wise preordering. Th us Preord is an exp onen- tial ideal in RGph , and so the reflection preserv es finite pro ducts b y [ 7 ]. On the other hand, b y Lemma 4.3 b elo w, the reflection cannot b e semi-left-exact since P reord is not lo cally cartesian closed. T o see this, consider the preorder X = { x, y , y ′ , z } with x ≤ y and y ′ ≤ z , and the tw o ma ps 1 → X pic king out y and y ′ . Their co equalizer is { x ≤ y ≤ z } , but this is not preserv ed b y pulling bac k a long the inclusion of { x ≤ z } in to X , so P reord cannot b e lo cally cartesian closed. Our c haracterization of G rothendiec k quasitop oses, in Theorem 6.1 b elo w, inv olve s three conditio ns on a reflection: that it b e semi-left- exact, that it preserv e finite pro ducts, and t hat it preserv e monomor- phisms. By the three examples ab o v e, w e see tha t none of these three conditions can b e o mitted. 4. Quas itoposes In this section w e tak e a slight detour to study conditions under whic h a r eflectiv e sub category is a quasitop os. First of all, a reflectiv e sub category E of S has any limits or colimits whic h S do es, so of course w e ha v e: Prop osition 4.1. I f L : S → E is any r efle ction, then E h as finite limits and finite c olimits if S do es so. T o deal with the remaining parts of the quasitop os structure w e require some assumptions on the reflection. Prop osition 4.2. If L : S → E is F r ob enius then E is c artesian close d if S is so. Pr o of. Suppose that L satisfies the F rob enius condition. W e shall sho w that if A, B ∈ E , then [ A, B ] is also in E . GROTHENDIECK QUASITOPOSES 13 The comp o site L [ A, B ] × A ϕ − 1 / / L ([ A, B ] × A ) L ev / / LB ǫ / / B induces a morphism c : L [ A, B ] → [ A, B ]. If w e can sho w that cℓ : [ A, B ] → [ A, B ] is the iden tit y , then c will mak e [ A, B ] into an L -algebra a nd so [ A, B ] will lie in E . No w comm utativit y of [ A, B ] × A ℓ × 1 / / ℓ + + ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ev ' ' ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ L [ A, B ] × A ϕ − 1 ( ( ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ B ℓ ( ( ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ 1 " " ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ L ([ A, B ] × A ) L ev   LB ǫ   B sho ws that ev ( cℓ × 1) = ev and so that cℓ = 1.  Lemma 4.3. If L : S → E is sem i-left-exact, then E is lo c al ly c arte- sian close d i f S is so. Pr o of. F or any ob ject A ∈ E , the reflection L induces a reflection of S / A in to E / A , whic h is F rob enius. It follows b y Prop o sition 4.2 that E / A is cartesian closed.  Remark 4.4. As observ ed in the previous section, there are conv erses to the previous tw o results. If S is cartesian closed, a nd E is a full reflectiv e sub categor y closed under exp onen tials, then the reflection is F rob enius. And if S is lo cally cartesian closed, and E is a full reflectiv e sub category closed under exp onentials in the slice categories, then the reflection is semi-left-exact. W e no w turn to the existence of weak sub o b ject classifiers . F or this, w e consider one further condition on our r eflection L , w eak er than preserv ation of finite limits. W e say , follo wing [ 5 ], that L is quasi-lex if, fo r each finite diagr a m X : D → S , the canonical comparison map L (lim X ) → lim ( LX ) in E is b oth a monomorphism and an epimor- phism. W e may then say that L “quasi-preserv es ” the limit. The pro of of the next result closely follo ws that of [ 5 , Theorem 1.3.4], although the assump tions made here are rather w eak er. When w e sp eak of unions of regular sub ob jects, w e mean unions o f subo b jects which 14 RICHARD GARN ER AND STEPHEN LACK happ en to b e regular: there is no suggestion that the union itself m ust b e regular. W e sa y that suc h a union is e ff e ctive when it is constructed as the pushout ov er the in tersection. Prop osition 4.5. L et S b e finitely c omplete and r e gular, and supp ose further that S has (epi, r e gular m o no) f a ctorizations of monomor- phisms, and effe ctive unions of r e gular sub obj e cts; for examp l e , S c o uld b e a quasitop os. If the r efle ction L : S → E pr eserves fin ite p r o ducts and monomo rphisms then it is quasi-lex. Pr o of. W e kno w that L preserv es finite pro ducts, thus it will suffice to sho w that it quasi-preserv es equalizers. Consider an equalizer diagram X e / / Y f / / g / / Z in S . Since L pr eserv es finite pro ducts, quasi-preserv ation of this equalizer is equiv alen t to quasi-preserv ation of the equalizer X e / / Y ( f Y ) / / ( g Y ) / / Z × Y in whic h now the pa rallel pair has a common retraction, given b y the pro jection Z × Y → Y . This implies that the exterior of the diagram X e / / e   Y f ′   ( f Y )   ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ Y g ′ / / ( g Y ) ( ( ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ Z ′ m # # ● ● ● ● ● ● ● ● ● Z × Y is a pullback . By effectiv enes s of unions, w e can form the union of  f Y  and  g Y  b y constructing the pushout square as in the in terior o f the diagram. Then the induced map m : Z ′ → Z × Y will b e the union, and in particular is a monomorphism. No w a pply the reflection L to GROTHENDIECK QUASITOPOSES 15 this last diagram, to get a diagram LX Le / / Le   LY Lf ′   ( Lf LY )   ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ LY Lg ′ / / ( Lg LY ) ) ) ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ LZ ′ Lm % % ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ LZ × LY in S . The inte rior square is still a pushout, and Lm and Le are still monomorphisms. F a cto r ize Le as an epimorphism k : LX → A follo w ed by a regular monomorphism d : A → LY . Then d , lik e an y regular monomorphism, is the equalizer of its coke rnel pair. Since k is an epimorphism, d and d k = Le hav e the same cok ernel pair , namely Lf ′ and Lg ′ . Th us d is the equalizer of Lf ′ and Lg ′ , and k is the canonical comparison. It is an epimorphism b y construction, and a monomorphism b y the standard cancellation prop erties. Th us L quasi- preserv es t he equalizer of f ′ and g ′ , and so also the equalizer of  f Y  and  g Y  , and so finally that of f and g .  Remark 4.6. In fact there is also a partia l conv erse to the preceding result: if L is quasi-lex and has strongly epimorphic units, then it preserv es finite pro ducts and monomor phisms; indeed an y quasi-lex L preserv es monomorphisms: see [ 5 ]. Lemma 4.7. If L is quasi-lex, then E has a we ak sub obje ct classifier if S do es so. Pr o of. Let t : 1 → Ω b e the w eak sub ob j ect classifier of S . Now Lt : L 1 → L Ω is a strong (in f act split) sub o b ject, so there is a unique map χ : L Ω → Ω suc h that L 1 Lt / /   L Ω χ   1 t / / Ω is a pullback. F orm the equalizer Ω ′ i / / Ω χℓ / / 1 / / Ω in S . 16 RICHARD GARN ER AND STEPHEN LACK Observ e that χ.ℓ .χ = χ.Lχ.ℓL Ω = χ.Lχ.Lℓ Ω, and so χ.ℓ.χ.Li = χ.Lχ.Lℓ Ω .Li = χ.Li ; th us χ.Li factorizes as i.χ ′ for a uniq ue χ ′ : L Ω ′ → Ω ′ . F urthermore, i.χ ′ .ℓ Ω ′ = χ.Li.ℓ Ω ′ = χ.ℓ.i = i and so χ ′ .ℓ = 1. This pro v es that Ω ′ ∈ E . F urthermore χ.ℓ.t = χ.Lt.ℓ = t and so t = it ′ for a unique t ′ : 1 → Ω ′ . W e shall sho w that t ′ : 1 → Ω ′ is a w eak sub ob j ect classifier for E . Supp ose then that m : A → B is a strong sub ob ject in E . The inclusion, b eing a rig h t adj o in t, preserv es strong sub ob jects, so there is a unique f : B → Ω for which the diag r a m A m / /   B f   1 t / / Ω is a pullbac k. W e shall sho w that f f actorizes as f = if ′ ; it then fo llows that f ′ : B → Ω ′ is the unique map in E classifying m . T o do so, it will suffice to sho w that χ.ℓ.f = f , or equiv alently χ.Lf .ℓ = f . Now consider the diagram A m / / ℓ   B ℓ   LA Lm / / L !   LB Lf   L 1 Lt / /   L Ω χ   1 t / / Ω in whic h the top square is a pullbac k since ℓA and ℓB are in v ertible, and the b otto m square is a pullbac k, b y definition of χ . Th us if the middle square is a pullbac k, then t he comp osite will b e, and so χ.Lf .ℓ m ust b e the unique map f classifying m . No w w e kno w that the comparison x from LA to the pullbac k P of Lf and Lt is b oth a n epimorphism and a monomorphism in E . But Lm , lik e m , is a strong monomorphism, and factorizes as sx , whe re s is the pullbac k of Lt ; th us x is a stro ng monomorphism and an epimorphism, and so inv ertible. This completes the pro of.  Com bining the main results of this section, w e ha v e: GROTHENDIECK QUASITOPOSES 17 Theorem 4.8. I f the r efle ction L : S → E is sem i - left-exact and quasi-lex, then E is a quasitop os if S is o ne. Corollary 4.9. If the r e fle ction L : S → E is semi-left-exact and pr ese rv e s finite pr o ducts and monomo rphisms, and so a lso if it has stable units and pr ese rv e s mo n omorphisms, then E i s a quasitop o s if S is one. Pr o of. Com bine the previous theorem with Theorem 3.5 and Prop osi- tion 4.5 .  5. Subquasitoposes As recalled in the in tro duction, a subtop os of a t o p os is a f ull reflec tiv e sub category for whic h the r eflection preserv es finite limits. These can b e c haracterized in v arious w a ys, for example using La wv ere-Tierney top ologies, or univ ers al closure op erat o rs. By analo gy with this, w e de- fine a sub quasitop os of a quasitop os S to b e a full reflectiv e sub category for whic h the reflection has stable units and preserv es monomorphisms. By Corollary 4.9 we kno w that the sub cat ego ry will indeed b e a qua- sitop os. In this section, we giv e a classification of sub quasitop oses of S using pr op er universal c l o sur e op er ators . A closure op erator j , on a category C with finite limits, assigns to eac h sub ob j ect A ′ ≤ A a sub ob ject j ( A ′ ) ≤ A in suc h a w a y that A ′ ≤ j ( A ′ ) = j ( j ( A )) and if A 1 ≤ A 2 ≤ A then j ( A 1 ) ≤ j ( A 2 ) ≤ A . The closure op erator is said to b e universal if for eac h f : B → A and eac h A ′ ≤ A w e ha v e f ∗ ( j ( A ′ )) = j ( f ∗ ( A ′ )). It is said to b e pr op er , esp ecially in the case where C is a quasitop os, if j ( A ′ ) ≤ A is strong sub ob ject whenev er A ′ ≤ A is one; of course this is automatic if C is a to p os, so that all sub ob jects are strong. If j ( A ′ ) ≤ A is a strong sub ob j ect for al l sub ob jects A ′ ≤ A , then j is said to b e a strict univ ersal closure op erator . Giv en a univers al closure op erator j o n C , a sub ob ject m : A ′ → A is said to b e j -dense if j ( A ′ ≤ A ) = A . An ob ject X of C is said to b e a j -she af if it is orthogonal to eac h j -dense monomorphism, and j -sep ar ate d if it is separated with resp ect to eac h j -dense morphism. Recall, f o r example from [ 12 , Theorem A4.4.8 ], that for a quasitop os S there is a bijection b et w ee n lo calizations of S and pro p er univ ersal closure op erators on S . Explicitly , the bijection asso ciates to a prop er univ ersal closure op erator j the sub category Sh( S , j ) of j -shea v es; while for a lo calization L : S → E , the corresp onding closure op erator sends a sub ob ject A ′ ≤ A to the pullbac k of LA ′ ≤ LA a long the unit ℓ : A → LA . F urthermore, b y [ 12 , Theorem A4.4 .5], if j is strict then Sh( S , j ) is a top os. Con v erse ly , if j is a prop er univ ersal closure 18 RICHARD GARN ER AND STEPHEN LACK op erator for whic h Sh( S , j ) is a top os, then for an y subob ject A ′ ≤ A in S , the reflection LA ′ ≤ LA is a sub ob ject in a top os, hence a strong sub ob ject; th us j ( A ′ ) ≤ A to o is a strong sub o ject, and j is strict. F or an y quasitop os Q , there is a strict univ ersal closure o p erator sending a sub ob ject A ′ ≤ A to its strong closure A ′ ≤ A , giv en by factorizing the inclusion A ′ → A as an epimorphism A ′ → A ′ follo w ed b y a strong monomorphism A ′ → A . An ob ject of Q is said to be c o arse if it is a sheaf for this closure op erator, and w e write Cs( Q ) for the full subcategory consisting of the coarse ob jects; this is a top o s, and is reflectiv e in Q via a finite-limit-preserving reflection Q → Cs( Q ) which in v erts precisely those monomorphisms whic h are also epimorphisms; see [ 12 , A4.4]. W e now supp ose tha t S is a quasitop os, and L : S → E a reflection on to a sub quasitop o s. As b efore, w e write R for the full sub catego ry of S consisting of tho se ob j ects X ∈ S for whic h t he unit ℓ : X → LX is a monomorphism. W e sa w in Prop osition 3.3 that R is reflectiv e in S , a nd w e saw in Corolla ry 3.4 that this reflection has stable units and preserv es monomorphisms; th us by Corollary 4.9 the cat ego ry R , lik e E , is a quasitop o s. Prop osition 5.1. Th e r e is a strict universa l closur e op er ator k on S whose she a ves ar e the c o arse obj e cts i n E and whose sep ar ate d obje cts ar e the obje cts of R . The class K of k -dense monomorphisms c onsists of al l those m onomorphisms m : X → Y for whic h the monomorp h ism Lm : LX → LY is a l s o an e p imorphism in E . Pr o of. W rite H : E → Cs( E ) for the reflection; b y the remarks ab ov e it preserv es finite limits. Recall from Prop osition 4.5 that L : S → E is quasi-lex; since H preserv es finite limits and inv erts the epimorphic monomorphisms, the comp osite H L : S → Cs( E ) is a finite-limit- preserving reflection. It follo ws that there is a prop er univ ersal closure op erator k on S whose shea v es ar e the coarse ob jects in E . Sinc e Cs( E ) is a top os, k is strict. A monomorphism m : X → Y in S is k - dense just when it is in- v erted b y H L ; that is, just when the monomorphism Lm is also an epimorphism. An ob ject A ∈ E is certainly separated with resp ect to suc h an m , since for any a : X → A the induced La : LX → A has at most one factorization through the epimorphism Lm : LX → LY . F urthermore, the m -separated ob jects a r e closed under sub o b jects, so that ev ery o b ject of R is k -separated. Con v erse ly , suppo se that X is k -separated; that is, separated with resp ect t o eac h k -dense m . W e m ust sho w that ℓX : X → LX is a monomorphism. Let d, c : K ⇒ X b e the k ernel pair of ℓX , and GROTHENDIECK QUASITOPOSES 19 δ : X → K the diagonal. If X is δ -separated, then since dδ = 1 = cδ , the t w o morphisms d and c m ust b e equal, whic h is to sa y that ℓX is a monomorphism. Th us it will suffice to sho w t ha t δ is k -dense. Since L preserv es finite pro ducts and monomorphisms, it also preserv es join tly monomorphic pairs; thus Ld and Lc are, like d and c , join tly monomorphic. On the other hand LℓX is in v ertible, and LℓX.Ld = LℓX.Lc , and so Ld = Lc ; thus in fact Ld is monomorphic. But Lδ is a section of Ld , and so b oth maps are inv ertible. In pa rticular, since Lδ is inv ertible, δ is k - dense, and so X ∈ R .  W e are now ready t o prov e our characterization of sub quasitop o ses. Theorem 5.2. Sub quasitop oses o f a quasitop os S ar e in bije ction wi th p airs ( h, k ) , wher e k is a strict universal closur e op er ator on S , and h is a pr op er unive rsal closur e op er ator on Sep( S , k ) with the pr op erty that every h -dense sub obje ct is also k -dense; the sub quasitop os c orr e- sp on ding to the p air ( h, k ) is Sh(Sep ( S , k ) , h ) . Pr o of. If k is a strict univ ersal closure op era t or o n S , then the categor y Sh( S , k ) of k -shea v es is reflectiv e in S via a finite-limit-preserving reflection M . The catego ry Sep( S , k ) of k -separated o b jects is also reflectiv e, and w e may obtain the reflection M ′ b y factorizing the unit m : X → M X of M as a strong epimorphism m ′ : X → M ′ X fo llow ed b y a monomorphism κ : M ′ X → M X , exactly as in Prop osition 3.3 . By Corollary 3.4 w e kno w that M ′ has stable units and preserv es monomorphisms. No w Sh (Sep( S , k ) , h ) is reflectiv e in Sep( S , k ) via a finite-limit-preserving r eflection, and so the comp osite reflection S → Sh(Sep( S , k ) , h ) has stable units and preserv es monomorphisms. Con v erse ly , let L : S → E b e a reflection on to a sub quasitop o s. As ab ov e, w e define k to b e the strict univ ersal closure op erator whose shea v es are the coarse ob jects in E . By Corollary 3.4 , w e kno w that the r estriction L : R → E of L to R preserv es finite limits, and so corresp onds to a prop er univ ersal closure op erator h on R , whose cat- egory of shea v es is E . Since ev ery k -sheaf is an h -sheaf, ev ery h -dense monomorphism is k - dense. It remains to pro v e the uniqueness of the h and k giving rise to L : S → E a s in the first paragraph. W e constructed M ′ ab ov e by factorizing X → M X as a strong epimorphism m ′ : X → M ′ X fo l- lo w ed b y a monomorphism κ : M ′ X → M X . Since ev ery h -dense monomorphism is k - dense, certainly every k -sheaf is an h -sheaf. Thus κ : M ′ X → M X factorizes through LX b y some ν : M ′ X → LX , nec- essarily monic, and now ℓ : X → LX factorizes as a strong epimorphsm m ′ : X → M ′ X follo w ed b y a monomorphism ν : M ′ X → LX . Th us 20 RICHARD GARN ER AND STEPHEN LACK Sep( S , k ) is uniquely determined b y L . In general, there can b e sev- eral differen t prop er univ ersal closure o p erators with a giv en category of separated ob j ects, but b y the discus sion after [ 12 , Theorem A4.4.8], there can b e at most o ne strict univ ers al closure o p erator with a giv en category of separated ob jects. Th us k is uniquely determined. Unlik e the case of separated o b jects, a prop er univ ersal closure op erator is uniquely determined by its shea ves , and so h is also uniquely deter- mined.  Observ e that in our characterization the t w o prop er univ ersal closure op erators liv e on differen t catego r ies. In t he next section, we shall see that when S is a presheaf to p os, there is a n alternative c haracteriza- tion in terms of tw o univ ersal closure op erators on S . In fact, ev en fo r a general quasitopo s, w e may giv e a characterization purely in terms of structure existing in S provid ed that w e prepared to w ork with stable classes of mono mo r phisms r a ther than univ ersal closure op erator s. As in Prop osition 5.1 , we let K denote the class of monomorphisms m : X → Y in S for whic h Lm is an epimorphism as w ell a s a monomorphism; as there, these ar e the dense monomorphisms for a univ ersal closure op erator, and so in particular are stable under pull- bac k. Now we let J b e the class of monomorphisms m : X → Y in S , ev ery pullback of whic h is in v erted by L . This is clearly the largest stable class of monomorphisms inv erted b y L . As we saw in Prop osi- tion 3.3 , the unit ℓX : X → LX is a mo no morphism for any X ∈ R ; furthermore since L has stable units, it preserv es the pullbac k o f ℓX along an y map, and so L in v erts not just ℓX but a lso all of its pullbac ks. Th us ℓX lies in J for all X ∈ R ; more generally , since L preserv es all pullbac ks ov er ob jects in R by Remark 3.6 , an y monomorphism f : X → Y in R whic h is inv erted b y L will lie in J . Theorem 5.3. L et S b e a quasitop os , and L : S → E a r efle ction onto a ful l sub c ate gory. If L has stable units and pr eserves monom orphisms, then (i) an obje ct X of S lies in R just when it is K -sep ar ate d; (ii) an obj e ct X of S lies in E just when it is K -sep a r ate d and a J -she af. Pr o of. W e ha v e already prov ed pa rt ( i ) in Prop osition 5.1 . F o r part ( ii ), first observ e that if A ∈ E then A is orthog onal to all morphisms in v erted b y L , not just those in J . It is of course also separated with resp ect to K . GROTHENDIECK QUASITOPOSES 21 Con v erse ly , if A is K -separated then it is in R ; but then ℓA : A → LA is in J , and so if A is a J -sheaf then ℓA mu st b e in v ertible a nd so A ∈ E .  A t the curren t lev el o f generalit y , there seems no reason wh y J need b e the dense monomorphisms f or a prop er univ ersal closure o p erator on S . In t he following sec tion w e shall see that this will b e so if S is a presheaf top os. 6. Grothendieck quasitop oses In this final section w e suppose tha t S is a presheaf top os [ C op , Set ], as w ell as the standing assum ption that L : S → E is a reflec tion whic h preserv es monomorphisms and has stable units. Recall that J consists of the monomorphisms whic h a r e stably inv erted b y the reflection L , and that K consists of the monomorphisms m for whic h Lm is an epimorphism in E as w ell as a monomorphism. By Theorem 5.2 , the class K consists of t he dense monomorphisms for a (prop er) univ ersal closure op erator k on S ; a nd by our new assumption that S is a presheaf top o s, k corresp onds to a Grothendiec k t o p ology with the same sheav es and separated ob jects. Since at this stage we are really only in tere sted in the shea ves and sep arated ob jects, w e tak e the lib ert y of using the same name k for the top ology as for the univ ersal closure op erator. As for J , since it is a stable system of monomorphisms, it can b e seen as a cov erage, in the sense of [ 12 ], and so generates a Grothendiec k top ology j whose shea v es are the o b jects o rthogonal to J . Theorem 6.1. F or a r efle ction L : [ C op , Set ] → E o nto a ful l sub c at- e gory o f a pr eshe af c ate gory, the fo l lowing c onditions a r e e quivalent: (i) The sub c ate gory E has the fo rm Sep ( k ) ∩ Sh( j ) for top olo gies j and k on C with k c ontaining j ; (ii) L is se mi-left-exact and pr eserves finite pr o ducts and monomor- phisms; (iii) L has stable units and pr eserves monomorphism s . An E as in the theorem is called a Grothendiec k quasitop os; as w e saw in the introduction, a category E has this f orm for some C , j , and k if and only if it is a lo cally presen table quasitop os [ 2 ]. Pr o of. The equiv a lence of ( ii ) and ( iii ) w as sho wn in Theorem 3.5 . The fact that t hese imply ( i ) now follow s from Theorem 5.3 . Th us it will suffice to suppo se ( i ) and show that ( iii ) follows. 22 RICHARD GARN ER AND STEPHEN LACK W e hav e adjunctions Sep( k ) ∩ Sh( j ) L 2 s s 3 3 ⊥ Sh( j ) L 1 s s 3 3 ⊥ [ C op , Set ] and L 1 preserv es a ll finite limits. It will clearly suffice to show that L 2 preserv es monomorphisms as we ll as pullbacks ov er an ob ject of Sep( k ) ∩ Sh( j ). No w Sep( k ) ∩ Sh( j ) is just the category o f separated ob jects in the top os Sh( j ) f or a (Law v ere-Tierney) top o logy k ′ in Sh( j ). Thus it will suffice t o show that fo r a top os S and a top ology k , the reflection L : S → Sep( k ) preserv es monomorphisms as w ell as pullbac ks ov er separated ob jects. This is the sp ecial case o f (one direction of ) Theo- rem 5.2 , where h is trivial.  As w e sa w in the previous section, the top ology k can b e recov - ered from Sep( k ) ∩ Sh ( j ), since Sh( k ) is the top os of coarse ob jects in Sep( k ) ∩ Sh( j ), whic h can b e obtained b y inv erting all those morphisms in Sep( k ) ∩ Sh( j ) whic h are b oth monomorphisms and epimorphisms. Unlik e the case of the (prop er) univers al closure o p erator h of the pre- vious section, j need not b e uniquely determined, as we no w explain. There exist non- trivial top olog ies k for whic h ev ery separated ob ject is a sheaf; these w ere studied b y Johnstone in [ 11 ]. In this case, for any top ology j contained in k we ha v e Sep( k ) ∩ Sh( j ) = Sh( k ) ∩ Sh( j ) = Sh( k ) , where the last step holds since Sh( k ) ⊆ Sh( j ). In pa rticular w e could tak e j to b e either trivial or k and obtain t he same subcatego ry Sh( k ) as Sep( k ) ∩ Sh( j ). Example 6.2. F or example, as explained in [ 12 , Example A4.4.9], w e could tak e the category Set M of M -sets, where M is the tw o - elemen t mo no id M = { 1 , e } , with e 2 = e , or equiv alently the category of sets equipp ed with an idemp o ten t. Then Set can b e seen as t he full reflectiv e sub catego ry of M -sets with trivial action. The reflection L : Set M → Set splits the idempo ten t; this preserv es all limits and so is certainly a lo calization. Since the unit of the a djunction is epimorphic, ev ery separated ob ject for the induced to p ology k is a sheaf. Remark 6.3. Theorem 6.1 can b e generalized to the case of a Gr o then- diec k t o p os S in place of [ C op , Set ]; then j and k would b e Lawv ere- Tierney to p ologies on S . It can f ur t her b e generalized to the case where S is a Gro thendiec k quasitop os, pro vided that w e are willing to work with prop er univ ersal closure op erators j and k rather tha n GROTHENDIECK QUASITOPOSES 23 top ologies. In either case, E will still b e a quasitop os b y Corollary 4.9 , and is in fact a G rothendiec k quasitop os. In the case o f a quasitop os or to p os S whic h is not lo cally presen table, how ev er, there seems no reason wh y the ob jects orthog onal to J should b e the shea v es, either for a t o p ology or a univ ersal closure op erator. Reference s [1] John C. B aez and Alexander E. Hoffnung. Conv enient categor ies of smo oth spaces. T r ans. Amer. Math. 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Vol. 2 , v olume 44 of O xfor d L o gic Guides . The Cla rendon Press O xford University Press, Oxford, 2002 . 2 [14] Jacques Penon. Sur les qua s i-top os. Cahiers T op olo gie G´ eom. Diff´ er entiel le , 18(2):181 –218 , 1977. 2 24 RICHARD GARN ER AND STEPHEN LACK Dep ar tment of Computing, Macquarie University, NSW 2109 Aus- tralia. E-mail add r ess : rich ard.g arner@ mq.edu.au Dep ar tment of Ma thema tics, Ma cquarie University, N SW 2109 Aus- tralia. E-mail add r ess : stev e.lac k@mq.e du.au