On principally generated Q-modules in general, and skew local homeomorphisms in particular

On principally generated quan taloid-mo dules in general, and sk ew lo cal homeomorphisms in particular Hans Heymans ∗ and Isar Stubb e † W ritten: Decem ber 18, 2007 Submitted: Marc h 5, 2008 Revised: Marc h 31, 2009 Abstract Ordered sheav es on a small quantaloid Q have been deﬁned in terms o f Q - enriched categorical structures; they form a lo c a lly ordere d category Ord ( Q ). The free-coc o mpletion KZ- doc tr ine on Ord ( Q ) ha s Mo d ( Q ), the quantaloid o f Q -mo dules, as category of Eilenber g-Mo ore algebra s. In this pap er we give an int rinsic description of the Kleisli algebra s: w e call them the lo c al ly princip al ly gener ate d Q -mo dules . W e deduce that Ord ( Q ) is biequiv ale n t to the 2-catego ry of lo cally pr incipally genera ted Q -mo dules and left adjoint mo dule morphisms. The exa mple of lo cally principa lly generated modules on a locale X is w orked out in full detail: relating X -mo dules to ob jects of the slice catego ry Lo c / X , we show that ordere d sheav es o n X corres p ond with skew lo c al home omorphi sms into X (lik e sheaves on X cor resp ond with lo cal ho meomorphisms into X ).  . Intr oduction Lo cales and quan tales, sheav es and logic. A lo c ale X is a complete lattice in whic h ﬁnite in ﬁma distrib ute o ver arbitrary supr ema. A particular class of examples of lo cales comes f rom top ology: the op en su bsets of any top ologi cal space form a lo cale. But n ot ev ery lo cale arises in this w a y , wh en ce the slogan that lo cales are “p oin tfree top ologies” [Johnstone, 1983]. There is a “p oin tfree” wa y to do sheaf th e- ory: a sheaf F on a locale X is a functor F : X op / / Set satisfying gluing conditions. The collection of all suc h functors, together with natural transformations b et w een them, forms the topos Sh ( X ) of she aves on X . One of the m any close ties betw een logic and sheaf theory , wh ich is of particular in terest to u s, is that the internal lo gic ∗ Department o f Mathematics and Computer Science, Un iversit y of Ant w erp, Middelheimlaa n 1, 2020 A nt werpen, Belgium, hans.heym ans@ua.ac.b e † P ostdoctoral F ellow of the Research F oundation Flanders (FW O ), Department of Mathemat- ics and Computer Science, Un iversit y of Ant w erp, Middelheimlaan 1, 202 0 Ant w erpen , Belg ium, isar.stubb e@ua.ac.be 1 of Sh ( X ) is an in tu itionistic higher-order predicate logic with X as ob ject of truth v alues [M ac Lan e and Mo erd ij k , 1991 ; Borceux, 1994; Johnstone, 2002]. T o b orro w a p hrase from [Re y es, 197 7] and others, sheaf theory th us serv es as algebr aic lo gic . The d eﬁnition of lo cale can b e restated: X is a complete latti ce and ( X, ∧ , ⊤ ) is a monoid suc h that the m ultiplication distributes on b oth sides o ver arbitrary suprema. It is natural to generalise this: a quantale Q = ( Q, ◦ , 1) is, b y deﬁnition, a monoid str u cture on a complete lattice such that the multiplica tion d istributes on b oth sides o v er arbitrary s u prema [Mulv ey , 1986; Rosenthal, 1992]. Beca use the monoid stru cture of a locale is obviously commutativ e, b ut the one for a q u an tale need not b e, on e can think of quan tales as “p oin tfr ee n on-comm utative top ologies”. Examples of quant ales, other than lo cales, can b e found in algebra and geometry [Mulv ey and P elletier, 2001; R esend e, 2007 ], in logic [Y etter, 1990], in compu ter science [Abramsky and Vic kers, 1993; R osicky , 200]. In the spirit of (enric hed) catego ry theory [Kelly , 1982] , it is n ot hard to see th at a quan tale is p recisely a monoid in the symm etric monoidal closed category Sup of complete latt ices and suprema-preservin g functions. And a quantal oid Q is th en d eﬁned as a category enric hed in Sup (so a quan tale is the same thing as a quan taloid with one ob j ect, precisely a s a group is a group oid with one ob ject) [Rosen thal, 1996]. The success of sheaf theory to study logic from lo cales, and the u seful generali- sation fr om lo cales to quantale s (and ev en quantalo ids), mak e one wonder ab out the “logic of shea ves on quan tales”. Ho w ever, it is not at all straigh tforw ard to deﬁ ne “shea ves on a qu an tale”! Man y diﬀerent deﬁnitions hav e b een prop osed b y many diﬀeren t authors, e.g. [Borceux and V an den Bossc h e, 1986 ; Mulv ey and Na waz, 1995; Gylys 2001; Garra w a y , 2005], but often only for particular classes of quan- tales. In previous w ork we h a ve ta k en the foll o w ing stance on the matt er: whereas shea ves on a locale X can b e describ ed in te rms o f sets equipp ed with an X -v alued equalit y r elation [La wv ere, 1973 ; F ourman and Scott, 1 979; Borceux, 1994], the non-comm utativit y of the quant ale m ultiplication forces shea ves on a qu an tale Q to b e sets equipp ed with a Q -v alued ine qu ality relation 1 [Borceux and Cruciani, 1998 ; Stubb e, 2 005b]. In other w ords, ou r atten tion go es to the cate gory of or der e d she aves on a quantale (or ev en quan taloid), which we s ee as “algebraic non-commutat iv e logic” . More precisely , Stu b b e [2005b] stud ied ord ered shea v es on a quant aloid Q in terms of Q -enric hed categorie s, th us generali sing to the non-comm u tative case the w ork of [W alters, 1981; Borceux and Cru ciani, 1998] on lo cales. In th is pap er w e shall sho w that they can equiv alen tly b e describ ed as particular Q -mod ules. (If Q is a quan taloid, then a Q -mo dule is b y deﬁnition a S up -enric hed functor F : Q op / / Sup . F or a quan tale Q this reduces to a complete la ttice on which Q acts.) Th is in turn can b e applied to a locale X , and w e ﬁ nd a c haracterisation of the relev ant X -mo dules as p articular lo cale morphism s with co domain X . W e sp eak of princip al ly gener ate d 1 W e make a remark ab out Q -val ued e qualities at the end of th is Introd uction. 2 Q -mo dules in general, and ske w lo c al home omo rphisms into X in p articular, as we shall introdu ce next. Principally generated Q -mo dules. T o introd u ce this n o ve l notion in Q -mo dule theory , around which this article is cen tered, we ﬁ rst recall a simple fact from order theory: The well-kno wn adjunction b etw een the category Ord of ordered sets and order-preserving functions on the one hand, and the categ ory S up of complete lattices and s u prem um-preservin g fun ctions on the other, Ord ⊥ F ( ( U h h Sup , (1) has the remark able feature that b oth fun ctors in v olv ed are em b eddings. This allo ws us to view Sup as a part of Ord , but also Ord as a part of Sup . The ﬁrst viewp oint corresp onds to the common understanding that a complete la ttice is a n ordered set in whic h all su prema exist and that a sup-morphism is an order-pr eserving fu nction that preserves suprema. More tec hnically: Sup is the category of Eilenberg-Mo ore algebras for the ‘free-co completion KZ-do ctrine’ on Ord , whic h sends an ordered set to the set of its downclosed sub sets ordered b y inclusion. The second p oin t of view is wh at the notion of totally alg ebraic complete lattice is all ab out. Recall that an elemen t a of a complete lattice L is total ly c omp act (a.k.a. sup er c omp act ) when, for an y do wnclosed A ⊆ L , a ≤ W A implies a ∈ A ; and a complete lattice L is total ly algebr aic (a.k.a. sup er algebr aic ) w hen eac h elemen t is the supr em u m of totally compact ones [Gierz et al. , 1980]. It turns out that th e replete image of the left adjoin t in the ab o ve adju nction is precisely the sub categ ory of S up of total ly algebraic ob jects and left adjoin t morph isms; thus Ord is describ ed in trinsically (and up to equiv alence) as a part of Sup . W e wan t to broaden the situation d epicted ab ov e: instead of studying ordered sets, i.e. ord ered s hea ves on the t w o-elemen t Bo olean algebra 2 , we wan t to consider ordered shea ves on an y sm all quan taloid Q . The latter form a 2-categ ory Ord ( Q ) that we d eﬁne 2 as Cat cc ( Q si ): its ob jects are the Cauc h y-complete categories en- ric h ed in th e sp lit-idemp oten t completion of the quan taloid Q , and its morphisms are the Q -enric hed f unctors. It was p ro ved b y I. Stubb e [2007b] that the cate- gory of “internal sup-lattices and sup-morp h isms” in Ord ( Q ) is (biequiv alen t to) Mo d ( Q ) := QUANT ( Q op , Sup ), the quan taloid of Q -mo dules. That is to sa y , in analogy w ith the situat ion in (1) abov e, there is a biadju nction Ord ( Q ) ⊥ F ( ( U h h Mo d ( Q ) (2) 2 See [Stubb e, 2005b] for a more “elementary” deﬁnition of ordered sheav es on a quantaloid. 3 that sp lits the free-co completion KZ-do ctrine on Ord ( Q ) and such that moreo v er Mo d ( Q ) is (biequiv alen t to) the category of Eilen b erg-Mo ore alge bras for that do c- trine. This d escrib es Mo d ( Q ) as p art of Ord ( Q ), and for Q = 2 we thus reco v er exactly half of the situation describ ed in the ﬁrst paragraph ab o ve. But w hat ab out the other half: Can w e also int rinsically characte rise Ord ( Q ) as part of Mo d ( Q )? Can w e giv e a m o d ule-theoretic condition on an ob ject of Mo d ( Q ) that make s it equiv alen t to the free co completion of an ob ject of Ord ( Q )? And how d o morph isms then r elate ? With our Deﬁnition 5.3 and our Theorem 5.9 w e answ er these questions in th e aﬃrmativ e: w e prov e that Map ( Mo d lpg ( Q )), deﬁned as the sub category of Mod ( Q ) of lo c al ly princip al ly gener ate d Q -mo dules and left adjoint Q -mo dule morphisms , is precisely the replete imag e of the left biadjoin t functor in (2) a b o v e: Ord ( Q ) F / / K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K Mo d ( Q ) Map ( Mo d lpg ( Q )) ?  O O The tec hnology th at we use to solv e this pr oblem is Q -enric hed catego rical alg e- bra, as pioneered (in greater generalit y) b y [B ´ enab ou, 1967 ; W alters, 198 1; Street, 1983a ] and more recentl y survey ed by [Stubb e, 2005a]. More particularly , in this pap er we bu ild further on r esults from [Stub b e, 2007a], whic h treats totally algebraic co complete Q -categories, and [Stubb e, 2006], where an explicit comparison is giv en b et w een co complete Q -cat egories and Q -mo dules. Sk e w local homeomo rphisms. Th e n otion of ordered sheaf on a qu an taloid Q is so d evised that, when taking Q to b e the one-ob ject susp en s ion of a lo cale X (i.e. Q has one ob ject, the elemen ts of X are view ed as arro ws of Q , comp osition of which corresp ond s to ﬁ nite meets in X , the identit y arr o w th us b eing the top elemen t ⊤ of X ), Ord ( X ) is equiv alen t to the category of ord ered ob jects and ord er- preserving morphisms in the to p os Sh ( X ) of s h ea ve s on X [W alters, 1981; Borce ux and Cruciani, 1998; Stub b e, 2005b]: Ord ( X ) ≃ Ord ( Sh ( X )). Our general results on Q -mod ules from the ﬁrst part of this pap er surely sp ecialise to the localic case: ordered shea v es on X can b e d escrib ed equ iv alen tly a s lo cally p r incipally generated X -mo dules, order-preserving morp hisms then corresp ond to left adjoin t X -mo d u le morphisms. In analo gy with ring theory it is v ery natural to r egard a lo cale morphism f : Y / / X as a left X -mo du le ( Y , ◦ f ) w ith action y ◦ f x := y ∧ f ∗ ( x ) [Joy al and Tier- ney , 1984]. This construction extends to a (con trav ariant) em b edding of the s lice catego ry Lo c / X in Mo d ( X ). Thus it is natural to try to c h aracterise the sub category of Loc / X whic h corresp onds, under this embedd ing, to the lo cally p rincipally gen- erated X -mo d ules and the left adjoin t X -mo dule morphism s b et we en them, or in 4 other words, to Ord ( X ). In Deﬁnitions 7.1 and 7.7 w e introd uce the lo cale theoretic notions of skew op en morphism and skew lo c al home omorphism , and in Theorem 7.10 w e then pro ve that ( Lo c / X ) o slh , b y deﬁnition the (n on-full) sub category of Loc / X of sk ew lo cal homeomorphism s as ob jects and sk ew open morph isms b et ween them, is the sought-a fter equiv alen t of Ord ( X ). A local h omeomorphism is necessarily a s k ew lo cal h omeomorphism; and an op en lo cale morphism is alw a ys sk ew op en to o. Thus the category LH / X of lo cal homeomorphisms o v er X is n atur ally a full sub category of ( Lo c / X ) o slh . This situation to o can b e stated in terms of X -mo du les: in Deﬁnition 7.11 we in tro d uce ´ etale X -mo dules as a particular kin d of lo cally p rincipally generated X -mo dules, suc h that in Theorem 7.12 w e can pro v e that the f ull sub category M ap ( Mod ´ et ( X )) of Map ( Mo d lpg ( X )) deﬁned by the ´ etale X -mod ules is indeed equiv alent to LH / X . The ca tegory LH / X is a we ll-kno w n equiv alen t o f the to p os Sh ( X ), see e. g. [Mac Lane and Mo erdijk, 1992, p. 524]; th us th e inclusion of lo cal homeomorphisms o ver X in to skew lo cal homeomorphisms o v er X , or equiv alen tly th e inclusion of ´ etale X -mo dules in to lo cally p rincipally generated X -mo du les, is precisely the same thing as the inclusion of sh ea ve s on X into ordered sheav es on X . Moreo v er, in the realm of enric hed catego rical structures it is a matter of f act that the “ X -sets” of M. F ourman and D. Scott [1979] (see also [Borceux, 1994, V ol. 3; Borceux and Cru ciani, 1998; Johnstone, 2002, p . 502–513]) are included in X -ord ers. All this establishes the follo wing unifying diagram of equiv alen t emb eddings of categories of s y m metric (or discrete) ob jects into 2-cate gories of asymmetric (or o rdered) ob j ects: Ord ( Sh ( X )) Ord ( X ) ( Lo c / X ) o slh Map ( Mo d lpg ( X )) Sh ( X ) ?  O O Set ( X ) ?  O O LH / X ?  O O Map ( Mo d ´ et ( X )) ?  O O This sh o ws the relation b et ween (ordered) shea ves on a lo cale X as (i) functors on X satisfying g luing conditions, (ii) X -enric hed categ orical structures, (iii ) locale morphisms in to X and (iv) X -mo dules. Ov erview of con t en t s. Sections 2 through 6 of this p ap er are concerned with the translation of the notion of or der d she af on a smal l qu antalo id Q fr om its origi- nal deﬁnition in terms of Q -enric h ed cate gorical structur es [W alters, 1981; Stub b e, 2005b] to the language of Q -modu les. T o mak e this paper self-con tained, we there- fore start with an o ve rview of the Q -enric hed categorica l algebra that w e need: i n Section 2 we recall the deﬁnition of Q -catego ries, functors and d istributors; we sp eak of weig h ted colimit s in a Q -category and of Cauch y-completeness of a Q -category; and we end with the deﬁnition of Q -order. W e ha v e tried to includ e th e r elev ant “historical” references, but in practice we refer mostly to the more r ecen t [Stubb e, 2005a , 2005b, 2006] wh ose notations we follo w. In Section 3 we r ecall some material 5 on totally algebraic co complete Q -categories f rom [Stub b e, 2007a ]; in fact, we re- cast the d eﬁ nition of a totally compact ob ject in a w ay that suits our needs further on. (In Section 8 we explain a biadju nction in v olving tota lly algebraic co complete Q -categ ories: str ictly sp eaking it is of no tec hnical imp ortance for the rest of th is pap er, but since it ma y b e of indep en d en t in terest we ha v e c hosen to add it as an Addendu m.) S ection 4 then con tains the crucial translation fr om Q -enric h men t to Q -v ariation – to b orr o w a term from [Betti et al. , 1983], later pick ed up b y [Gordon and P o wer, 1997] and [S tubb e, 2006] – wh er e we intro d uce the notion of princip al ly gener ate d Q -mo dule . Our ﬁrst main theorem is that Cauc hy-complete Q -cate gories and functors b etw een them f orm a category which is equiv alen t to that of prin cipally generated Q -mo dules and left adjoin t mo du le m orp hisms. Finally , in Section 5 w e explain ho w a so-called lo c al ly princip al ly ge ne r ate d Q -mo dule is the same thing as a principally generated modu le on the split-i demp otent completion of Q , th u s pa ving the wa y for our second main result: Q -orders (meaning Cauch y-complete categories enric hed in the split-idemp oten t completion of Q ) and th eir morph isms are essen- tially the same thing as lo cally pr incipally generated Q -mo du les and left adjoin t mo dule morph isms. W e discuss s ome examples in Section 6. The s econd part of this pap er, con tained in Section 7, is d ev oted to the appli- cation of the ab o ve to the sp eciﬁc case where Q is the one-ob ject susp ens ion of a lo cale X (view ed as monoid ( X , ∧ , ⊤ )). Lo cally p rincipally generated X -mo du les are then equ iv alen t to ord ered shea v es on X , whic h this time can really b e under- sto o d as ordered ob jects in the top os Sh ( X ) [W alters, 198 1; Borce ux and Cru ciani, 1998; S tubb e, 2005b]. (If one tak es f or gran ted that lo cally pr incipally generated X -mo dules are ordered shea v es on X , then one can start r eading Section 7 righ t a wa y; this second p art of the p ap er is tec hnically sp eaking rather indep enden t from the ﬁrst p art.) But lo cally principally generated X -mo dules can also b e expressed in terms of certain lo cale morphism s into X , and it is t heir stud y that we deal with here. Thus, w e b egin by brieﬂy explaining, taking h ints from [Joy al and Tierney , 1984], ho w an y lo cale morph ism int o X can b e regarded as an X -mo dule; we d eﬁne skew op en morphisms in the slice category Lo c / X to corresp ond to left adjoin t X - mo dule m orphisms. A detailed study of lo cally principally ge nerated X -mo d u les is carried out thereafte r; w e sho w in p articular that an y suc h X -mo dule is necessarily induced by a locale morphism int o X . This w ork b eing done, w e come to our third main resu lt of th e pap er: we deﬁne skew lo c al home omorphisms in terms of co v er- ings by skew op en se ctions , and pr o ve that th e su b category of Lo c / X with sk ew lo cal homeomorphisms as ob jects an d skew op en morp h isms b et we en them, is equiv alen t to the category of locally principally generated X -mo dules and le ft adjoin t mo du le morphisms, viz. ordered s hea ves on X . Remarking that local homeomorph isms are necessarily sk ew lo cal homeomorphisms, we end with the iden tiﬁ cation of ´ etale X - mo dules as those lo cally p rincipally generated X -mo dules whic h corresp ond to local homeomorphisms, viz. shea v es on X . 6 F urther w ork. In this pap er w e do not sp eak of the “inte rnal logic” of the category of o rdered shea ve s on a quan taloid – it is still an op en problem – but we hop e that our con trib ution here will b e helpful to inv estigate this. In this r esp ect it s h ould b e in teresting to inv estigate links with examples of noncommutativ e logics dev elop ed from a more logical (rather than algebraic, i.e. sheaf theoretic) p oint of view: for example, R. Goldblatt’s [2006] enco ding of p r edicates in some non-comm utativ e logic as quan tale-v alued fu nctions on a set (whic h can b e seen as elemen ts of a principally generated mo dule!); or [Baltag et al. , 2007], who us e quan tale-modu les in their treatmen t o f epistemic lo gic; or K. Rosen thal’s [1 994] m od el for the “bang” op erator in linear logic via mo du les on a quant ale; the construction in [Coniglio and Miraglia, 2001] of a logic from a v ery particular notion of sheaf on a restricted class of quan tales; or the qu an tale based seman tics for prop ositional normal modal logic in [Marcelino and Resend e, 2008]; and man y others. In [Resende and Ro drigues, 2008] lo cal homeomorph isms in to X are shown to corresp ond to Hilb ert X -mo du les with a Hilb ert basis (a sp ecial case of Hilbert Q -mo dules, f or Q an in volutiv e qu antale [Pasek a, 1999]). Their results and our results in Section 7 (particularly Deﬁn ition 7.11) are s imilar in that w e b oth pr o vide a description of lo cal homeomorphisms in to a lo cale X in terms of particular X - mo dules. In [Heymans and S tubb e, 2008] we explain, in the generalit y of mod ules on an inv olutiv e qu an tale Q , the p recise r elatio nship b et wee n Hilb ert Q -mo dules admitting a Hilb ert basis on the one hand , and lo cally principally generated Q - mo dule satisfying a suitable symmetry c ondition on its lo cally principal elements on the other hand; we argue th at the latter are p r ecisely sets with a Q -v alued e quality , i.e. a “ Q -sets” rather than a “ Q -orders”. Ap p lied to a lo cale X this give s (ordinary) shea ves on X ; b etter still, applied to suitably constructed in vol utiv e qu an tales we can describ e all Grothend iec k top oses, in a manner closely related to [W alters, 1982]. Our cur r en t w ork cont in ues along this line and fo cuses on Q -v alued equalities in the generalit y of an inv olutive quantal oid Q , more sp eciﬁcally on the interpla y b et w een symm etric and non-symmetric Q -categories and their Cauc h y completions. This is directly r elate d to [W alters, 1982; Betti and W alters, 198 2; F reyd and S ce- dro v, 1990] but also has ties with [Gylys, 2 001; Heymans, 2009].  . Preliminaries Quan taloids. Let Sup denote the category of complete lattices and maps that pre- serv e arbitrary sup r ema ( suplattic es and supmorphisms ): it is symm etric monoidal closed for the usual tensor pro d u ct. A quantaloid is a Sup -enric h ed category; a one- ob ject qu an taloid is most often thought of as a monoid in Sup : it is a quantale . A Sup -fu nctor b etw een quan taloids is sometimes called a homomorph ism ; QUANT denotes th e (“illegit imate”) category of qu an taloids and their homomorphisms. T h e standard reference on catego ries enric hed in a symm etric mon oidal category in gen- eral is [K elly , 1 982]; f or quan tales and quanta loids in p articular there is [Rosen thal, 7 1990, 1996]. Comp osition with a morphism f : X / / Y in a qu an taloid Q giv es r ise to adju nc- tions, one for eac h A ∈ Q , Q ( A, X ) ⊥ f ◦ − ( ( [ f , − ] h h Q ( A, Y ) and Q ( Y , A ) ⊥ − ◦ f ( ( { f , − } h h Q ( X, A ) . These righ t adjoints are resp ectiv ely called lifting and extension (through f ). W e shall ke ep the notations “[ − , − ]” and “ {− , −} ” for liftings and extensions in an y quan taloid that follo ws; no confu sion shall arise 3 . Giv en a quan taloi d Q w e write Q si (“ si ” stands for “split the idemp otents” ) for the n ew q u an taloid whose ob jects are the idemp oten t arrows in Q , and in whic h an arro w from an idemp oten t e : A / / A to an idemp otent f : B / / B is a Q -arro w g : A / / B satisfying g ◦ e = g = f ◦ g . Comp osition in Q si is d on e as in Q , the iden tit y in Q si on some id emp oten t e : A / / A is e itself, and the lo cal order in Q si is that of Q . T here is an obvious inclusion j : Q / / Q si , mapping f : A / / B to f : 1 A / / 1 B , whic h expr esses Q si as the univ ersal split-idemp otent completion of Q in QUANT . When Q is a sm all quant aloid, Mo d ( Q ) sh all b e shorthand f or QUA NT ( Q op , Sup ): the ob jects of this (la rge) quan taloid are called the mo dules on Q , or b rieﬂy Q - mo dules. Since idemp oten ts sp lit in Sup , and noting that Q si is small wh enev er Q is, it follo ws that Mo d ( Q ) is equiv alen t to Mo d ( Q si ). W e shall come bac k to m o d ules on Q and on Q si in S ectio ns 4 and 5. A qu an taloi d is in particular a lo cally ordered category , and therefore we can straigh tforwardly d eﬁne adjoint p airs in an y qu an taloid: f : X / / Y i s left adjoin t to g : Y / / X (and g is r ight adjoint to f , wr itten f ⊣ g ) when 1 X ≤ g ◦ f and f ◦ g ≤ 1 X . If a morphism f : X / / Y has a right adjoint, then the lat ter is u nique, and we shall often u se f ∗ as its n otatio n. Because left adj oints are sometimes called “maps”, Ma p ( Q ) is ou r notation f or the (lo cally ordered) category of left adjoin ts in a qu an taloid Q . Quan taloid-enric hed categories. A quan taloi d is a bicategory and therefore it ma y serv e itself as base for enrichmen t [B ´ enab ou 1967; W alters, 1981; S treet, 1983a]. The theory of quan taloid-enric hed categ ories, functors and distributors is survey ed in [Stubb e, 2005a] where many more appropriate references are giv en; here w e can only pro vide a brief summary , but we f ollo w the notatio ns of op. cit. for easy cross reference. T o a v oid size issues w e w ork from no w on with a smal l quantaloid Q . A Q -c ate gory A consists of a set A 0 of ‘ob jects’, a ‘t yp e’ function t : A 0 / / Q 0 , and for any a, a ′ ∈ A 0 a ‘hom-arro w ’ A ( a ′ , a ): ta / / ta ′ in Q ; these data are required 3 These right adjoints also go by the name of r esiduations when Q is a quantale, i.e. a one-ob ject quantaloid. W h ereas our notations are the usual ones in category theory (for closed monoidal categories or bicategories ), other notations instead of [ f , g ] and { f , g } that can b e found in the literature includ e f → g and g ← f , or f → r g and f → l g , or f /g and f \ g , or g f and f g . 8 to satisfy A ( a ′′ , a ′ ) ◦ A ( a ′ , a ) ≤ A ( a ′′ , a ) and 1 ta ≤ A ( a, a ) for all a, a ′ , a ′′ ∈ A 0 . A fu nctor F : A / / B is a map A 0 / / B 0 : a 7→ F a that satisﬁes ta = t ( F a ) and A ( a ′ , a ) ≤ B ( F a ′ , F a ) for all a, a ′ ∈ A 0 . A Q -category A h as an underlying or der 4 ( A 0 , ≤ ): f or a, a ′ ∈ A 0 deﬁne a ≤ a ′ to mean ta = ta ′ =: A and 1 A ≤ A ( a, a ′ ). If a ≤ a ′ and a ′ ≤ a w e w r ite a ∼ = a ′ and s ay that these are isomorphic obje cts in A . F or parallel functors F , G : A / / / / B w e no w put F ≤ G w hen F a ≤ Ga for ev ery a ∈ A 0 . With the obvio us comp osition and iden tities w e thus obtain a lo cally ordered category Ca t ( Q ) of Q -catego ries and fun c- tors. Pr ecisely b ecause Cat ( Q ) is a 2-cate gory , w e can fr om no w on unambiguously use 2-categorical notions su c h as adjoint functors, Kan extensions, and so on. T o giv e a distributor (or mo dule or pr ofunctor ) Φ: A ❝ / / B b et w een Q -categ ories is to sp ecify , for all a ∈ A 0 and b ∈ B 0 , arrows Φ( b, a ): ta / / tb in Q su ch that B ( b, b ′ ) ◦ Φ( b ′ , a ) ≤ Φ( b, a ) and Φ( b, a ′ ) ◦ A ( a ′ , a ) ≤ Φ( b, a ) for ev ery a, a ′ ∈ A 0 , b, b ′ ∈ B 0 . Tw o distribu tors Φ: A ❝ / / B , Ψ: B ❝ / / C comp ose: w e write Ψ ⊗ Φ: A ❝ / / C for the d istr ibutor with e lemen ts  Ψ ⊗ Φ  ( c, a ) = _ b ∈ B 0 Ψ( c, b ) ◦ Φ( b, a ) . The identi t y d istributor on a Q -category A is A : A ❝ / / A itself, i.e. th e distributor with elemen ts A ( a ′ , a ): ta / / ta ′ . F or parallel distr ib utors Φ , Φ ′ : A ❝ / / ❝ / / B w e deﬁ ne Φ ≤ Φ ′ to mean that Φ( b, a ) ≤ Φ ′ ( b, a ) for ev ery a ∈ A 0 , b ∈ B 0 . It is easily seen that Q -categories and distr ibutors form a quant aloid Dist ( Q ). Ev ery fun ctor F : A / / B b etw een Q -c ategories represen ts an adjoint pair of dis- tributors: - th e left adjoin t B ( − , F − ): A ❝ / / B h as elemen ts B ( b, F a ): ta / / tb , - th e right adjoint B ( F − , − ): B ❝ / / A h as elemen ts B ( F a, b ): tb / / ta . The assignment F 7→ B ( − , F − ) is a (bijectiv e and ) faithful 2-fun ctor from Cat ( Q ) to Dist ( Q ); thus, whenever a distrib utor Φ: A ❝ / / B is represente d by a functor F : A / / B , this F is essen tially unique. 4 By an or der w e mean a reﬂexiv e and transitive relation, i.e. a (small) ca tegory wi th at most one arro w b etw een any tw o ob jects; some call this a preorder. W e shall sp eak of a p artial or der or an antisymmetric or der if we requ ire moreo ver antisymmetry . 9 W eigh ted colimits. In a Q -enric hed category C we can sp eak of w eigh ted limits and coli mits, as int ro duced b y R. Street [19 83a] for general bicateg ory-enric h men t. F or our short accoun t here we use [Stubb e, 2005a, 2006] as references, b ut w e sh ould certainly also men tion the wo rk of [Gordon and P ow er, 1997] on conical (co)limits and (co)tensors (to b e explained b elo w). Giv en a distributor Φ: A ❝ / / B and a fu nctor F : B / / C , a functor K : A / / C is the Φ - weig hte d c olimit of F when it sat isﬁes C ( K − , − ) = [Φ , C ( F − , − )] in Dist ( Q ) (a nd in that case it is essenti ally uniqu e, so we write it as colim(Φ , F )). The Q -cate gory C is c o c omplete wh en it admits all s uc h w eigh ted colimits. A functor H : C / / C ′ is c o c ontinuous when it preserv es all col imits that happ en to exist in C : H ◦ colim(Φ , F ) ∼ = colim(Φ , H ◦ F ). A left adjoint functor is alwa ys coconti n uous; con versely , if the domain of a co con tin u ous functor is cocomplete, then that functor is left adjoint. C o complete Q -categories an d co con tin uous functors form a su b-2- catego ry Co cont ( Q ) of Ca t ( Q ). (Dually one can sp eak of weighte d limits , c omplete Q -c ate gories and c ontinuous functors ; w e shall not explicitly need these fu r ther on, ho w ever, it is a matter of f act that a Q -c ategory is complete if and only if it is co complete [Stu bb e, 2005a, 5. 10].) Ev ery ob ject X of a quan taloid Q determines a one-ob j ect Q -catego ry ∗ X whose single h om-arro w is 1 X . A c ontr avariant pr eshe af of typ e X on a Q -category A is a distributor φ : ∗ X ❝ / / A ; these are the ob jects of a co complete Q -cate gory P A w hose hom-arro ws are giv en by lifting in Dist ( Q ). Ev er y ob ject a ∈ A 0 determines, and is determined b y , a functor ∆ a : ∗ ta / / A ; th us a ∈ A 0 also represen ts a (left adjoint) presheaf A ( − , a ): ∗ ta ❝ / / A . The Y one da emb e dding Y A : A / / P A : a 7→ A ( − , a ) is a fully faithful 5 con tinuous fun ctor. The pr esheaf constru ction A 7→ P A extends to a 2-functor P : Ca t ( Q ) / / Co cont ( Q ) wh ich is left b iadj oin t to the inclusion 2- functor, with th e Y oneda em b eddings as unit; thus pr esheaf categories are the fr eely co complete ones. In fact, a Q -category C is co complete if and only if the Y oneda em b edding admits a left adjoint in Cat ( Q ); if this is the case we write sup C : P C / / C for that left adjoint: it maps a presheaf φ o n C to the we igh ted colimit sup C ( φ ) := colim( φ, 1 C ). Note b y the wa y that sup C ◦ Y C ∼ = 1 C ; actually , Y C admits a left adjoin t if and only if it admits a left in v erse. Giv en an y X ∈ Q we shall write ( C X , ≤ ) for the (p ossibly empty) sub-order of ( C 0 , ≤ ) con taining all c ∈ C 0 for whic h tc = X . If for eac h X ∈ Q the order ( C X , ≤ ) is a complete lattice , we sa y that C is or der-c o c omplete . On the other hand, for eac h morphism f : A / / B in Q w e can consid er the one-elemen t d istributor ( f ): ∗ A ❝ / / ∗ B . Supp ose that c ∈ C is of t yp e tc = B , and th at colim(( f ) , ∆ c ) exists: it is itself a functor from ∗ A to C , an d can th us b e identiﬁed with an elemen t of C of type A . W e write that elemen t as c ⊗ f and call it the tenso r of c with f ; C has al l tensors 5 A funct or F : A / / B is f ul ly f aithful when A ( a ′ , a ) = B ( F a ′ , F a ) for every a, a ′ ∈ A 0 . 10 when all suc h colimits with one-eleme n t w eights exist. T h e dual notio n is c otensor . It has b een prov ed in [Stu bb e, 2006 , 2.13] that a Q -category C is co complete if and only if it is order-co complete and has all tensors and cotensors; moreo ver, for an y Φ: A ❝ / / B and F : B / / C , the we igh ted colimit colim(Φ , F ): A / / C is the fun ctor deﬁned by a 7→ W b ∈ B 0 F b ⊗ Φ( b, a ). Ordered shea v es on a quan t aloid. The imp ortance of the f ollo wing notion h as ﬁrst b een recognised b y B. La w v ere [19 73] in the cont ext of categories enriched in a monoidal category . A Q -category C is Cauchy c omplete if for an y other Q -category A the m ap Cat ( Q )( A , C ) / / Map ( Dist ( Q ))( A , C ): F 7→ C ( − , F − ) is surjectiv e, i.e. when an y left adjoin t distrib utor (also call ed Cauchy distributor ) in to C is repr esen ted by a fun ctor. This is equiv alent to the requir ement th at C admits an y coli mit weig h ted by a Cauc hy distributor; and moreo ver such weigh ted colimits are absolute in the sense that th ey are p reserv ed by any fu nctor [Street, 1983b]. W e write Cat cc ( Q ) for the full s u b category of Cat ( Q ) wh ose ob jects a re the Cauc h y complete Q -ca tegories. No w we hav e ev erything ready to deﬁne the cen tral notion of this pap er [Stubb e, 2005b]. Deﬁnition 2.1 F or a smal l quantaloid Q , we write O rd ( Q ) for the lo c al ly or der e d c ate gory Cat cc ( Q si ) , and c al l its obje cts ordered sh ea ve s on Q , or simply Q -orders . When taki ng Q to b e the (one-ob ject su sp ension of ) a lo cale X , Ord ( X ) is the catego ry of ordered ob jects and order -p reserving morphisms in the to p os Sh ( X ), as ﬁrst (imp licitly) ob s erv ed by B. W alters [1981] (but see also [Borceux and Cru ciani, 1998] f or the locale-sp eciﬁc notion, and [S tu bb e, 20 05b] for the generalisation to quan taloids and the comparison b etw een [W alters, 1981] and [Borceux and Cruciani, 1998]) . Ob viously , this example ins pired our terminolog y .  . Tot al algebr aicity revisited W e sh all review and expand the material that w e n eed from [Stubb e, 2007a]. Deﬁnition 3.1 (Stubb e, 2007a) L et A b e a c o c omplete Q -c ate gory. The totally b elo w distribu tor on A is the right extension of A ( − , sup A − ) thr ough P A ( Y A − , − ) 11 in Dist ( Q ) : P A ❝ P A ( Y A − , − ) / / ❝ A ( − , sup A − )   A A ❝ Θ A := n A ( − , sup A − ) , P A ( Y A − , − ) o . > > ≤ An obje ct a ∈ A is totally compact when 1 ta ≤ Θ A ( a, a ) . Writing i A : A c / / A for the f u l l emb e dding of the total ly c omp acts, A i s totally algebraic when the left Kan extension of i A along itself is isomorphic to 1 A . In the simp lest p ossible case, when Q is the (one-ob ject susp ension of ) th e t w o- elemen t Boolean algebra ( 2 , ∧ , ⊤ ), a Q -categ ory A is an ord ered set ( A, ≤ ), it is co complete p recisely w hen ( A, ≤ ) is a su p-lattice , and the distribu tor Θ A is the follo wing “totally b elo w” relation: a ′ ≪ a when, for eve ry down-clo sed s ubset D ⊆ A , a ≤ W D implies a ′ ∈ D . A totally compact elemen t is one which is totally b elo w itself, an d ( A, ≤ ) is totally algebraic if and only if ev er y elemen t is the suprem um of the totally compacts b elo w it. These notions are related to, but stronger than, the “wa y b elo w” relation and the “algebraic” sup-lattices [Gierz et al. , 1980]. Theorem 3.2 (Stubb e, 2007a) The 2-functor P : Dis t ( Q ) / / Co cont ( Q ):  Φ: A ❝ / / B  7→  Φ ⊗ − : P ( A ) / / P ( B )  is lo c al ly an e quivalenc e, and its c or estriction to the ful l sub- 2-c ate gory of tota l ly algebr aic c o c omplete Q -c ate gories is a bie qui v alenc e: Dist ( Q ) ≃ Co cont ta ( Q ) . W e ma y restrict the b iequiv alence of whic h this theorem sp eaks, to left adjoin ts: we then obtain the biequiv alence (whic h we write with the s ame letter) P : Ma p ( Dist ( Q )) / / Map ( Co cont ta ( Q )) . But the deﬁnition of Cauc hy completeness for Q -categories implies that also Cat cc ( Q ) / / Map ( Dist ( Q )):  F : A / / B  7→  B ( − , F − ): A ❝ / / B  is a biequiv alence, hence comp osing these t w o w e get a third biequiv alence (whic h w e still write with the same lett er): Corollary 3.3 The 2-functor P : Cat cc ( Q ) / / Map ( Co cont ta ( Q )):  F : A / / B  7→  B ( − , F − ) ⊗ − : A ❝ / / B  is a bie quivalenc e. 12 The inv erse biequiv alence is giv en b y “taking tota lly compact ob jects”. More pre- cisely , if F : A / / B is in Map ( Co cont ( Q )) then it maps totally compact ob jects of A to totally compact ob jects of B , hen ce we get a functor F c : A c / / B c out of it. If A and B are moreo ver totally algebraic, then A c and B c are Cauc h y complete. This describ es a 2-fun ctor ( − ) c : Map ( Co cont ta ( Q )) / / Cat cc ( Q ), whic h turns out to b e the sough t-after in v er s e. In fact, the biequ iv alence in Corollary 3.3 can also b e seen as resu lting f rom (co)restricting the follo wing biadjunction to the ob jects for whic h the (co)unit is an equiv alence: Cat ( Q ) ⊥ P ( ( ( − ) c h h Map ( Co cont ( Q )) . This is without im p ortance for what follo ws, so w e shall n ot includ e the details here; but since this m a y b e of indep end en t in terest, w e h a ve written the details in a tec hn ical Add endum at the end of this pap er. Sev eral equiv alen t expressions for th e deﬁn ition of totally compact ob ject are giv en in [Stubb e, 2007a] but for the pu rp oses of this p ap er the follo wing are partic- ularly u s eful 6 : Prop osition 3.4 L et a ∈ A b e an obje ct, of typ e A ∈ Q say, of a c o c omplete Q - c ate gory. The fol lowing c onditions ar e e quivalent: i. a i s tota l ly c omp act, ii. for al l φ ∈ P A : φ ( a ) = A ( a, sup A ( φ )) , iii. the functor H a : A / / P ( ∗ A ): x 7→ A ( a, x ) (“homming with a ”) is c o c ontinuous, iv. the functor T a : P ( ∗ A ) / / A : f 7→ a ⊗ f (“ tensoring with a ”) is c o c ontinuous and admits a c o c ontinuous right adjoint . Pr o of : (i ⇐ ⇒ ii) I t is easily seen (and sp elled out in [Stubb e, 2007a, 5.2 ]) that, f or an y x, y ∈ A , Θ A ( x, y ) = ^ φ ∈P A { A ( y , sup A ( φ )) , φ ( x ) } and h ence, straigh tforw ardly , 1 A ≤ Θ A ( a, a ) ⇐ ⇒ ∀ φ ∈ P A : 1 A ≤ { A ( a, sup A ( φ )) , φ ( a ) } ⇐ ⇒ ∀ φ ∈ P A : A ( a, sup A ( φ )) ≤ φ ( a ) . But b ecause φ ≤ Y A ( sup A ( φ )) is automatic, an d thus φ ( a ) ≤ A ( a, sup A ( φ )) alw a ys holds, a b eing totally compact is indeed equiv alent to the clause in statemen t (ii). 6 Especially cond ition (iii) in Proposition 3.4 is reminiscent of the notion of atom deﬁ ned by M. Bunge [1969] and th at of smal l-pr oje ctive obje ct deﬁn ed by M. Kelly [1982]. 13 (ii ⇐ ⇒ iii) By a straigh tforward calculatio n (e.g . using [Stubb e, 2006, Corollary 2.15]) it is easily seen that, for an y φ ∈ P ( A ), φ ( a ) is the φ -w eigh ted colimit of H a : colim( φ, H a ) = _ x ∈ A H a ( x ) ⊗ φ ( x ) = A ( a, − ) ⊗ φ = φ ( a ) . On the other h and it is clear by d eﬁnition of H a that A ( a, sup A ( φ )) = H a ( sup A ( φ )). Th us the form ula in the second sta temen t of the lemma can b e rewritten as for all φ ∈ P A : colim ( φ, H a ) = H a ( sup A ( φ )) . As foll o w s straigh tforw ard ly from [Stubb e, 2005a, 5.4], this in turn is equiv alen t to H a preserving a l l weigh ted colimits. (iii ⇐ ⇒ iv) Du e to A ’s co completeness, we surely ha v e that all tensors with the ob ject a exist in A ; thus (the dual of ) Prop osition 3.2 in [Stubb e, 2006] sa ys that H a is th e righ t adjoint to T a in Cat ( Q ). (A d irect v eriﬁcation is v ery easy to o.) Because the co con tin uous fu nctors b et ween co complete Q -categorie s are p r ecisely the left adjoint ones in Cat ( Q ) [S tubb e, 200 5a, Prop osition 6.8], the result follo ws directly . ✷  . Princip all y ge nera ted modules A Q -mo d u le F : Q op / / Sup d etermines a co complete Q -catego ry A F whose set of ob jects is ` X F X , with t yp es given by tx = X if and only if x ∈ F X , and hom- arro ws A F ( y , x ) = _ { f : tx / / ty | F ( f )( y ) ≤ x } . (3) Similarly , a mo dule morphism α : F + 3 G determines a cocontin uous functor F α : A F / / A G : x 7→ α tx ( x ) . This sets u p the biequiv alence of Mo d ( Q ) with Co cont ( Q ), as studied in great detail in [Stubb e, 2006 , Section 4] based on w ork by R. Gordon and J. P o wer [1997]. W e wish to charac terise, purely in terms of Q -mo du les, those F ∈ Mo d ( Q ) for which the corresp onding A F ∈ Co cont ( Q ) is totally algebraic. In order to do so, w e m ust in tro duce some notations. Let F : Q op / / Sup b e a Q -mo dule, an d supp ose that a ∈ F ( A ) for some A ∈ Q . W e sh all wr ite Q ( − , A ) τ a + 3 F for the Sup -natural transform ation th at a ∈ F ( A ) corresp onds w ith by the S up - enric hed Y oneda Lemma. Of ten w e shall lo osely sp eak of “an elemen t a of F ”, a nd ev en write “ a ∈ F ”, where actually we should b e m ore precise and stipulate that a ∈ F ( A ) for some A ∈ Q . 14 Because M o d ( Q ) is a (large) quan taloid, we can compute extensions and liftings. It is straigh tforward to v erify , with th e aid of the Sup -enr ic hed Y oned a Lemma, that for a Q -mod ule F : Q op / / Sup and x ∈ F ( X ) and y ∈ F ( Y ), the righ t lifting of τ x through τ y , Q ( − , X ) τ x % - R R R R R R R R R R R R R R R R R R R R [ τ y , τ x ]   F Q ( − , Y ) τ y 1 9 l l l l l l l l l l l l l l l l l l l l ≤ (4) is pr ecisely represented b y the Q -morphism A F ( y , x ) in (3). Of course it mak es sense to sp eak of adjoin ts in M od ( Q ); if a Q -mo dule morp h ism α : F + 3 G has a righ t adjoint 7 , w e shall usually d enote it as α ∗ : G + 3 F . Prop osition 4.1 L et F : Q op / / Sup b e a Q -mo dule, and A F the asso c iate d c o c om- plete Q -c ategory. We have the fol lowing: i. a ∈ A F is total ly c omp act if and only if a ∈ F and τ a is a left adjoint. ii. A F is total ly algebr aic if and only if for e ach x ∈ F , τ x = _ n τ a ◦ [ τ a , τ x ]    a ∈ F and τ a is a left adjoint o . (5) Pr o of : (i) A repr esen table m o d ule Q ( − , A ): Q op / / Sup corresp onds under the biequiv alence Mo d ( Q ) ≃ Co cont ( Q ) with P ( ∗ A ), and a Sup -natural transform a- tion τ a : Q ( − , A ) + 3 F corresp ond s to the “tensoring with a ” co con tinuous fun ctor T a : P ( ∗ A ) / / A F : f 7→ a ⊗ f = F ( f )( a ). Thus the fourth statemen t in Prop osition 3.4 pr o ve s the claim made here. (ii) With suitable application of the S up -enric hed Y oneda Lemma, it is easily deduced from (i) that th e form ula in (5), whic h is s tated in terms of the Q -mo d ule F , sa ys precisely the same thing as for all x ∈ A F , x = _ { a ⊗ A F ( a, x ) | a ∈ ( A F ) c } , whic h is s tated in terms of th e co complete Q -category A F . But the righ t hand side in this la tter f orm u la is the explicit w a y of writing the v alue in x of the (p oint wise) left K an extension of i A F : ( A F ) c / / A F along itself (see e.g. [Stu bb e, 2005a, p. 26] com b in ed with [Stub b e, 2006, Corollary 2.15]): hence it sa ys that this left Kan extension is the identit y functor o n A F . ✷ The p receding result promps the follo wing deﬁnition. 7 Whether α has a right adjoin t in Mo d ( Q ) or not, each of its components certainly h as a right adjoin t in O rd , sa y α ′ X : G ( X ) / / F ( X ), and these alwa y s fo rm a lax n atural Ord -transfo rmation α ′ : G + 3 F . If α has a righ t a djoint in Mo d ( Q ), then – fo r reasons of un icit y of adjoints in Ord – it m ust b e α ′ . In other w ords, α has a right adjoin t in Mod ( Q ) if and only if the lax n atural Ord -transformation α ′ is strictly natu ral and its comp onen ts preserve suprema. 15 Deﬁnition 4.2 L et F : Q op / / Sup b e a Q -mo dule. An a ∈ F is a principal elemen t if τ a is a left adjoint in Mo d ( Q ) . Writing the set of princip al elements of F as F p r , F is prin cipally generated if for e ach x ∈ F , τ x = _ n τ a ◦ [ τ a , τ x ]    a ∈ F p r o . (6) W e can add th e follo wing succinct characte risation. Prop osition 4.3 L et F : Q op / / Sup b e a Q -mo dule. The set of princip al elements of F is F p r = n τ (1 X ) | X ∈ Q and τ : Q ( − , X ) + 3 F is a left adjoint o , and F is princip al ly gener ate d if and onl y if id F = _ n τ ◦ τ ∗    X ∈ Q and τ : Q ( − , X ) + 3 F is a left adjo int o . (7) Pr o of : Th e ﬁrst p art of the Pr op osition is an application of th e Sup -enriched Y oneda Lemma. As for th e second part, for an y x ∈ F w e can compute, by general calculation rules f or liftings in a quan taloid 8 , that  _ n τ a ◦ τ ∗ a    a ∈ F p r o ◦ τ x = _ n τ a ◦ τ ∗ a ◦ τ x    a ∈ F p r o = _ n τ a ◦ [ τ a , τ x ]    a ∈ F p r o . If we assume (7), i.e. if the brac k eted expression on the far left equ als id F , then clearly (6) follo ws. Con v ersely , assuming (6), i.e. assuming that the far r ight expression equals τ x = id F ◦ τ x , a nd this for ev ery x ∈ F , implies – b ecause th e rep r esen tables are generators in Mo d ( Q ) – that th e brac kete d expression on the f ar left m u st b e equal to id F . ✷ F rom this we can no w deduce an elegan t c haracterisatio n of p rincipally generated Q - mo dules, entirely in terms of un iv ers al constructions 9 in the co complete quant aloid Mo d ( Q ) (we thank S. Lac k for a stim ulating discussion on t his to pic). Corollary 4.4 An F ∈ Mo d ( Q ) is princip al ly gener ate d i f and only if i t is the adjoint r etr act of a dir e ct sum of r epr esentable Q -mo dules. Pr o of : This h olds b y application of Lemma 4.5 b elo w to th e family of all left adjoin t Q -mo dule morp hisms from represen table Q -mod ules to F ; compare with (6). ✷ 8 In any quantaloid, if f : A / / B is left adjoint to f ∗ : B / / A , then for any g : C / / B we hav e [ f , g ] = f ∗ ◦ g . 9 In any q uantal oid, pro ducts coincide with copro ducts, and are often called dir e ct sums . An ob ject Y is an adjoi nt r etr act of an ob ject X when there exists a left adjoint p : X / / Y whose right adjoin t s : Y / / X is its splitting ( p ◦ s = 1 Y ). 16 Lemma 4.5 L et F b e an obje ct in a quantaloid with dir e ct sums and c onsider a family of left adjoints with c o domain F : n F ⊥ f ∗ i 6 6 f i v v R i o i ∈ I . Writing f : F / / ⊕ i R i and f ∗ : ⊕ i R i / / F for the unique f actorisa tions of the f i ’s and the f ∗ i ’s we have that f ◦ f ∗ = W i ( f i ◦ f ∗ i ) and f ∗ ◦ f ≥ 1 ⊕ i R i . Henc e 1 F = W i ( f i ◦ f ∗ i ) if and only if f and f ∗ expr ess F as an adjoint r etr act of ⊕ i R i . Pr o of : Recall that, in any quantal oid, an ob ject ⊕ i R i is the direct sum of a family of ob jects ( R i ) i ∈ I if and only if there are morphisms s j : R j / / ⊕ i R i and p j : ⊕ i R i / / R j , for all j ∈ I , suc h that W i ( s i ◦ p i ) = 1 ⊕ i R i and p j ◦ s i = δ i,j (where δ i,j : X i / / X j is zero when i 6 = j and the identit y otherwise). It follo ws that also p j ◦ f ∗ = f ∗ j and f ◦ s j = f j hold. T o pro ve the equalit y , w e compute that f ◦ f ∗ = f ◦ ( _ i ( s i ◦ p i )) ◦ f ∗ = _ i (( f ◦ s i ) ◦ ( p i ◦ f ∗ )) = _ i ( f i ◦ f ∗ i ) . The in equalit y is (b ecause w e are dealing with a 2-cat egorical (co)pro duct) equiv- alen t to requiring p j ◦ 1 ⊕ i R i ◦ s k ≤ p j ◦ f ∗ ◦ f ◦ s k for all j, k ∈ I . But the left hand side equals δ j,k and the r igh t h and side equals f ∗ j ◦ f k , and sin ce it is true that 1 R j ≤ f ∗ j ◦ f j , we are done. ✷ W e sh all write Mo d pg ( Q ) for th e full sub quan taloid of Mo d ( Q ) determined by the pr incipally generated Q -modu les, and th us obtain: Theorem 4.6 F or any sma l l quanta loid Q , the bie quivalenc e Mo d ( Q ) ≃ Co cont ( Q ) r estricts to a bie quiv alenc e Mod pg ( Q ) ≃ Co cont ta ( Q ) . Com bined w ith the ea rlier observ ation in Corollary 3.3 w e get: Corollary 4.7 F or any smal l qu antalo id Q , the lo c al ly or der e d c ate gories Ca t cc ( Q ) and Map ( Mo d pg ( Q )) ar e bie q uivalent. Although slight ly oﬀ-topic, w e ﬁ nd it imp ortan t to remark that Corollary 4.4 implies that the principally generated mo dules on Q form a close d class of c olimit weights in the sense of [Alb ert and Kelly , 1988; Kelly and Sc hmitt, 2005]; in fact, this class is the closur e of the class of (weigh ts for) d ir ect su ms and adjoint retracts. Th e general theory explained for V -enric hed catego ries in the cited references implies that, for an y small quant aloid Q , Mo d pg ( Q ) is precisely the fr ee co completion of Q for direct sum s and adjoin t retracts, or equiv alently , the f r ee co completion of Q for all colimits weig hed b y a principally generated mo dule. Since we know that Dist ( Q ) ≃ Co cont ta ( Q ) ≃ Mo d pg ( Q ), this at once d escrib es the un iv ers al pr op ert y of the distr ibutor quantal oid to o. In [S tubb e, 2005a] it is s ho wn that Dist ( Q ) is 17 the universal “ direct su m and split monad” completion of Q ; but it is trivial that, in a quanta loid, splitting monads are the same thing as adjoint retracts. In the latter reference it is m oreo v er sho w n that direct sums and splitting monads suﬃce to admit all lax limit s and all lax colimits. Com bining all this, it th us follo ws th at the pr incipally generated mo dules, as a class of w eigh ts, describ e precisely the lax (co)co mpletion of Q .  . Locall y princip all y gene ra ted modules It is w ell-kno wn that id emp oten ts in S up split: for an idemp otent e : L / / L w e let L e s e / / L p e o o denote the obvious splitting w ith L e := { x ∈ L | e ( x ) = x } ; of cours e any other splitting of e is isomorphic to this one. Giv en a quan taloid Q , let Q si denote its split-idemp oten t completio n; note that Q si is small b ecause Q is. W riting j : Q / / Q si for the ob vious in clusion, − ◦ j : M o d ( Q si ) / / Mo d ( Q ) is an equiv alence of quantalo ids. W e w ish to describ e the full s u b category of Mo d ( Q ) that is equiv alen t to Mo d pg ( Q si ) under the action of − ◦ j . Th ereto we sh all ﬁ rst ﬁx some n otatio ns. W e sh all wr ite ( − ) si : Mo d ( Q ) / / Mo d ( Q si ) for the in v erse equiv alence to − ◦ j : it sends a Q -mo du le F to the Q si -mo dule F si deﬁned (up to isomorph ism) by: - f or an o b ject e : A / / A of Q si , F si ( e ) := F ( A ) F ( e ) , - f or a morphism f : e / / e ′ in Q si , F si ( f ) := F ( f ). If e : A / / A is an idemp oten t in the quanta loid Q , then the repr esen table Sup - natural transform ation Q ( − , e ): Q ( − , A ) + 3 Q ( − , A ) is idemp oten t to o. All idem- p oten ts in Mo d ( Q ) split, so this one do es too: we shall write F e σ e + 3 Q ( − , A ) π e k s for the ob vious sp litting o ve r F e := Q si ( − , e ) ◦ j = Q si ( j − , e ), and w e refer to suc h a Q -mo du le F e as th e ﬁxp oint Q - mo dule for e : A / / A . Prop osition 5.1 L et F : Q op / / Sup b e a Q - mo dule, and F si the asso ciate d Q si - mo dule. 18 i. Given an idemp otent e : A / / A in Q , a ∈ F si ( e ) is a princip al element of the Q si -mo dule F si if and only if a ∈ F ( A ) satisﬁes F ( e )( a ) = a and the Q -mo dule morphism τ a ◦ σ e : F e + 3 F is a left adjoint. ii. F si is a princip al ly gener ate d Q si -mo dule if and only if for e ach x ∈ F , τ x = _      τ a ◦ [ τ a , τ x ]    ther e exists an idemp otent e in Q such that F ( e )( a ) = a and τ a ◦ σ e : F e + 3 F is a left adjo int      . (8) Pr o of : (i) Because we pu t F si ( e ) = F ( A ) F ( e ) , it is trivial that a ∈ F si ( e ) if and only if a ∈ F ( A ) satisfying F ( e )( a ) = a . By the Sup -enric hed Y oneda Lemma, we know that a ∈ F ( A ) corresp onds un iquely to a Q -mo dule morphism, which we called τ a : Q ( − , A ) + 3 F ; but also a ∈ F si ( e ) corresp onds uniqu ely to a Q si -mo dule mor- phism: let u s ca ll it ρ a : Q si ( − , e ) + 3 F si . By Deﬁnition 4.2, a ∈ F si ( e ) is a prin cipal elemen t if ρ a is a left adjoin t (in Mo d ( Q si ), th at is). Because of the equiv alence of Mo d ( Q ) and Mo d ( Q si ), expr essed by ( − ) si and − ◦ j , and b ecause F e = Q si ( − , e ) ◦ j , ρ a corresp onds u niquely to a left adjoint Q -mo dule morp hism ζ a : F e + 3 F . But b e- cause F e is the Q -mo du le o ver which the idemp otent Q ( − , e ): Q ( − , A ) + 3 Q ( − , A ) splits (and recall that we write the splitting with inclusion σ e and p ro jection π e ) we necessarily ha ve th at ζ a = τ a ◦ σ e (and τ a = ζ a ◦ π e ). Thus it is indeed suﬃcient and necessary that τ a ◦ σ e b e a left adjoin t Q -mod ule morph ism . (ii) Again follo w ing Deﬁnition 4.2, and with the notations that we int ro duced in the ﬁr st part o f this pro of, F si is a principally generated Q si -mo dule if for eac h x ∈ F si , ρ x = _ n ρ a ◦ [ ρ a , ρ x ]    a ∈ ( F si ) p r o . This supremum of Q si -mo dule morphism s can b e written in terms of Q -mo dule morphisms, f or similar reasons as in the ﬁr st part of the p ro of: for eac h x ∈ F si , ζ x = _ n ζ a ◦ [ ζ a , ζ x ]    a ∈ ( F si ) p r o . Using the notation α := ( ( a, e )    e : A / / A a n id emp oten t in Q and a ∈ F such that F ( e )( a ) = a and τ a ◦ σ e is left adjoin t ) w e can sp ell this out as: for eac h idemp oten t d : X / / X in Q and eac h x ∈ F s uc h that F ( d )( x ) = x , τ x ◦ σ d = W n ( τ a ◦ σ e ) ◦ [( τ a ◦ σ e ) , ( τ x ◦ σ d )]    ( a, e ) ∈ α o . (9) 19 Assume n o w that (9) holds. T aking in particular d = 1 A (in wh ic h case σ d = σ 1 A is the iden tit y transformation) it implies th at for all x ∈ F τ x = _ n ( τ a ◦ σ e ) ◦ [( τ a ◦ σ e ) , τ x ]    ( a, e ) ∈ α o = _ n ( τ a ◦ σ e ) ◦ π e ◦ [ τ a , τ x ]    ( a, e ) ∈ α o = _ n τ a ◦ [ τ a , τ x ]    ( a, e ) ∈ α o = _ n τ a ◦ [ τ a , τ x ]    a ∈ β o b y a suitable application of Lemma 5.2 stated b elo w (and the pro of of whic h is straigh tforward) to p ass from the ﬁrs t to the second line, by the fact that τ a ◦ σ e ◦ π e = τ a ◦ Q ( − , e ) = τ a (whic h is the equiv alent of F ( e )( a ) = a ) to p ass f rom the second to the third line, and where we in tro duced another auxiliary nota tion in the la st line: β := n a    there exists an idemp otent e : A / / A in Q s uc h that ( a, e ) ∈ α o . A priori an a ∈ F ma y b e lo cally principal at sev eral diﬀerent idemp oten ts, in wh ich case α co n tains seve ral “copies” of a (one for eac h idemp oten t it is lo cally pr incipal at) but β con tains a only once. But b ecause an expr ession lik e τ a ◦ [ τ a , τ x ] do es not con tain an y reference to the idemp oten ts at whic h a is locally principal, w e can mak e the last step in th e ab o v e series of equalities. Hence we derived the condition expressed in (8). Con v ersely , assume th e v alidit y of (8). Then, for every idemp otent d : X / / X in Q and x ∈ F su c h that F ( d )( x ) = x , w e c an compute in a similar f ash ion that τ x ◦ σ d =  _ n τ a ◦ [ τ a , τ x ]    a ∈ β o ◦ σ d = _ n τ a ◦ [ τ a , τ x ] ◦ σ d    a ∈ β o = _ n τ a ◦ ( σ e ◦ π e ) ◦ [ τ a , τ x ] ◦ σ d    ( a, e ) ∈ α o = _ n ( τ a ◦ σ e ) ◦ [( τ a ◦ σ e ) , ( τ x ◦ σ d )]    ( a, e ) ∈ α o . Th us we obtain the condition exp ressed in (9). ✷ Lemma 5.2 In any quantalo id, for a diagr am like X p ◦ i = 1 X 7 7   i   Y   j   1 Y = q ◦ j f f E i ◦ p = e 2 = e 8 8 p O O O O a ◦ e = a   ; ; ; ; ; ; ; ; ; ; ; ; F f = f 2 = j ◦ q f f q O O O O b = b ◦ f               A 20 we have q ◦ [ b, a ] ◦ i = [ b ◦ j, a ◦ i ] . The resu lt in Prop osition 5.1 su ggests a n ew deﬁnition, to b e compared with Deﬁnition 4 .2: Deﬁnition 5.3 L et F : Q op / / Sup b e a Q -mo dule. An a ∈ F is a lo cally principal elemen t (at an idemp oten t e : A / / A in Q ) if (ther e is an idemp otent e : A / / A in Q such th at) F ( e )( a ) = a an d τ a ◦ σ e is a left adjoint in Mod ( Q ) . Writing F lp r for the set o f lo c al ly princip al elements of F , we say that F is locally pr incipally generated if for e ach x ∈ F , τ x = _ n τ a ◦ [ τ a , τ x ]    a ∈ F lp r o . Th us, a lo cally p rincipal elemen t of F : Q op / / Sup at an identity of Q is the same thing as, simply , a pr in cipal element of F : idemp otents in Q are v iewed as lo c alities (or “op ens” ). It follo ws f rom the comparison of Deﬁn itions 4.2 and 5.3 that a principally generated Q -mo dule is necessarily also lo cally pr incipally generated; but the conv erse is n ot true in general, as the next example sh o ws. Example 5.4 A ﬁxp oin t Q -mod ule F e : Q op / / Sup for an idemp oten t e : A / / A in Q trivially has e ∈ F e ( A ) as lo cally prin cipal elemen t (at e , as a matter of fact); it follo ws straigh tforw ardly that F e is lo cally principally generated. Ho w ev er, F e need not ha v e any principal elemen t and th us n eed not b e principally generated; f or a concrete example, let Q b e the one-ob ject susp ension of the three-elemen t chain { 0 ≤ e ≤ 1 } . W e ha ve the follo wing useful c h aracterisati on of lo cally principally generated Q -mo dules. Prop osition 5.5 L et F : Q op / / Sup b e a Q -mo dule. The set of lo c al ly princip al elements is F lp r = ( ζ ( e )    e is an idemp otent in Q and ζ : F e + 3 F is a left adjoint ) , and F is lo c al ly princip al ly gener ate d if and only if id F = _ ( ζ ◦ ζ ∗    e is an idemp otent in Q and ζ : F e + 3 F is a left adjoint ) . (10) Pr o of : T h is follo ws straight forwa rdly fr om Prop osition 4.1 and (the pro of of ) P rop o- sition 5.1. ✷ Muc h lik e Corollary 4.4 d o es f or principally generated Q -mo du les, we can n o w giv e a c h aracterisat ion in terms of u n iv ers al constructions in Mo d ( Q ) of lo cally principally generated Q -mo du les. Corollary 5.6 An F ∈ Mod ( Q ) i s lo c al ly princip al ly g ener ate d if and only if it is an adjoint r etr act of a dir e ct su m of ﬁxp oint Q -mo dules. 21 Pr o of : Apply Lemma 4.5 to the family of left adjoin t Q -mo dule morphisms fr om all ﬁx p oin t Q -mod u les to F ; compare with (10). ✷ Remark 5.7 A Q -mo du le F : Q op / / Sup is pr oje ctive when the represen table Sup - functor M od ( Q )( F , − ): Mo d ( Q ) / / Sup p reserv es epimorph isms. It is kn o wn (see e.g. [Stubb e, 2007a, Pr op osition 9.5]) that this is equiv alen t to F b eing smal l- pr oje ctive (in the sense of [Kelly , 1982]: Mo d ( Q )( F , − ) preserv es al l small weig h ted colimits), and equiv alent to F b eing a retract of a direct sum of representa ble Q -mo dules. It th us follo ws from Corollarie s 4.4 a nd 5.6 that an y (lo cally) principally generated Q -mo dule is necessarily pro jectiv e in Mo d ( Q ). But the diﬀerence b et we en “pro- jectiv e” and “(lo cally) prin cipally generate d” lies precisely in the strictly stronger requirement that, for the latter to hold, F needs to b e an adjoint retract of a direct sum of represen table Q -mo du les (ﬁxp oint Q -mod ules). Let M o d lpg ( Q ) denote the full sub qu an taloid of Mod ( Q ) whose ob jects are the lo cally-principally g enerated Q -modules. Theorem 5.8 F or a smal l quantaloid Q , the bie qui v alenc e Mod ( Q si ) ≃ M od ( Q ) r estricts to a bie quiv alenc e Mod pg ( Q si ) ≃ Mo d lpg ( Q ) . The biequiv alences in Corollary 3.3 and Theorem 4.6 app ly to any small quan taloid, hence in particular to the sp lit-idemp oten t completion Q si of a small qu an taloid Q . The com b in ation w ith the biequ iv alence in Theorem 5.8 th en sho ws that the follo wing diagram commutes: Co cont ( Q si ) ∼ / / Mo d ( Q si ) ∼ / / Mo d ( Q ) Cat cc ( Q si ) ∼ / / P 8 8 r r r r r r r r r r r r r r r r Map ( Co cont ta ( Q si )) ∼ / / ?  O O Map ( Mo d pg ( Q si )) ∼ / / ?  O O Map ( Mo d lpg ( Q )) ?  O O Up to the id en tiﬁcation of Ord ( Q ), the lo cally ord ered category of ordered sheav es on a small quan taloid Q , with the lo cally ordered c ategory Cat cc ( Q si ), the comp osition of the 2-functors Ord ( Q ) = Cat cc ( Q si ) P / / Co cont ( Q si ) ∼ / / Mo d ( Q si ) ∼ / / Mo d ( Q ) is p recisely th e left biadjoin t 2-functor F : Ord ( Q ) / / Mo d ( Q ) in (1) (see also [Stubb e, 2007b, 3.3]). Hence we pr o ved that this 2-fun ctor factors as a (comp osition of ) biequiv alence(s) follo w ed b y an inclusion. Theorem 5.9 The lo c al ly or der e d c ate gory Ord ( Q ) of or der e d she aves on a smal l quantaloid Q is bie quivalent to Map ( Mo d lpg ( Q )) , the lo c al ly or der e d c ate gory of lo- c al ly princip al ly gener ate d Q -mo dules a nd lef t a djoint Q - mo dule morph isms b etwe en them. 22  . Examples In the three examples that f ollo w w e shall consider a one-ob ject quantalo id Q . In th is situation w e prefer to view Q as monoid Q := ( Q ( ∗ , ∗ ) , ◦ , 1) in Sup , and a Q -mo dule M : Q op / / Sup as the ob ject M := M ( ∗ ) o f Sup together with an ac tion 10 M × Q / / M : ( x, q ) 7→ x ◦ q := M ( q )( x ) . With sligh t abuse of notation we sh all w r ite [ x, y ] for th e elemen t of Q that represents the lifti ng [ τ x , τ y ] in Mo d ( Q ); that is to sa y , [ x, y ] = W { q ∈ Q | x ◦ q ≤ y } (compare with (3) and (4)). An elemen t a ∈ M is principal if and only if [ a, − ]: M / / Q is a Sup -morphism that p reserv es th e action of Q . No w sup p ose moreo v er th at the underlying complete lattice of Q is totally al- gebraic (in the classical sense, as recalled in the Int ro duction). Principalit y of an elemen t a ∈ M is then equiv alent to the follo wing t w o requiremen ts: i. [ a, x ◦ c ] ≤ [ a, x ] ◦ c , for all x ∈ M and all totall y compact c ∈ Q , ii. [ a, W i ∈ I x i ] ≤ W i ∈ I [ a, x i ], f or all ( x i ) i ∈ I ∈ M . This is equiv alen t to asking, for all x, ( x i ) i ∈ I ∈ M and tota lly compact c, d ∈ Q , i. if a ◦ d ≤ x ◦ c then there exists a tota lly compact k ∈ Q such th at a ◦ k ≤ x and d ≤ k ◦ c , ii. if a ◦ d ≤ W i ∈ I x i then ther e exists an i ∈ I suc h that a ◦ d ≤ x i . In particular, for a p rincipal elemen t a ∈ M and a totally compact elemen t d ∈ Q , the element a ◦ d ∈ M is totally compact in (the underlyin g complete lattice of ) M . Let Q c , resp. M c , denote the partially ord erd sets of totally compact elements of the underlying complete lattices of Q , resp. M . F or an elemen t x of a Q -mo du le M w e ha v e x = _ n a ◦ [ a, x ]    a ∈ M p r o ⇐ ⇒ x = _ n a ◦ q    a ∈ M p r , q ∈ Q , a ◦ q ≤ x o ⇐ ⇒ x = _ n a ◦ d    a ∈ M p r , d ∈ Q c , a ◦ d ≤ x o = ⇒ x = _ n b    b ∈ M c , b ≤ x o . Hence, eve ry principally generate d mo dule on a totally algebraic quan tale has a totally algebraic underlying sup-lattic e. W e sh all now sp ell out three quite diﬀerent app licatio ns. 10 As mentioned earlier, Q - modules are essentially “the same thing” as co complete Q - categories ; actions of Q correspond with tensors in Q -categories. In previous w ork [Stubb e, 2006] we therefore denoted actions with “ ⊗ ”, the usual symb ol for tensors in enriched categories. How ever to a void any confusion with pure tensors in a tensor pro duct of sup-lattices, we here adopt a “ ◦ ” as notation. 23 Example 6.1 (C omplete latt ices.) The quantal oid Sup ca n b e iden tiﬁed with Mo d ( 2 ), where 2 stands for the tw o-elemen t c hain with its obvi ous quan tale struc- ture, ( { 0 ≤ 1 } , ∧ , 1). F or a complete latt ice/ 2 -modu le S , the conditions ab o v e sa y that: i. a 6 = 0 S , ii. a is totally compact. Hence, the principal elemen ts of S are the non-zero totally compact elemen ts, while 0 S is not principal but (the u nique elemen t th at is) lo c al ly principal at 0. Thus S is a tota lly algebraic complete lattice if and only if it is a p rincipally generated 2 -mo dule, if and only if it is a lo cally principally generated 2 -mo dule. Example 6.2 (Aut omata.) Let ( N , · , 1) b e a monoid (in Set ), then the p ow erset of N can b e equip ed with the point wise m ultiplication A · B := { a · b | a ∈ A, b ∈ B } for A, B ⊆ N and thus (2 N , · , { 1 } ) is a quan tale (it is th e free quantal e on ( N , · , 1), see [Rosenthal, 1990]) . The totally compact elemen ts of 2 N are the s in gletons and the empty su bset. An element a ∈ M of a 2 N -mo dule M is pr incipal if and only if, for all x , ( x i ) i ∈ I ∈ M and n , p ∈ N , ia. a ◦ n 6 = 0 M , ib. if a ◦ n ≤ x ◦ p then there exist s a k ∈ N such that a ◦ k ≤ x and n = k ◦ p , ii. if a ◦ n ≤ W i ∈ I x i then ther e exists an i ∈ I suc h that a ◦ n ≤ x i . (W e wrote a ◦ n instead of a ◦ { n } for notational con venience.) Note that (ia) + (ii) are equiv alent to a ◦ n b eing principal in the underlying complete lattice of the mo dule M . This example can b e interesting for the theory of automata (or lab elled tr ansition systems): b y Corollary 4.7, the principally generate d 2 N -mo dules can b e iden tiﬁed with Cauc hy complete 2 N -enric h ed categorie s. It is well- kno wn th at cate gories en- ric h ed in a free quantale a re precisely n on-deterministic automata with N as s et o f lab els [Betti, 1980 ; Rosen thal, 1990]. Example 6.3 ( R el ( S, S ) -mo dules.) L et Rel d enote the quan taloid of sets and re- lations, th en s u rely f or an y set S , Q S := Rel ( S, S ) is a totally algebraic qu an tale: its totally compact elements are the empty set and th e sin gletons ( s, t ) ∈ S × S (w e omit the curly brac k ets for cla rit y). A Q S -mo dule M is “the same thing” as the sk eletal (i.e. ha ving no non-identical isomorp hic ob jects) co complete Q S -categ ory A M (as explained in the b eginning of S ectio n 4). On the other hand , to giv e a Q S -categ ory 24 A , with ob ject set A 0 , is equiv alen t to giving an order relat ion 4 on the set A 0 × S ; the corresp ond ence is giv en b y: ( a, s ) 4 ( b, t ) if and only if ( s, t ) ∈ A ( a, b ) . W riting the equiv alence relation on A 0 × S induced b y the ord er 4 as ≈ , it can b e v eriﬁed that A is sk eletal and co complete if and only if 4 satisﬁes: i. f or all s ∈ S , ( a, s ) ≈ ( b, s ) implies a = b , ii. for all ( a i ) i ∈ I ∈ A 0 there exist s a W i ∈ I a i ∈ A 0 suc h that ( W i ∈ I a i , s ) 4 ( b, t ) if and only if ( a i , s ) 4 ( b, t ) for all i ∈ I , iii. for all a ∈ A 0 and s, t ∈ S there exists b ∈ A 0 suc h that ( b, t ) ≈ ( a, s ) and moreo ver, wh enev er u 6 = t , ( b, u ) is a b ottom element for the ord er ( A 0 × S, 4 ). Conditions (ii ) and (iii) are easily d educed from the equiv alence of cocomplete Q - catego ries to conically co complete and tensored Q -cate gories [Kelly , 1982; Stubb e, 2006]: W i ∈ I a i in (ii) is the conica l colimit of ( a i ) i ∈ I in A (and thus its order th e- oretical join in ( A 0 , ≤ )), w h ile b in (iii) is the tensor pro duct a ◦ ( s, t ). Moreo ve r, these cond itions imply ii’. F or all ( b i ) i ∈ I ∈ A 0 there exist s a V i ∈ I b i ∈ A 0 suc h that ( a, s ) 4 ( V i ∈ I b i , t ) if and only if ( a, s ) 4 ( b i , t ) for all i ∈ I . Hence, a Q S -mo dule M can b e give n in terms of an order 4 on M × S , satifying conditions (i–iii). An element a ∈ M is then principal if and only if, f or all s, t ∈ S , i. ( a, s ) 6≈ ⊥ , where ⊥ d enotes a b ottom elemen t for ( M × S , 4 ), ii. if ( a, s ) 4 ( W i ∈ I x i , t ) then there exists an i ∈ I such that ( a, s ) 4 ( x i , t ). The order 4 on M × S , corresp onding to a Q S -mo dule M h as the charact eristics of an en tailmen t (esp ecial ly (ii) and (ii’) ab o v e): a couple ( a, s ) ∈ M × S can b e though t of as an o ccurrence of an even t a ∈ M at a p lace s ∈ S . Quantale s of th e form R el ( S, S ) arise in the con text of relat ional represen tations of spatial quan tales Q , i.e. quantal e homomorphisms ρ : Q / / End (2 S ) = Rel ( S, S ) [Mulv ey and Resend e, 2005]. In the next section we shall dw ell on the case w here Q is the one-ob ject susp en- sion of a lo cale X . The formulat ion of ordered sheav es on X b y means of lo cally principally generated X -mo dules allo ws for a neat translation to “sk ew lo cal home- omorphisms” in to X . 25  . Skew local homeomorphisms Induced modules on a locale X . In w hat follo w s, Loc denotes the (2- )catego ry of locales. W e follo w the notat ional con v en tion of [Johnstone, 1982, p. 40] for mor- phisms in Loc : th us a lo cale morphism f : Y / / X is an adjoint pair Y ⊥ f ∗ 4 4 X f ∗ t t in the 2-category of p artially ord er ed s ets such that the left adjoin t p r eserv es ﬁ- nite inﬁma. W e do not follo w the conv en tion of [Johnstone, 1982; Mac Lane and Mo erdijk, 1992 ] when it comes to d eﬁ ning an order on the hom-sets in Lo c : for f , g : Y / / / / X in Lo c we deﬁn e that f ≤ g if f ∗ ≤ g ∗ . T hat is to say: w e ha ve that Lo c ∼ = F rm coop as 2-categories (whereas th e cite d references h a ve Lo c ∼ = F rm op ) 11 . Considering a lo cale X as a monoid ( X , ∧ , ⊤ X ) in Sup it mak es sense to write Mo d ( X ) for the quantalo id of mo dules on the lo cale. Instead o f w riting these mo d - ules as con trav ariant S up -enric h ed presh ea v es on the one-ob ject susp ension of the lo cale, w e rather consider them as ob jects of Sup on whic h ( X, ∧ , ⊤ X ) acts on the righ t: w e write ( M , ◦ ) for a Sup -ob j ect M together with the action ( m, x ) 7→ m ◦ x . In the same vein, an X -mo d ule morphism α : ( M , ◦ ) / / ( N , ◦ ) is a Sup -morph ism α : M / / N w hic h is equ iv arian t for the resp ectiv e actions. Giv en an f : Y / / X in Loc , it is easily seen that putting y ◦ f x := y ∧ f ∗ ( x ) (11) for y ∈ Y and x ∈ X resu lts in an action of the monoid ( X, ∧ , ⊤ X ) on Y in Sup . In other w ords, from f : Y / / X in Loc we get an ob ject ( Y , ◦ f ) ∈ Mo d ( X ). Moreo ver, supp ose that Y h / / f A A A A A A A Z g ~ ~ ~ ~ ~ ~ ~ ~ ~ X (12) is a comm u tativ e triangle in Loc , th en h ∗ : Z / / Y is a m orphism in Sup satisfying h ∗ ( z ◦ f x ) = h ∗ ( z ) ◦ g x , for all x ∈ X , z ∈ Z . Th at is to sa y , h ∗ : ( Z, ◦ g ) / / ( Y , ◦ f ) is 11 The reason for our preference is in th e ﬁrst place notational conv en ience, esp ecially in the 2-functors considered furt h er on. How ev er, there is maybe a deep er reason why this d iﬀeren t ordering of locale morphisms is natural here: In the cited references lo cale morphisms are studied as inducing geometric morphisms b etw een top oses of shea ves; th e ordering of lo cale morph isms is chos en to correspond with th e usu al notion of natural transformation b etw een geometric morphisms. W e ho wev er shall study locale morphisms (or rather, morph isms in the slice category Lo c / X ) as inducing order-preserving morphisms b etw een the (ordered) sh ea v es themselves; and the ordering of the locale morphisms is chosen to correspond with the n atural ordering of those morphisms b et w een sheav es. 26 a m orp hism in Mo d ( X ). All this adds up to an in jectiv e and faithful (but n ot full) 2-functor ( Lo c / X ) coop / / Mo d ( X ) . (13) W e are no w int erested in left ad j oin t X -mo dule morphisms: Deﬁnition 7.1 A morphism h : f / / g in Lo c / X as in (12) is skew op en if the c orr e - sp onding or der-pr e se rvi ng function h ∗ : Z / / Y has a left adjoint h ! : Y / / Z satisfying the “b alanc e d F r ob eniu s identity 12 ”: for al l y ∈ Y and x ∈ X , h ! ( y ∧ f ∗ ( x )) = h ! ( y ) ∧ g ∗ ( x ) . (14) Example 7.2 F or an h : Y / / Z in Loc the follo win g are equiv alen t: i. h : Y / / Z is op en in Lo c (according to th e “usual” d eﬁnition of op en ness as in e.g. [M ac Lane and Mo erd ij k , 199 2, p. 500]), ii. for an y f : Y / / X and g : Z / / X in Lo c such that g ◦ h = f , the morp h ism h : f / / g in Loc / X is skew op en, iii. considerin g h : Y / / Z and 1 Z : Z / / Z as ob jects in Lo c / Z , the (unique) mor- phism h : h / / 1 Z in Lo c / Z is skew op en. Clearly the identit y morphisms in Lo c / X are sk ew op en, and th e comp osition of sk ew op en morp h isms is again ske w op en ; it thus m ak es sense to s p eak of the sub-2- catego ry ( Lo c / X ) o of Lo c / X with the same ob jects but only its skew op en morphisms . Up on insp ection it is easil y seen that, for any t wo lo cale morph ism s f : Y / / X and g : Z / / X , there is an isomorph ism of ordered sets ( Lo c / X ) o ( f , g ) ∼ = Map ( Mo d ( X ))(( Y , ◦ f ) , ( Z, ◦ g )) giv en by sending a sk ew op en morphism h to the X -mo dule morphism h ! with right adjoin t h ∗ . Sending sk ew op en morphisms in Lo c / X to their utmost left adj oints (i.e. h 7→ h ! ) thus giv es rise to an injectiv e and fu lly faithful 2-functor ( Lo c / X ) o / / Map ( Mo d ( X )) . (15) In the co domain catego ry of this functor we are n o w in terested in th e lo cally principally generated ob jects. In th e n ext subsection we dev elop that n otion further. 12 Putting Z = X and g = 1 X this red uces to what is called th e “F rob enius identit y” in [Mac Lane and Moerdijk, 1992, p. 500] ; we cal l this generalisation “balanced” b ecause we ge t th e (“un bal- anced”) F robenius identit y by plu gging in a terminal ob ject. 27 Lo cally principally generated X -mo dules. Let X b e a lo cale. As is customary in lo cale th eory , see e.g. [Mac Lane and Moerd ijk, 1992, p. 486], for an y u ∈ X we generically write i : ↓ u / / / / X for the corresp onding op en su blo cale of X , i.e. it is the op en Lo c -morphism deﬁ ned b y i ∗ ( v ) := ( u ⇒ v ), i ∗ ( x ) := ( x ∧ u ) and i ! ( v ) := v . As n oted b efore, it is therefore also sk ew op en in Lo c / X as (un ique) morp hism fr om i : ↓ u / / / / X to the termin al ob ject 1 X : X / / X , ↓ u ! ! i ! ! C C C C C C C C / / i / / X 1 X ~ ~ } } } } } } } } X (16) All elements of the Sup -monoid ( X, ∧ , ⊤ X ) are idemp oten t, thus eac h u ∈ X giv es rise to an idemp oten t represent able X -mo dule morp hism on the (only) repr esen table X -mo dule ( X, ∧ ). Th e image und er the functor in (15) of the ( Loc / X ) o -morphism in (16) is precisely the splitting of this idemp oten t: ( ↓ u, ∧ ) i ! / / ( X, ∧ ) . i ∗ o o − ∧ u   (17) It is notewo rthy that this is actually an adjo int splitting , since i ! ⊣ i ∗ in Mo d ( X ), and th at – b ecause 1 X is te rminal in Lo c / X and th e functor in (14) is fully fait hful – th is is the only adjunction in Mo d ( X ) b et ween ( ↓ u, ∧ ) and ( X , ∧ ). Applying Deﬁnition 5.3 to an X -mo du le ( M , ◦ ) w e get th e follo w ing. An element p ∈ M is lo c al ly princip al at u ∈ X if and only if p ◦ u = p and the comp osite X - mo dule morph ism ( ↓ u, ∧ ) i ! / / ( X, ∧ ) p ◦ − / / ( M , ◦ ) admits a right adjoin t in M o d ( X ). Let ( M , ◦ ) lp r denote the set of elemen ts of M whic h are locally principal at s ome u ∈ X . Then ( M , ◦ ) is lo c al ly princip al ly gener- ate d if and only if, for eac h m ∈ M , m = _ { p ◦ [ p, m ] | p ∈ ( M , ◦ ) lp r } , where [ p, m ] := W { u ∈ X | p ◦ u ≤ m } . W e sh all recast the latter deﬁnition in a more plea san t form. Prop osition 7.3 L et ( M , ◦ ) b e an X -mo dule. i. If p ∈ M is lo c al ly princip al at u ∈ X , then for any m ∈ M , p ◦ [ p, m ] is lo c al ly princip al at u ∧ [ p, m ] . ii. F or any m ∈ M , { p ◦ [ p, m ] | p ∈ ( M , ◦ ) lp r } = ↓ m ∩ ( M , ◦ ) lp r . 28 iii. ( M , ◦ ) is lo c al ly princip al ly gener ate d if and only if for al l m ∈ M , m = _ ( ↓ m ∩ ( M , ◦ ) lp r ) . (18) Pr o of : (i) F or sh orthand w e in tro duce q := p ◦ [ p, m ] and v := u ∧ [ p, m ]. Th en it is easily veriﬁed that q ◦ v = ( p ◦ [ p, m ]) ◦ ( u ∧ [ p, m ]) = p ◦ ([ p, m ] ∧ u ∧ [ p, m ]) = ( p ◦ u ) ◦ [ p, m ] = p ◦ [ p, m ] = q . Moreo ve r the diag ram ( ↓ v , ∧ ) i ! / /   ( X, ∧ ) q ◦ − / / ( M , ◦ ) ( ↓ u, ∧ ) i ! / / ( X, ∧ ) p ◦ − / / ( M , ◦ ) in M od ( X ), w h ere the left down wa rd arro w is the ob vious in clus ion of ↓ v in to ↓ u , comm u tes: f or w ≤ v w e can compute that q ◦ w = ( p ◦ [ p, m ]) ◦ w = p ◦ ([ p, m ] ∧ w ) = p ◦ w. But ( ↓ v , ∧ ) / / ( ↓ u, ∧ ): w 7→ w is a left adjoin t in Mod ( X ), hence the top comp osite morphism is a left adjoin t whenev er the b ottom comp osite morphism is. (ii) Because p ◦ − ⊣ [ p, − ] as order-pr eservin g maps b et wee n X and M , it is trivial that p ◦ [ p, m ] ≤ m and ( p ≤ m ⇒ p ◦ [ p, m ] = p ), for any p, m ∈ M . W e h a ve just shown that if p is lo cally pr in cipal then so is p ◦ [ p, m ]. Hence the equalit y of these s ets. (iii) Is no w immediate. ✷ W e can also translate to an X -mo dule ( M , ◦ ) the condition in Prop osition 5.5 that exp r esses that it is lo cally principally generated if and only if id ( M , ◦ ) = _ { ζ ◦ ζ ∗ | u ∈ X , ζ ∈ Map ( Mo d ( X ))(( ↓ u, ∧ ) , ( X , ∧ )) } , (19) where we write ζ ∗ for th e right adj oin t to ζ . This f act allo w s us to pro v e the follo wing remark able pr op ert y : Prop osition 7.4 L et ( M , ◦ ) b e a lo c al ly princip al ly g ener ate d X -mo dule. i. F or m, n ∈ M , m = n if and only if for every u ∈ X and e v ery left adjoint X -mo dule morp hism ζ : ( ↓ u, ∧ ) / / ( M , ◦ ) we have ζ ∗ ( m ) = ζ ∗ ( n ) . 29 ii. F or every m, n ∈ M and x ∈ X , ( m ∧ n ) ◦ x = m ∧ ( n ◦ x ) . iii. M is a lo c ale and f ∗ : X / / M : x 7→ ⊤ M ◦ x is the inv e rse image of a lo c ale morphism f : M / / X for which ( M , ◦ f ) = ( M , ◦ ) . Pr o of : (i) One direction is trivial; for th e other one expands m = id ( M , ◦ ) ( m ) an d n = id ( M , ◦ ) ( n ) b y means of t he f ormula give n ab ov e. (ii) Let u ∈ X and ζ ∈ Map ( Mo d ( X ))(( ↓ u, ∧ ) , ( X , ∧ )) with r igh t adjoin t ζ ∗ . Then ζ ∗ (( m ∧ n ) ◦ x ) = ζ ∗ ( m ∧ n ) ∧ x = ( ζ ∗ ( m ) ∧ ζ ∗ ( n )) ∧ x b ecause ζ ∗ is a mo dule morp hism (and thus turn s the “ − ◦ x ” in to a “ − ∧ x ”) and b ecause it is a righ t adjoint (and thus preserves inﬁ ma). But “for the same reasons” we also ha v e that ζ ∗ ( m ∧ ( n ◦ x )) = ζ ∗ ( m ) ∧ ζ ∗ ( n ◦ x ) = ζ ∗ ( m ) ∧ ( ζ ∗ ( n ) ∧ x )). Th us ζ ∗ (( m ∧ n ) ◦ x ) = ζ ∗ ( m ∧ ( n ◦ x )) for all u and all ζ , an d w e conclude b y the ab o v e that ( m ∧ n ) ◦ x = m ∧ ( n ◦ x ). (iii) Let m, ( m i ) i ∈ I b e elements of M . Let u ∈ X and ζ : ( ↓ u, ∧ ) / / ( X, ∧ ) a left adjoin t in Mo d ( X ) with right adjoin t ζ ∗ . Using that ζ ∗ is b oth a left and a r igh t adjoin t in Ord one computes that ζ ∗ ( m ∧ _ i m i ) = ζ ∗ ( m ) ∧ _ i ζ ∗ ( m i ) but also that ζ ∗ ( _ i ( m ∧ m i )) = _ i ( ζ ∗ ( m ) ∧ ζ ∗ ( m i )) . In b oth r igh t hand sides we n o w ﬁnd elemen ts of the locale ↓ u , w here ∧ distrib u tes o ver W , a nd hence th ese expressions are equal. This holds f or all u and all ζ , so by the the ﬁrst statemen t w e ob tain m ∧ W i m i = W i ( m ∧ m i ), w hic h means that M is a lo cale. Finally , the function f ∗ : X / / M is certainly a Sup -morphism: b ecause the action of ( X , ∧ , ⊤ X ) on M p reserv es su prema “in b oth v ariables”. But moreo ve r, for x, y ∈ X , we ma y compute – u sing the formula in (ii ) with m = ⊤ M ◦ x and n = ⊤ M to p ass from the seco nd line to the third – that f ∗ ( x ∧ y ) = ⊤ M ◦ ( x ∧ y ) = ( ⊤ M ◦ x ) ◦ y = ( ⊤ M ◦ x ) ∧ ( ⊤ M ◦ y ) = f ∗ ( x ) ∧ f ∗ ( y ) . Th us f ∗ is indeed the in verse imag e part of a locale morp hism f : Y / / X . Putting n = ⊤ M in the form ula in (ii) it follo ws that moreo v er m ∧ f ∗ ( x ) = m ∧ ( ⊤ M ◦ x ) = m ◦ x, that is to sa y , ( M , ◦ f ) = ( M , ◦ ) as cla imed. ✷ W e n o w go on to deﬁne the notion of “sk ew lo cal homeomorphism”. 30 Sk e w lo cal homeomorphisms. Let f : Y / / X b e in Lo c and u ∈ X ; we k eep the n otatio n i : ↓ u / / / / X f or the corresp ond ing op en sub locale of X . Recall fr om [Mac L ane and Moerdijk, 199 2, p. 524] that the el emen ts of the set S f ( u ) := Lo c / X ( i, f ) are the se ctions of f at u . Th is d eﬁnes a s heaf S f : X op / / Set , and th is construction extends to a functor Lo c / X / / Sh ( X ) whose r estriction to lo cal homeomorphisms is an equiv alence of categories. A particular f eature of lo cal h omeomorphisms is that, whenev er f = g ◦ h in Lo c , if f and g are lo cal homeomorphisms then so is h ; recall also that a lo cal homeomorphism is alwa ys op en in Lo c (see lo c. ci t. ). Thus, if f : Y / / X is a lo cal homeomorphism then every s ∈ S f ( u ) is an op en se ction in the sense that s : ↓ u / / Y is an op en lo cale morphism . With this in mind th e follo wing is a natural generali- sation. Deﬁnition 7.5 F or f : Y / / X in Lo c and i : u / / / / X , we put S o f ( u ) := ( Lo c / X ) o ( i, f ) and c al l its elements the sk ew op en sections of f at u . Example 7.6 Ev ery op en s ection s : ↓ u / / Y of a lo cale map f : Y / / X is necessar- ily s k ew op en to o; but the co n v erse need n ot hold. Ho w ev er, if f : Y / / X is a local homeomorphism then S f ( u ) = S o f ( u ) for all u ∈ X . A m orp hism f : Y / / X in Lo c is a lo cal h omeomorphism if and only if Y can b e co vered by its op en sections [Johns tone, 2002, v ol. 2, p. 50 3], i.e. ⊤ Y = _ n s ! ( u )    u ∈ X , s ∈ S f ( u ) and s is open in Lo c o . In this case, ev ery y ∈ Y can b e co v ered by op en secti ons of f , b y taking the restrictions of the op en sections of f to y . This motiv ates our main deﬁn ition in this section: Deﬁnition 7.7 A morphism f : Y / / X in Loc is a ske w local homeomorphism if 1 Y = _ { s ! ◦ s ∗ | u ∈ X , s ∈ S o f ( u ) } . F or the record w e immediatel y add: Example 7.8 Ev ery lo cal homeomorph ism is a sk ew lo cal h omeomorphism. A sk ew lo cal homeomorphism is a lo cal homeomorphism if and only if its (skew op en) sections are all op en. Sk ew lo cal homeomorphisms can b e c haracterised in diﬀerent w ays: 31 Prop osition 7.9 L et f : Y / / X b e in Lo c . i. Ther e is a bije ction b etwe en skew op en se ctions of f at u ∈ X and lo c al ly princip al elements of the X -mo dule ( Y , ◦ f ) at the idemp otent u ∈ ( X , ∧ , ⊤ X ) ; if s ∈ S o f ( u ) then s ! ( u ) ∈ Y is the c orr esp onding lo c al ly princip al element. ii. The f ol lowing statements ar e e q uivalent: (a) ( Y , ◦ f ) is a lo c al ly princip al ly gener ate d X - mo dule, (b) for al l y ∈ Y , y = W ( ↓ y ∩ { s ! ( u ) | u ∈ X, s ∈ S o f ( u ) } ) , (c) f is a skew lo c al home omorp hism. Pr o of : (i) By the fully faithful 2-functor in (15) w e kn o w, for eac h i : ↓ u / / / / X , that ( Lo c / X ) o ( i, f ) ∼ = Map ( Mo d ( X ))(( ↓ u, ∧ ) , ( Y , ◦ f )); the le ft hand side is precisely S o f ( u ), and the bijectio n is give n from left to righ t b y sending an s ∈ S o f ( u ) to s ! . As in Prop osition 5.5, the right hand side is in bijection with the set of elemen ts of ( Y , ◦ f ) which are locally principal at u , by sending s ! to s ! ( u ). (ii) Imm ed iate from (ii i) in Prop osition 7.3, and (1 9). ✷ Let ( Lo c / X ) o slh denote the full sub category of ( Loc / X ) o whose ob jects are the sk ew lo cal homeomorph isms. I t follo ws from the ab o ve r esults that the fully faithful 2-functor in (15) (co)restricts to a fully faithful 2-functor ( Lo c / X ) o slh / / Map ( Mo d lpg ( X )) . (20) This 2-functor is easily seen to b e in j ectiv e on ob jects; b ut due to Prop osition 7.4 it is su rjectiv e to o: for ev ery lo cally prin cipally generated X -mo du le ( M , ◦ ) the lo cale morphism f : M / / X with inv erse image f ∗ ( x ) = ⊤ M ◦ x , w hic h satisﬁes ( M , ◦ ) = ( M , ◦ f ), is a skew lo cal homeomorphism. The consequence of o ur w ork is then the follo wing result. Theorem 7.10 F or any lo c ale X , the 2-functor in (20) is an isomorphism of lo c al ly or der e d c ate gories: ( Lo c / X ) o slh ∼ = Map ( Mo d lpg ( X )); b oth of these ar e thus e quiv alent to Ord ( X ) ≃ Ord ( Sh ( X )) , the or der e d she aves on X v iewe d as enriche d c ate goric al structur es, r e sp. the i nternal or ders in the top os Sh ( X ) . W e ha v e seen in Example 7.8 that any lo cal homeomorphism is necessarily a sk ew lo cal homeomorphism; and we ha v e seen in Example 7.2 that an y op en lo cale morphism is necessarily sk ew op en too. It follo ws that LH / X is a full s u b category of ( L o c / X ) o slh . W e j ust pro ve d the latter to be isomorphic to Map ( Mo d lpg ( X )), th us it mak es sense to determine those lo cally prin cipally generate d X -mo dules which, under th is isomorphism, corresp ond to local homeomorphisms. 32 Deﬁnition 7.11 A lo c al ly princip al ly gener ate d X - mo dule ( M , ◦ ) is an ´ etale X - mo dule when every left adjoint X -mo dule morp hism ζ : ( ↓ u, ∧ ) / / ( M , ◦ ) satisﬁes, for al l v ∈ ↓ u and m ∈ M , ζ ( v ∧ ζ ∗ ( m )) = ζ ( v ) ∧ m. It is straigh tforward from Example 7.8 and Prop osition 7.9 that a sk ew lo cal home- omorphism f : Y / / X is a lo cal homeomorphism if and only if ( Y , ◦ f ) is an ´ etale X -mo dule. Letting Mo d ´ et ( X ) stand for the full sub-2-category of Mo d lpg ( X ) con- sisting o f ´ etale X -mod ules, we can co nclude with the follo wing summary . Theorem 7.12 F or any lo c ale X ther e is a c ommuting squar e ( Lo c / X ) o slh Map ( Mo d lpg ( X )) LH / X ?  O O Map ( Mo d ´ et ( X )) ?  O O in which the e qualities denote i somorphisms of (lo c al ly or der e d) c ate gories and the upwar d arr ows ar e ful l emb e ddings. The c ate gories in the b ottom r ow ar e e quivalent to Sh ( X ) , th e lo c al ly or der e d c ate gories in the top r ow ar e e q u ivalent to Ord ( Sh ( X )) , and the inclusions view “sets as discr ete (or symmetric) or ders”.  . Addendum: biadjunction, biequiv alence It w as shown in [Stubb e, 2007 a, Section 8] that, f or an y Q -catego ry C , the totally compact ob jects of the presheaf category P C form p recisely the Cauc hy-completio n of C : ( P C ) c = C cc . That is to sa y , a con tra v ariant presheaf φ : ∗ A / / C is totally compact in P C if and only if it has a right adjoint in Dist ( Q ) (“ φ is Cauch y”). Repre- sen table con tra v arian t preshea ves certainly are Cauc hy , th us the Y oneda embed ding Y C : C / / P C , whic h sends an ob ject c ∈ C to the r ep resen table C ( − , c ): ∗ tc ❝ / / C , corestricts to ( P C ) c : Y C : C / / ( P C ) c : c 7→ C ( − , c ) . (21) The f ollo wing is a mere trivialit y . Lemma 8.1 F or any Q - c ate gory C , the functor Y C : C / / ( P C ) c is ful ly faithful, and it is an e quivalenc e if and only if C is Cauchy-c omplete. On the other hand w e can compute, for any cocomplete Q -ca tegory A and any φ ∈ P ( A c ), the φ -weig h ted colimit of the in clus ion i A : A c / / A . Th is deﬁnes a functor R A : P ( A c ) / / A : φ 7→ colim( φ, i A ) (22) ab out wh ic h w e record some auxiliary resu lts. 33 Lemma 8.2 F or any c o c omplete Q -c ate gory A , the functor R A : P ( A c ) / / A is c o- c ontinuous and admits a c o c ontinuous right adjoint. M or e over, R A is always ful ly faithful, and it is an e quivalenc e if and only if A is total ly algebr aic. Pr o of : W e claim that R A is left adjoin t to H : A / / P ( A c ) : a 7→ A ( i A − , a ). Ind eed, for φ ∈ P ( A c ) an d x ∈ A , A ( R A φ, x ) = A (colim( φ, i A ) , x ) = h φ, A ( i A − , x ) i = P ( A c )( φ, H x ) . Next we p r o ve that H itself is cocon tin uous; it suﬃces to s ho w that it pr eserv es suprema of con tra v ariant presheav es: f or any φ ∈ P A , H ( sup A ( φ )) = colim( φ, H ). Note ﬁrs t that, b y Prop osition 3.4, H ( sup A ( φ )) = A ( i A − , sup A ( φ )) = φ ( i A − ) = A ( i A − , − ) ⊗ φ. But then also P ( A c )  H ( sup A ( φ )) , −  = h A ( i A − , − ) ⊗ φ, − i = h φ, [ A ( i A − , − ) , − ] i = h φ, P ( A c )( H − , − ) i whic h is the un iv ersal prop ert y that we had to c h ec k. T o see that R A is f ully faithfu l it is (necessary and ) su ﬃ cien t to sh o w that the unit of the adju nction R A ⊣ H is an isomorph ism (cf. [Stub b e, 2007, 2.3] for example). Th us, for c ∈ A c and φ ∈ P ( A c ) we compute that (( H ◦ R A )( φ ))( c ) = A ( i A c, colim( φ, i A )) = H c (colim( φ, i A )) (with nota tions as in Prop osition 3.4) = colim( φ, H c ◦ i A ) (b ecause of Prop osition 3.4) = colim( A ( − , i A − ) ⊗ φ, H c ) . By an argument in th e pro of of Proposition 3.4, w e kno w that a presheaf-w eigh ted colimit of H c is the v alue of that we igh t in c ; here this all o w s u s to equate colim( A ( − , i A − ) ⊗ φ, H c ) = A ( c, i A − ) ⊗ φ = A c ( c, − ) ⊗ φ = φ ( c ) , taking in to account th at c ∈ A c and i A is a f u ll em b edding. Th is indeed prov es that H ◦ R A = 1 P ( A c ) . 34 Finally , kn o wing th at R A is alw a ys fully faithful, it is an equ iv alence if and only if also the counit of the adjunction R A ⊣ H is an isomorphism . Sp elled out this means that, for ev ery a ∈ A , a ∼ = colim( A ( i A − , a ) , i A ), whic h pr ecisely sa ys that 1 A is the (p oin t wise) left Kan extension o f i A along itself. ✷ Theorem 8.3 Ther e is a biadjunction Cat ( Q ) ⊥ P ( ( ( − ) c h h Map ( Co cont ( Q )) wher e the involve d 2-functors ar e deﬁne d as: P  C F / / D  := P C D ( − , F − ) ⊗ − / / P D ,  A F / / B  c := A c F / / B c and c ounit and unit ar e given by the functors in (21) and (22). Pr o of : W e shall p ro ve that, for an y co complete Q -cate gory A ,  A c , R A : P ( A c ) / / A  is a biun iv ersal righ t reﬂection along the 2-fun ctor P : Cat ( Q ) / / Map ( Co cont ( Q )). The latter is ind eed a 2-fun ctor: for an F : C / / D in Cat ( Q ) we ha v e adjoin ts P C ⊥ ⊥ D ( − , F − ) ⊗ − ! ! [ D ( F − , − ) , − ] = = [ D ( − , F − ) , − ] = D ( F − , − ) ⊗ − o o P D in Cat ( Q ) (with all comp ositions and liftings computed in Dist ( Q )) so P lands in the 2-catego ry Map ( Co cont ( Q )); and 2-functorialit y is ob vious. Moreo v er, Lemm a 8.2 pro vides the information that R A is a morphism of Ma p ( Cocont ( Q )). So all w e n eed to sh o w, is that R A has the required 2-universal p r op ert y , i.e. the order-preserving function Cat ( Q )( C , A c ) / / Map ( Co cont ( Q ))( P C , A ): F 7→ R A ◦ P F (23) is an equiv alence of ordered s ets. W e shall pro v e ﬁrst th at it is essen tially su rjectiv e, and th en that it is ord er-reﬂecting. 35 Essential su rje ctivity. S upp ose giv en a left adjoint G : P C / / A in Co cont ( Q ), or equiv alen tly , sup p ose giv en adjoints P C ⊥ ⊥ G   K @ @ H o o A in Cat ( Q ) . F or ψ ∈ P A we can compute, with str aigh tforward argu m en ts in v olving liftings and comp ositions in the quan taloid Dist ( Q ), th at ψ ( G ◦ Y C − ) is the ψ -w eigh ted colimit of H : h ψ , P C ( H − , − ) i = h ψ , [ P C ( Y C − , H − ) , − ] i = h ψ , [ A ( G ◦ Y C − , − ) , − ] i (b ecause G ⊣ H ) = h A ( G ◦ Y C − , − ) ⊗ ψ , − i = h [ A ( − , G ◦ Y C − ) , ψ ] , − i = h ψ ( G ◦ Y C − ) , − i (b y Y o neda Lemma for Q -ca ts) = P C ( ψ ( G ◦ Y C − ) , − ) . But for an y x ∈ C , we can also compute that A ( GY C ( x ) , s up A ( ψ )) = P C ( Y C ( x ) , H ( sup A ( ψ ))) (by adjun ction G ⊣ H ) = ( H ◦ s up A ( ψ ))( x ) (by Y oneda Lemma for Q -ca ts) = colim( ψ, H )( x ) (b ecause H is coconti n uous) . Putting these toget her w e h a ve, for an y ψ ∈ P A and x ∈ C , that ψ ( GY C ( x )) = A ( GY C ( x ) , s up A ( ψ )) whic h, acco rding to Prop osition 3.4, means that for any x ∈ C the ob j ect GY C ( x ) of A is totally compact . In other words, the giv en G : P C / / A factors as P C G / / A C Y C O O G / / A c i ? ? ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ where G ( x ) := G ( Y C ( x )). It is a matter of calculations, using co cont in uit y of G amongst other things, to see that G = R A ◦ P ( G ): for φ ∈ P C , ( R A ◦ P ( G ))( φ ) = colim( P ( G )( φ ) , i A ) 36 = colim( A c ( − , G − ) ⊗ φ, i A ) = colim( A ( − , i A − ) ⊗ A c ( − , G − ) ⊗ φ, 1 A ) = colim( A ( − , i A ◦ G − ) ⊗ φ, 1 A ) = colim( A ( − , GY C − ) ⊗ φ, 1 A ) = colim( φ, G ◦ Y C ) = G ◦ colim( φ, Y C ) = G ( φ ) . Th us we prov ed that the function in (23 ) is essen tially surjectiv e. Or der-r eﬂe ction. Remark ﬁr st that for an y F : C / / A c in Cat ( Q ), the outer diagram P C / / P ( F ) / / P ( A c ) R A / / A C Y C O O F / / A c Y A c O O i = = { { { { { { { { { { { { { { comm u tes: b oth th e left h and square and the righ t hand triangle a re easily c hec ked b y computation. No w supp ose th at some F , G : C / / / / A c in Cat ( Q ) are su c h that R A ◦ P ( F ) ≤ R A ◦ P ( G ) in Map ( Co cont ( Q )). T hen w e can deduce fr om the ab o v e that i A ◦ F = R A ◦ P ( F ) ◦ Y C ≤ R A ◦ P ( G ) ◦ Y C = i A ◦ G. But b ecause i A : A c / / A is a f u ll em b edding, it follo ws that necessarily F ≤ G from the start. This pro v es th at the f unction in (23) is also order-reﬂecting. It is no w a matter of rou tin e computations to v erify that the righ t biadjoin t to the 2-fun ctor P : Ca t ( Q ) / / Map ( Co cont ( Q )) is ind eed giv en by “restricting to totally compacts”: ( − ) c : Map ( Co cont ( Q )) / / Cat ( Q ):  F : A / / B  7→  F : A c / / B c  , and that th e unit of the b iadjunction is in deed giv en b y those corestrictions of Y oneda e m b eddings as in (21). ✷ The (co)restriction of biadjoint 2-functors to th ose ob jects for whic h the (co)unit is an equiv alence, is a biequiv alence of 2-cate gories. 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