A dualizing object approach to non-commutative Stone duality

A DUALIZING OBJECT APPRO A CH TO NON- COMMUT A TIVE STONE DUALITY GANNA KUDR Y A VTSEV A Abstract. The aim of the presen t pap er is to extend the dualizing ob ject approac h to Stone duality to the non-commut ative setting of ske w Bo olean algebras. This cont i nues the study of non-commuta tive generalizations of dif- ferent forms of Stone duality initiated in recent pap ers by Bauer and Cvetk o- V ah, La wson, Lawson and Lenz, Resende, and also the current author. In particular, we const r uct a seri es of dual adjunctions betw een the categories of B o olean spaces and skew Bo olean algebras, unital versions of whi c h are induced by dualizing ob jects { 0 , 1 , . . . , n + 1 } , n ≥ 0. W e describ e Eilenberg- Mo ore categories of the monads of our adjunction s and construct easily under- stoo d non-commutat ive reflections of skew Bo olean algebras, where the latter can b e faithfull y embedded (if n ≥ 1) in a canonical wa y . As an application, we answer the question tha t arose in a rece nt pap er by Leec h and Spinks to describe the lef t adjoint to their ‘twisted pro duct’ functor ω . 2010 Mathematics S u bje ct Cla ssific ation: 1 8B30, 06 E 15, 0 6 E75, 54 B 40, 03 G05. Keywor ds and phr ases: Bo olean space, Bo olean alg ebra, Stone duality , skew Bo olean algebra, ´ etale space, dua lizing ob ject, schizophrenic ob ject, a djunction, monad, E ilenberg-Mo o re categor y , reflective sub categ ory . 1. Introduction Stone dua lity [3 3, 9 , 1 4] is pro bably the most fa mo us r esult a b o ut Bo olean alge- bras. It pro vides a subtle link betw een alg ebra and topology and has far-reaching consequences a nd n umerous imp orta nt generaliza tions. Recently , differ e n t v aria tions o f Stone duality hav e b een gener alized to non- commutativ e s e tting at the a lg ebraic side to skew Bo olean alg e bras by the a u- thor [1 6], to skew Bo olean a lgebras with intersections indep endently by Bauer and Cvetk o-V ah [2] a nd the author [16], to B o olean inv erse semigroups by Law- son [19, 20], to pseudogro ups by Lawson and Lenz [21] and to certain c la sses of quantales by Res e nde [30, 31]. V ery r ecently , Mar k Lawson and the a uthor [18] hav e found a common genera lization of results of [16] and [1 9, 20] to a duality where ´ etale actions of Bo olean in verse semig roups arise a t the a lgebraic side. The latter actions were intro duced in [13, 32] and play an imp ortant role in Morita theory of in verse semig roups [1 2, 32] as well as in interplay of semig roup and top os theories [10, 1 1, 13]. The study o f skew Bo olean algebra s, that are ob jects o f consideratio n o f this pap er, was initiated by Cor nish [8] and Leech [22, 23]. In [4] it was o bserved that sk ew B o olean alg ebras with in ter sections form a discriminator v arie ty and therefore to ols o f universal alg ebra can b e applied to their study . Stone duality for skew Bo o lean alg ebras [16] and skew Bo olean algebras with intersections [1 6, 2] op ens completely new p ersp ectives of lo oking at these algebr as. The dua lity o f [16] made the construc tio ns o f this pap er p os sible, a nd also, together with results o f [19, 20], led Mark Lawson and the author to the duality o f [18 ]. The la tter dua lit y , Author’s r esearc h i s partial l y supp orted by ARRS gr an t P1-0288. 1 2 GANNA KUDR Y A VTSEV A in its turn, witnesses that skew Boo le an algebras and Bo olean inv erse semigroups are closely rela ted. Putting them together leads to the construction of new ob jects that ar e c o nnected with impor tant concepts o f inv erse semigr oup theory and its applications. An imp orta nt fea ture of Stone duality is that (the unital version of ) it is induced by a dualizing obje ct (so metimes also c alled a schizo phr enic obje ct ) { 0 , 1 } , c o nsidered as a Bo ole an algebra or a s a dis crete top ologica l spa ce (see Subsection 2.1 for mo r e details). T o the c o ntrast, its g eneralizatio ns to skew Bo olean s e tting [2, 1 6] do not po ssess this prop er ty (se e Subsection 2.4 for mor e details ). The g o al of the pr esent pap er is to extend the dualizing ob ject approa ch to Stone duality to the non- commutativ e setting of skew Bo olean alge br as. Our motiv ation is well-established significance of dualizing ob jects, bo th from the universal alg ebra and the catego ry theory viewp oints, (see, e.g ., [6, 15, 29]), on the one hand, and r ecently revealed impo rtance of skew Bo olean algebra s, o n the other hand. The non-comm utative dualizing ob jects that a rise in this paper ar e primitive left- handed skew Bo olea n algebra s n + 2 = { 0 , 1 , . . . , n + 1 } , n ≥ 0 (see Subsectio n 2.2 for the definitions). Their ro le in o ur theory is simila r to that of the tw o-element Bo olean a lgebra 2 = { 0 , 1 } in Stone duality . These ob jects induce a ser ies of dual adjunctions Λ n ⊣ λ n , n ≥ 0, b etw een the c a tegories of Bo olean spaces and left- handed skew Bo olean algebra s . Given a left-ha nded skew Bo o lean algebr a S , the Bo olean algebr a S/ D r efle cts S in the sense that there is a functor from left-handed skew B o olean algebra s to Bo olean a lgebras sending S to S/ D that is the left adjo int to the inclusion functor in the reverse direction. Thu s S/ D is known as a c ommutative r efle ction of S . In this pap er w e construct a se r ies λ n Λ n ( S ), n ≥ 0 , of r e flections of S . If n = 0 then λ 0 Λ 0 ( S ) ≃ S/ D . If n ≥ 1 these r e fle c tions are non-c ommut a tive and the units o f the cons tructed a djunctions pr ovide a canonical wa y to faithfully repre s ent S in its very ‘Bo olean algebra like’ non-c ommutative r efle ction λ n Λ n ( S ). How ever, in o rder to decreas e the ‘degree of non-commutativit y’ of the env elo ping algebra one ha s to sacrifise the size of the underlying Bo olea n a lg ebra. Note that the p ossibility o f a n embedding of S into λ n ( X ) for some X can be easily deduced from [2 6, Coro llary 3.6] and Rema rk 3 of this pa per . But no specific construction o f an embedding of a skew Bo o le a n algebra into another skew Bo ole a n algebra with ‘low degree of non-commutativit y’ had b een k nown b efo r e. It is interesting that a v ariatio n of the functor λ 1 of this pap er has previously app eared in a nother disguis e in the pap er by Leech and Spinks [26] as the ‘twisted pro duct’ ω -functor. Inv oking the F reyd’s a djoint functor theorem, it was arg ued in [26] that the functor ω has a left adjoint , Ω, and the question to describ e Ω arose. As a first step, the action of Ω on finite ob jects was describ ed in [26]. The approach of [26] did no t a llow to go further than tha t. Our appro a ch provides a natural interpretation for b o th ω and Ω and leads to the full descriptio n of the functor Ω (se e Remarks 3 a nd 1 6). W e also study the monads induced by the a djunctions Λ n ⊣ λ n , n ≥ 0, and pro - vide their nice combinatorial description. W e then prov e that the E ilenberg-Mo o re categorie s o f thes e mona ds ar e equiv alent to the image of λ n , so that this imag e is a r eflective subc a tegory of the ca tegory o f left-handed skew Bo olean algebra s. Before stating in Sections 3, 4 and 5 the main co nstructions and r esults o f this pap er, we collect in Section 2 a ll necess ary preliminar ie s . In particular , we expla in what prec isely we mea n by a dualizing ob ject approach to the class ical Stone duality , then provide necessa ry background on skew Bo olean algebra s and ´ eta le spa ces and review the non-commutativ e Stone duality fro m [16] needed in this pap er . W e also explain why this duality is no t induced b y a dualizing ob ject. A DUALIZING OBJECT APPRO ACH TO NON-COMMUT A TIVE STONE DUALITY 3 Ackno wledgements I would like to thank the anonymous referee for suggesting that the topo logy on the spa ce Λ n ( S ) might c o incide with the top ology inherited by Λ n ( S ) from the pro duct space { 0 , . . . , n + 1 } S . This turned out to be the case and le d to a significant simplification of the pr o of that Λ n ( S ) is a Bo olean space in Section 4. I a m a lso grateful to the r eferee, Marcel Jackson a nd Mark Lawson for helpful c o mment s . 2. Preliminaries 2.1. The dualizing ob ject view of the classical Sto ne duali t y. By a Bo ole an algebr a we mean what is us ually c a lled a genera liz ed Bo olean algebra , that is a relatively complemented distributiv e lattice with a bottom ele ment. A Bo olean algebra with a top element will b e called a unital Bo ole an algebr a . A homomorphism ϕ : B 1 → B 2 of Bo olean a lgebras is called pr op er [9], provided that for a ny b ∈ B 2 there ex is ts a ∈ B 1 such that ϕ ( a ) ≥ b . In this pap er any homomorphism of Bo olean algebras is assumed to b e pr o p er. By BA we denote the category of Bo olean a lg ebras and pro p er ho momorphisms. By a Bo ole an sp ac e we mean what is usually ca lled a lo ca lly compa c t Bo olea n space, that is a Haus dorff s pace in whic h compact-op en sets form a base o f the top ology . Note that c omp act Bo ole an s p ac es a re usually refer red to as Bo olean spaces. By BS we denote the category o f Bo olean spa c es and co ntin uous pro p er maps (reca ll that a map o f top olo gical s pa ces is pr op er if inv erse images o f compact sets a re c o mpact se ts). Given a B o olean spa ce X , all contin uous maps X → { 0 , 1 } , wher e { 0 , 1 } is a discrete top olo g ical space , suc h that f − 1 (1) is a compact s et, form a Boolea n algebra denoted by X ∗ and ca lled the dual Bo ole an algebr a of X . The assignment X → X ∗ is the ob ject part of the contra v aria nt functor A : BS → BA . The restrictio n of A to the catego ry of compact Bo olean spac e s is an enriched contra v aria nt Ho m-set functor, b ecause given a compact B o olean s pace X , the unital Bo ole a n algebr a X ∗ consists o f al l contin uous maps X → { 0 , 1 } (note that such a map is automatica lly also pro p er). The sp e ctrum B ∗ of a B o olean a lg ebra B is the s e t of all non-zero homomor phisms from B to the t wo-element Bo o lean alg ebra 2 = { 0 , 1 } . The set B ∗ is e q uipped with the top o lo gy whose ba se is fo r med by the sets M ( a ) = { f ∈ B ∗ : f ( a ) = 1 } , a ∈ B . This s pa ce is a Bo olea n space and is called the dual sp ac e of B . It is well- known that the top o logy o n B ∗ can be also characteriz e d a s the subspace top olog y of the pro duct spac e { 0 , 1 } B . The a ssignment B → B ∗ is the ob ject par t o f the contra v ariant functor S : BA → BS . It is impo rtant for us tha t the restr iction of S to the categor y o f unital Bo olea n algebra s is an enriched co nt r av ar iant Hom-set functor, b eca us e given a unita l Bo olea n algebra B , the po int s of the space B ∗ are al l unital Bo olean algebr a homomor phisms B → 2 . Stone duality for unital Bo ole an algebr as [3 3] (se e also textb o ok s [7, 1 4]) states that the ab ov e describ ed contrav ar iant Hom-set functor s to { 0 , 1 } establish a dual equiv alence be tw ee n the catego ries of unital Bo olea n algebras and compact Bo o lean spaces. Ther e fore, { 0 , 1 } is called a du alizing obje ct , and this duality is induc e d by a dualizing obje ct . Stone duality for Bo ole an algebr as [3 3, 9] is an extension of the ab ov e duality . It states that the functor s A : BS → BA and S : BA → BS establish a dual equiv alence betw ee n the categ ories BS and BA . In this pap er, we will work at the lo cally compact and non-unital level of gener- ality . W e will construct functors, whose re s trictions to appro pr iate compact-unital sub c ategories are enriched Hom-se t functor s, in the same w ay as in the co mmutative case outlined in this s ubsection. 4 GANNA KUDR Y A VTSEV A 2.2. Sk ew Bo olean alge bras. F or a detailed intro duction to the theory o f s kew Bo olean algebr as we refer the reader to [4, 23, 25]. A skew lattic e is an alg ebra ( S ; ∧ , ∨ ) o f type (2 , 2) such that the op erations ∧ a nd ∨ are asso ciative, idempotent and satisfy the absor ption identities x ∧ ( x ∨ y ) = x = x ∨ ( x ∧ y ) a nd ( y ∨ x ) ∧ x = x = ( y ∧ x ) ∨ x . The natu r al p artial or der ≤ o n a skew lattice S is defined by x ≤ y if and only if x ∧ y = y ∧ x = x o r, eq uiv alen tly , x ∨ y = y ∨ x = y . A skew lattice S is symm et ric if x ∧ y = y ∧ x if and only if x ∨ y = y ∨ x . An ele ment 0 of S is called a zer o if x ∧ 0 = 0 ∧ x = 0 for all x ∈ S . S is Bo ole an if it is s ymmetric, ha s a zer o element and each principal subalgebr a ⌈ x ⌉ = { y ∈ S : y ≤ x } = x ∧ S ∧ x forms a Bo olean lattice. Let S b e a Bo olean skew lattice and x, y ∈ S . The r elative c omplement x \ y is the complement of x ∧ y ∧ x in the Bo olea n lattice ⌈ x ⌉ . A skew Bo ole an algebr a is a Bo olea n skew la ttice, w ho se signa tur e is enriched by the nullary op era tion 0 and the binary r elative complement op er ation, that is, it is an algebra ( S ; ∧ , ∨ , \ , 0). Skew B o olean a lgebras satisfy distributivity laws x ∧ ( y ∨ z ) = ( x ∧ y ) ∨ ( x ∧ z ) and ( y ∨ z ) ∧ x = ( y ∧ x ) ∨ ( z ∧ x ) [24, 2.5]. Let S b e a skew lattice. It is called r e ctangular if there exist tw o sets L and R such that S = L × R , and the op erations ∧ and ∨ are defined b y ( a, b ) ∧ ( c, d ) = ( a, d ) and ( a, b ) ∨ ( c, d ) = ( c, b ). Let D be the eq uiv alence r elation on S given by x D y if and only if x ∧ y ∧ x = x and y ∧ x ∧ y = y . It is known [22, 1.7] that D is a congruence relation, the D -cla sses of S ar e ma ximal rectangula r suba lgebras and the quotient S/ D is the maximal lattice image o f S . If S is a skew Bo ole an algebr a then S / D is the ma x imal Bo olean algebra imag e of S [4 , 3.1 ]. A skew lattice is called left-hande d ( right-hande d ) if it satisfies the identities x ∧ y ∧ x = x ∧ y and x ∨ y ∨ x = y ∨ x (r e sp ectively , x ∧ y ∧ x = y ∧ x and x ∨ y ∨ x = x ∨ y ). In a left-handed skew Bo o lean a lgebra the rectangular subalgebras are fl at in the sense that x D y if and o nly if x ∧ y = x and y ∧ x = y . If S is a left-handed s kew Bo olean algebra then the band ( S, ∧ ) is left nor mal. It can be shown (or see [17]) that for each a ∈ S and D ∈ S/ D such that [ a ] D ≥ D there is a unique b ∈ S such that [ b ] D = D a nd a ≥ b . This element b is called the r estriction of a to D and is denoted by a | D . A skew Bo olean a lg ebra S is ca lled primitive if it has o nly one non-zero D - class or , equiv alently , if S/ D is the Bo o lean algebr a 2 . Up to isomorphis m, finite primitive left-ha nded skew Bo olean alg ebras are the alg ebras n + 2 , n ≥ 0. Thes e algebras play a n imp ortant role in this pa p e r. The underlying set of n + 2 is the set { 0 , 1 , . . . , n + 1 } a nd its non-zero D -class is D = { 1 , . . . , n + 1 } . The op er ations on D are determined by lefthandedness: x ∧ y = x and x ∨ y = y for a ny x, y ∈ D . A skew Bo olean alg ebra has fi nite interse ctions if any finite set of its ele men ts has the greatest lower b ound called the interse ction with resp ect to the natural partial order. Since all intersections that we consider are finite, we will sometimes write just ‘int er sections’ for ‘finite int er sections’. Homo mo rphisms of skew Bo olean algebras with finite int er sections a r e r equired to pr eserve the finite intersections. Let ϕ : S 1 → S 2 be a homomorphism of skew Bo o le an algebras and let ϕ : S 1 / D → S 2 / D be the underlying ho mo morphism o f Bo olea n algebr as. W e call ϕ pr op er if ϕ is pro p e r . W e fix the notation Skew for the c ategory of left-handed skew Bo o lean algebr as and their a re pro p er homo morphisms. All s kew Boo lean alg ebras, considered in the sequel, are left-handed, so we take a conv ention to write ‘skew Bo o lean algebra ’ fo r ‘left-handed skew Bo olea n algebra’. By a morphism of skew Bo olean alge br as we will mean a mor phism in the ca tegory Skew . 2.3. ´ Etale s paces. Preliminar ies on ´ etale s paces can be found in any textb o ok on sheaf theory , e.g. in [5, 28]. An ´ etale sp ac e over X is a tr iple ( E , f , X ), where E , A DUALIZING OBJECT APPRO ACH TO NON-COMMUT A TIVE STONE DUALITY 5 X ar e top ologica l spaces and f : E → X is a surjective lo ca l homeomor phism. The po ints of E a r e called germs. A lo c al se ction o r just a se ction in E is a n o p en subs et A of E s uch that the restric tio n of the map f to A is injective. If U is an op en set in X then E ( U ) is the set o f all se ctions of E ov er U , where a sectio n A is over U provided that f ( A ) = U . F or x ∈ X w e denote the set of a ll y ∈ E suc h that f ( y ) = x by E x . This set is ca lled the stalk ov er x . If A ∈ E ( U ) then for x ∈ U b y A ( x ) we deno te the germ y ∈ A ∩ E x . Let ( A , g , X ) and ( B , h, Y ) b e ´ etale spa ces and f : X → Y be a contin uous map. A c ohomomorphism over f (or an f -c ohomomorphism ) k : B A is a collection of maps k x : B f ( x ) → A x for each x ∈ X suc h that for every sectio n s ∈ B ( U ) the function x 7→ k x ( s ( f ( x ))) is a section o f A ov er f − 1 ( U ). The maps k x are called the c omp onents o f k . W e intro duce the nota tion Etale for the ca tegory o f ´ etale spaces ov er Bo ole an spaces and their cohomo mo rphisms over co nt inuous pr op er maps. All ´ etale space s, considered in the sequel, are ov er Bo olea n spaces, therefore we take a conven tion to write ‘´ etale space’ for ‘´ etale space ov er a Bo o lean s pace’. By a morphism of ´ etale spaces we will mean a mor phism in the categ ory E tale . 2.4. Equiv alence of the categories of sk ew Bo olean algebras and ´ etale spaces. The co nstructions of this pa pe r rely on the g e ne r alization of Stone dua lity to left-handed skew Bo olean algebra s establishe d in [1 6], that is o utlined in this subsection. F or a detailed ex p o sition a nd pro ofs we refer the reader to [1 6]. Let ( E , π , X ) b e an ´ etale space. Denote b y E ∗ the s et of all compact-op en sections of E . Let A, B ∈ E ∗ . W e put ( A ∪ B )( x ) =  B ( x ) , if x ∈ π ( B ) , A ( x ) , if x ∈ π ( A ) \ π ( B ) , ( A ∩ B )( x ) = A ( x ) fo r all x ∈ π ( A ) ∩ π ( B ) . Then ∪ and ∩ are well-defined binar y o pe r ations o n E ∗ and ( E ∗ , ∪ , ∩ ) is a left- handed B o olean skew lattice. The a s so ciated skew Bo olean algebra E ∗ is c a lled the dual skew Bo ole an algebr a to the ´ etale space ( E , π , X ). Note that a D b in E ∗ if and only if π ( a ) = π ( b ) a nd the as s ignment [ a ] D 7→ π ( a ) establishes a n is o morphism betw ee n ( S/ D ) ∗ and X ∗ . Remark that E ∗ has finite intersections if and only if E is Hausdo rff. In the opp os ite dir e c tion, let S b e a skew Bo olean a lg ebra. The po int s of the dual ´ etale s pace of S are the ultrafilters of S . The latter can b e characterized as follows. Let F b e an ultrafulter of the Bo o lean a lgebra S/ D a nd let a ∈ S b e s uch that [ a ] D ∈ F . Then the se t X a,F = { b ∈ S : there is c ∈ S s uch that a, b ≥ c and [ c ] D ∈ F } is an ultrafilter in S , and a ny ultra filter in S is o f this form. The set S ∗ of all ultrafilters of S is called the sp e ctrum of S . L e t ˆ π : S ∗ → ( S/ D ) ∗ be the pr oje ction map given by X a,F 7→ F . F or a ∈ S we put M ( a ) = { F ∈ S ∗ : a ∈ F } . The top olo gy on S ∗ is given by a subbase , that is in fact a ba s e, co ns isting of the sets M ( a ), a ∈ S . W e have that S ∗ = ( S ∗ , ˆ π, ( S/ D ) ∗ ) is an ´ etale space called the dual ´ etale sp ac e of S . The space S ∗ is Hausdorff if and o nly if the skew Bo olean algebra S has finite intersections. W e now describ e the corresp o ndence for morphis ms . Let ( E , f , X ) , ( G, g , Y ) b e ´ etale space s and k : E → F a morphism. W e extend k to the map on s e ctions. This map takes compact-o pen sections to compact-op en sections and pr e s erves the op erations ∪ , ∩ and ∅ . This leads to the mor phism e k : E ∗ → G ∗ of skew Bo o lean 6 GANNA KUDR Y A VTSEV A algebras . Note that if the spaces E and G are Hausdorff and all comp onents of k are injective then e k preser ves finite intersections. Conv ersely , let S, T b e skew Bo olean a lgebras and k : S → T a morphism. The map k − 1 induces a morphism of Bo o lean spaces from ( T / D ) ∗ to ( S/ D ) ∗ . Let F ∈ ( T / D ) ∗ and V ∈ T ∗ F . Then the set k − 1 ( V ) is either empty o r a disjoint union of several ultrafilter s over k − 1 ( F ) . This a llows to define the comp onent e k F of the morphism e k : S ∗ → T ∗ . The domain of e k F is S ∗ k − 1 ( F ) and for U ∈ S ∗ k − 1 ( F ) we set e k F ( U ) = V if V ∈ T ∗ F and U ⊆ k − 1 ( V ). Note that if S, T hav e finite in ter s ections and k preserves them then all comp onents of e k : S ∗ → T ∗ are injective. W e no w state the theorem tha t pr ovides insights needed for establishing the main results of this pa p er . Theorem 1 ([16]) . The describ e d assignments ar e functors that establish an e quiv- alenc e b etwe en the c ate gories Eta le and Skew . If S is a skew Bo ole an algebr a t hen S is natu r al ly isomorphi c to S ∗∗ via the map β S given by β S ( a ) = M ( a ) , a ∈ S . If E is an ´ etale sp ac e over a Bo ole an sp ac e then it is natu r al ly isomorphic to E ∗∗ via the m ap γ E given by γ E ( A ) = N A = { N ∈ E ∗ : A ∈ N } , A ∈ E . Consider a n a pplication o f this theor em. Let n ≥ 0 and ϕ : S → n + 2 be a morphism o f skew Bo o le an algebra s (such morphisms will play an imp ortant role in Sectio n 4). W e interpret this top ologica lly . Observe that ( n + 2 ) / D = 2 and 2 ∗ is a one-element s pace, { a } . The cohomomo rphism e ϕ has ther efore only one comp onent e ϕ a : S ∗ ϕ − 1 ( a ) → { 1 , . . . , n + 1 } , that completely determines ϕ . Moreov er , any map φ : S ∗ ϕ − 1 ( a ) → { 1 , . . . , n + 1 } g ives rise to a morphims from S to n + 2 ov er ϕ − 1 . W e also hav e the following theor em. Theorem 2 ([16 , 2]) . The c ate gory of skew Bo ole an algebr as with interse ctions is e quivalent to the c ate gory of H ausdorff ´ etale sp ac es over Bo ole an sp ac es whose morphisms ar e c ohomomorphisms with inje ctive c omp onents. Let S, T b e sk ew Boo lean algebra s with in ters ections. Theorems 1 and 2 tell us that, unless S is commutativ e, the se t of all morphisms fro m S to T is muc h bigger than the set of intersection-preserving such morphis ms . F or exa mple, if S = 4 a nd T = 3 mo rphisms from S to T are given by functions from { 1 , 2 , 3 } to { 1 , 2 } . Intersection-prese r ving morphisms a r e given by injective such functions, which means that there ar e no intersection-preser v ing mo rphisms at a ll (note that we do not take into account the zero map s ince w e consider only prop er mo rphisms). F or the co nstructions of this pap er, it is cr ucial that we consider all co homomor- phisms b etw een appr opriate spaces , and not o nly ones with all compone nts injective. So it is impo rtant that the duality theorem we use in this pap er is Theo rem 1 and not Theorem 2 . W e now explain why the unital versions of dualities of Theorems 1 and 2 are not induced by a dua lizing ob ject. F or Theor em 1, consider the functor co ns tructing the sp ectrum o f a skew B o olean algebr a. As follows fro m Lemma 6 .3 of [16] p oints of the spectr um of a skew B o olean alg ebra S are in a bijective cor r esp ondence with morphisms S → n + 2 such that the inv erse image of 1 is non-empty and minimal p os s ible. This do es not pr o duce all morphisms S → n + 2 (a nd neither all morphisms from S to a nother skew Bo olea n algebra ), even in the case when S/ D is unital. F or the example fro m the prev ious pa ragr aph, there a r e ex a ctly 2 3 = 8 morphisms from 4 to 3 . But only three of thes e morphisms, that a re lis ted be low, A DUALIZING OBJECT APPRO ACH TO NON-COMMUT A TIVE STONE DUALITY 7 give rise to the p oints of the sp ectr um of 4 : ϕ 1 : 0 7→ 0 1 7→ 1 2 7→ 2 3 7→ 2 ; ϕ 2 : 0 7→ 0 1 7→ 2 2 7→ 1 3 7→ 2 ; ϕ 3 : 0 7→ 0 1 7→ 2 2 7→ 2 3 7→ 1 . F or Theorem 2, this follows from the fact that for a ny skew Bo olean alge bra with int er sections S , there exists another skew Bo olea n algebra with intersections T such that there a r e no in ter section-pres e rving morphisms from S to T at all (we hav e only to take car e that any stalk of T ∗ has cardinality sticktly less than the cardinality o f any stalk of S ∗ ). 3. The functors λ n , n ≥ 0 Let X b e a Bo ole a n space and let n ≥ 0 b e fixed. W e re gard the set { 0 , . . . , n + 1 } as a discrete top olog ical space. Let λ n ( X ) denote the set of all contin uous maps f from X to { 0 , . . . , n + 1 } such that the sets f − 1 (1), . . . , f − 1 ( n + 1) are compact. Define the binary op eratio ns ∧ and ∨ o n λ n ( X ) to b e induced by the op er ations ∧ and ∨ on the primitive skew Bo olea n algebra n + 2 . That is, for f , g ∈ λ n ( X ) we put ( f ∧ g )( x ) = f ( x ) ∧ g ( x ) , ( f ∨ g )( x ) = f ( x ) ∨ g ( x ) . With resp ect to ∧ and ∨ the set λ n ( X ) b ecomes a left-handed Bo olea n skew lattice. By adding to its signa ture the relative complement op era tion and the zero, we turn itno int o a left-handed skew Bo o lean alg e bra. Note that λ 0 ( X ) = X ∗ . It is immediate that the elements of λ n ( X ) ar e in a bijective co rresp ondence with o r dered ( n + 1)-tuples of pa irwise dis joint compact-o p en subsets of of S via the assignment f 7→ ( f − 1 (1) , . . . , f − 1 ( n + 1 )) . Remark 3 . Y et another r e alization of λ n ( X ) is by flags A n +1 ⊇ · · · ⊇ A 1 of c omp act-op en su bsets of X via the assignment f 7→ f − 1 ( { 1 , . . . , n + 1 } ) ⊇ f − 1 ( { 1 , . . . , n } ) ⊇ f − 1 (1) . L et A n +1 ⊇ · · · ⊇ A 1 and B n +1 ⊇ · · · ⊇ B 1 b e the flags c orr esp onding to f and g , r esp e ctively. I t is e asy t o verify, applying the defin itions of ∨ and ∧ , that the fl ag, c orr esp onding t o f ∨ g is C n +1 ⊇ · · · ⊇ C 1 , wher e C i = ( A i \ B n +1 ) ∪ B i for al l i , and the fl ag, c orr esp onding to f ∧ g is D n +1 ⊇ · · · ⊇ D 1 , wher e D i = A i ∩ B n +1 for al l i . F r om this description it fol lows that λ 1 ( X ) = ω ( A ( X )) , wher e A is the fun ctor fr om Subse ction 2.1 and ω : BA → Skew is the (version with pr op er morphisms of the) L e e ch-Spinks ω -functor [26] . Ther efo r e, the c onstru ction of λ 1 pr ovides a natur al interpr etation for the functor ω . Let us lo o k at the structure o f the skew Bo olean a lgebra λ n ( X ) in more detail. F or f ∈ λ n ( X ) by ˆ f : X → { 0 , 1 } we denote the ma p given by ˆ f − 1 (1) = f − 1 ( { 1 , . . . , n + 1 } ) . Lemma 4. L et f , g ∈ λ n ( X ) . The n f D g if and only if f − 1 ( { 1 , . . . , n + 1 } ) = g − 1 ( { 1 , . . . , n + 1 } ) . The Bo ole an algebr a λ n ( X ) / D is isomorphic t o the Bo ole an algebr a λ 0 ( X ) = X ∗ via the map [ f ] D 7→ ˆ f . Pr o of. By definition, f D g is equiv alent to f ∧ g = f and g ∧ f = g . Therefor e, f ( x ) ∧ g ( x ) = f ( x ) and g ( x ) ∧ f ( x ) = g ( x ) for all x ∈ X . Applying the description of the op era tion ∧ on n + 2 we se e that the latter equalities are equiv alent to the condition f ( x ) = 0 if and only if g ( x ) = 0 for all x ∈ X . This is clearly e q uiv alen t to f − 1 ( { 1 , . . . , n + 1 } ) = g − 1 ( { 1 , . . . , n + 1 } ). The s e c ond s tatement is straig htforward to verify .  8 GANNA KUDR Y A VTSEV A Lemma 5. The natur al p artial or der in λ n ( X ) is given by f ≥ g if and only if f − 1 ( i ) ⊇ g − 1 ( i ) for al l i = 1 , . . . , n + 1 . Pr o of. By definition, f ≥ g is equiv a lent to f ∧ g = g ∧ f = g . By the definition of the op eration ∧ o n λ n ( X ), the latter is equiv alent to f ( x ) ∧ g ( x ) = g ( x ) ∧ f ( x ) = g ( x ) for all x ∈ X , wher e the e qualities are in n + 2 . This is equiv alent to the condition g ( x ) = i implies f ( x ) = i for all i = 1 , . . . , n + 1.  Lemma 6. The skew Bo ole an algebr a λ n ( X ) has finite interse ctions. I f f , g ∈ λ n ( X ) then their interse ction f ∩ g is given by ( f ∩ g ) − 1 ( i ) = f − 1 ( i ) ∩ g − 1 ( i ) for al l i = 1 , . . . , n + 1 . Pr o of. Let h ∈ λ n ( X ) is given by h − 1 ( i ) = f − 1 ( i ) ∩ g − 1 ( i ) for all i = 1 , . . . , n + 1. Then f , g ≥ h by L e mma 5. Assume that f , g ≥ q and let i ∈ { 1 , . . . , n + 1 } . Then f − 1 ( i ) ⊇ q − 1 ( i ) and g − 1 ( i ) ⊇ q − 1 ( i ). It follows that h − 1 ( i ) ≥ q − 1 ( i ). This prov es that f ∩ g ex ists a nd e quals h .  It is well-kno wn that ultrafilters o f the Bo o lean algebra λ 0 ( X ) are the se ts { f ∈ λ 0 ( X ) : f ( x ) = 1 } . Consequently ultrafilters of the Bo olean algebra λ n ( X ) / D are the sets (1) N x = { [ f ] D ∈ λ n ( X ) / D : f ( x ) ∈ { 1 , . . . , n + 1 }} , x ∈ X . Let α : λ n ( X ) → λ n ( X ) / D b e the pro jection ma p. Lemma 7. F or every x ∈ X and i ∈ { 1 , . . . , n + 1 } t he set N x,i = { f ∈ λ n ( X ) : f ( x ) = i } is an ultr afilter of λ n ( X ) , and any ultr afilter is of this form. In addition, α ( N x,i ) = N x . Pr o of. Let f ∈ N x,i . Show that N x,i = X f ,N x . Let g ∈ N x,i . Since x ∈ f − 1 ( i ) ∩ g − 1 ( i ) then f ∩ g ∈ N x,i . Thus also [ f ∩ g ] D ∈ N x . This and f , g ≥ f ∩ g imply that g ∈ X f ,N x . Conv ersly , let g ∈ X f ,N x . Then there is some h ∈ λ n ( X ) with [ h ] D ∈ N x such that f , g ≥ h . Since f ≥ h and h ( x ) 6 = 0 it follows that h ( x ) = i by Lemma 5. Now, g ≥ h implies g ( x ) = i a gain by Lemma 5. It follows that g ∈ N x,i , as requir ed. The second claim is immediate.  Corollary 8. F or any ultra filter N x of λ n ( X ) / D we have α − 1 ( N x ) = n +1 [ i =1 N x,i . Conse quently, e ach stalk of the ´ etale sp ac e ( λ n ( X )) ∗ has c ar dinality n + 1 . Pr o of. Ultrafilters in λ n ( X ) that are over N x are of the form X f ,N x , where f ( x ) ∈ { 1 , . . . , n + 1 } . The sta temen t now follows from the equality N x,i = X f ,N x , where f ( x ) = i , establis hed in the pro of of Lemma 7.  Let X ( i ) = { N x,i } x ∈ X , i ∈ { 1 , . . . , n + 1 } . It follows from Lemma 7 that, as a set, ( λ n ( X )) ∗ is the union X (1) ∪ · · · ∪ X ( n +1) of ( n + 1) disjoint copies of X . Lemma 9. The sp ac e ( λ n ( X )) ∗ as a t op olo gic al sp ac e is a disjoint union of ( n + 1) c opies of the Bo ole an sp ac e X . Pr o of. Consider each X ( i ) as a top o logical s pa ce homeomor phic to X via the map N x,i 7→ x . The n ( λ n ( X )) ∗ can b e co nsidered as a dis joint union of the spaces X ( i ) . W e aim to show that this top olo gy on ( λ n ( X )) ∗ coincides with the dua l ´ eta le space topolo gy , whose construction is outlined in Subsection 2.4. T o show this, A DUALIZING OBJECT APPRO ACH TO NON-COMMUT A TIVE STONE DUALITY 9 it is enough to verify that the map π : ( λ n ( X )) ∗ → X given by N x,i 7→ x is a lo cal homeomor phism. Fix s o me N x,i and a compac t-op en set A in X such that x ∈ A . Cons ider the function f ∈ λ n ( X ) given by f − 1 ( i ) = A and f − 1 ( j ) = ∅ , j ∈ { 1 , . . . , n + 1 } \ { i } . It is clear that M ( f ) = ∪ y ∈ A N y ,i . This is a neig hborho o d of N x,i that is homeomor phic via π to A since a bas ic op en s ubs et of M ( f ) equals M ( g ), f ≥ g , and π ( M ( g )) = B ⊆ A , where g − 1 ( i ) = B .  W e now define λ n on morphis ms . Le t n ≥ 0 , g : X 1 → X 2 be a mor phism o f Bo olean s pa ces a nd f ∈ λ n ( X 2 ). W e put (2) λ n ( g )( f ) = f g It is ea sy to chec k that λ n ( g ) is a skew Bo o lean alg e br a morphism from λ n ( X 2 ) to λ n ( X 1 ) and that this makes λ n a contrav aria nt functor from the categ ory BS to the category Skew . W e finish this s ection by reco rding the following fact that will b e needed in Section 5 . Lemma 10. L et g : X 1 → X 2 b e a morphism of Bo ole an sp ac es. Then λ n ( g ) − 1 ( N x ) = N g ( x ) and λ n ( g ) − 1 ( N x,i ) = N g ( x ) , i for any x ∈ X 1 and i = 1 , . . . , n + 1 . Pr o of. W e hav e λ n ( g )([ f ] D ) = [ f g ] D . Therefor e, λ n ( g )([ f ] D ) ∈ N x if and only if [ f g ] D ∈ N x . By (1) this is equiv alent to f g ( x ) ∈ { 1 , . . . , n + 1 } . The latter means that [ f ] D ∈ N f ( x ) and implies the firs t equality . F or the second equality , observe that λ n ( g )( f ) ∈ N x,i ⇔ f g ( x ) ∈ N x,i ⇔ f ∈ N g ( x ) , i .  4. The functors Λ n and the ad junctions Λ n ⊣ λ n , n ≥ 0 Let S b e a s kew Bo olea n algebra , α : S → S / D the canonic a l pro jectio n and ˆ α the pro jection fr o m S ∗ to ( S/ D ) ∗ given by ˆ α ( X a,F ) = F . W e also fix n ≥ 0 . Let { 0 , 1 , . . . , n + 1 } S be the set of all maps from S to { 0 , 1 , . . . , n + 1 } . W e consider { 0 , 1 , . . . , n + 1 } a s a discr ete s pace and { 0 , 1 , . . . , n + 1 } S as a pro duct space. That is, a base of the top olo g y on { 0 , 1 , . . . , n + 1 } S is formed by the sets (3) U δ = \ t ∈ T { f ∈ { 0 , 1 , . . . , n + 1 } S : f ( t ) = δ ( t ) } , where T runs through the finite subsets of S and δ runs through the functions fr o m T into { 0 , 1 , . . . , n + 1 } . W e denote by Λ n ( S ) the set o f all mo r phisms from S to n + 2 in the categor y Skew . W e endove Λ n ( S ) with the subspace to p o logy inher ited fro m the pr o duct top ology o n the space { 0 , 1 , . . . , n + 1 } S . F or s ∈ S and i ∈ { 0 , 1 , . . . , n + 1 } put L ( s, i ) = { f ∈ Λ n ( S ) : f ( s ) = i } . Lemma 11. The set s L ( s, i ) , wher e s runs thr ough S and i ∈ { 1 , . . . , n + 1 } , form a su bb ase of the top olo gy on Λ n ( S ) . Pr o of. By the definition of subspace to p o logy and in view o f the base given by (3) we o btain that the sets V δ = U δ ∩ Λ n ( S ) = \ t ∈ T { f ∈ Λ n ( S ) : f ( t ) = δ ( t ) } , where T runs throug h the finite subsets of S and δ runs thro ugh the functions from T into { 0 , 1 , . . . , n + 1 } , form a base of the top ology on Λ n ( S ). Therefo r e, 10 GANNA KUDR Y A VTSEV A the top ology on Λ n ( S ) admits a subbase consisting from the sets L ( s, i ), s ∈ S , i ∈ { 0 , 1 , . . . , n + 1 } . Let s ∈ S . Show that the set L ( s, 0 ) can be expres s ed as a unio n of some of the sets L ( t, i ), where t ∈ S and i ∈ { 1 , . . . , n + 1 } . Le t A = { t ∈ Λ n ( S ) : α ( t ) ∧ α ( s ) = 0 } . W e aim to s how that (4) L ( s, 0 ) = [ t ∈ A n +1 [ i =1 L ( t, i ) . Let f ∈ L ( s, 0). Since f is non-z e ro, there is t ∈ S suc h that f ( t ) 6 = 0. Ob- serve that f ( s | α ( s ) ∧ α ( t ) ) = 0, sinc e s ≥ s | α ( s ) ∧ α ( t ) and f ( s ) = 0 . It follows that f ( t | α ( s ) ∧ α ( t ) ) = 0 as well since the tw o elements s | α ( s ) ∧ α ( t ) and t | α ( s ) ∧ α ( t ) are in the same D - class. It follows that f ( t | α ( t ) \ α ( s ) ) 6 = 0. W e obtain that t | α ( t ) \ α ( s ) ∈ A and f ∈ S n +1 i =1 L ( t | α ( t ) \ α ( s ) , i ), proving that L ( s, 0) ⊆ S t ∈ A S n +1 i =1 L ( t, i ). T o prove the r everse inclusio n, we let f ∈ S t ∈ A S n +1 i =1 L ( t, i ). W e have α ( s ∧ t ) = α ( s ) ∧ α ( t ) = 0, so that s ∧ t = 0 . It follows that f ( s ∧ t ) = f (0) = 0. On the other hand we hav e f ( s ∧ t ) = f ( s ) ∧ f ( t ). W e obtain that 0 = f ( s ) ∧ f ( t ) and f ( t ) 6 = 0 . It fo llows that f ( s ) = 0 and thus f ∈ L ( s, 0). This finis he s the pro of of (4).  Lemma 12. L et s ∈ S and i ∈ { 1 , . . . , n + 1 } . The set L ( s, i ) is a close d subset of the sp ac e { 0 , . . . , n + 1 } S . Conse quently, L ( s, i ) is a c omp act-op en subset of Λ n ( S ) . Pr o of. Let Y ∧ and Y ∨ denote the sets of functions fr o m S to { 0 , 1 , . . . , n + 1 } that preserve ∧ or ∨ , resp ectively . Let, further, Y 0 and Y s,i denote the sets of functions from S to { 0 , 1 , . . . , n + 1 } that map 0 to 0 or that ma p s to i , resp ectively . It can b e shown a pplying a standard arg ument (see, for example, pro o f o f Lemma 1, Chapter 34 , p. 326 of [14]) that each of the sets Y ∧ , Y ∨ , Y 0 and Y s,i is a closed subset of { 0 , 1 , . . . , n + 1 } S . It follows that L ( s, i ) is closed, to o, since L ( s, i ) = Y ∧ ∩ Y ∨ ∩ Y 0 ∩ Y s,i . F or the se cond cla im, observe that L ( s, i ) is compact in { 0 , . . . , n + 1 } S as a closed subset of a compact space. Therefore, L ( s, i ) is compac t- op en in Λ n ( S ).  Theorem 13. Λ n ( S ) is a Bo ole an sp ac e. Pr o of. Λ n ( S ) is Haus do rff as a subspace of the Hausdo e ff space { 1 , . . . , n + 1 } S . So, in view of Lemmas 11 and 1 2, we hav e that Λ n ( S ) is a Hausdo rff spa ce, in which compact-op en sets form a base of the top olog y . Thus Λ n ( S ) is a Bo olea n space.  W e now define Λ n on morphisms. Let h : S 1 → S 2 be a mo rphism o f skew Bo olean a lgebras. F or f ∈ Λ n ( S 2 ) we set (5) (Λ n ( h ))( f ) = f h. Lemma 14. Λ n ( h ) is a morphism of Bo ole an s p ac es fr om Λ n ( S 2 ) t o Λ n ( S 1 ) . Pr o of. It is immediate that for each f ∈ Λ n ( S 2 ) we have that (Λ n ( h ))( f ) ∈ Λ n ( S 1 ). T o show that Λ n ( h ) is a contin uous prop er ma p, it is eno ugh to verify that the se t (Λ n ( h )) − 1 ( L ( s, i )) is compact-o p en for any s ∈ S 2 and any i ∈ { 1 , . . . , n + 1 } . W e observe that f ∈ (Λ n ( h )) − 1 ( L ( s, i )) ⇐ ⇒ (Λ n ( h ))( f ) ∈ L ( s, i ) ⇐ ⇒ f h ∈ L ( s, i ) ⇐ ⇒ f ( h ( s )) = i ⇐ ⇒ f ∈ L ( h ( s ) , i ) , implying that (Λ n ( h )) − 1 ( L ( s, i )) = L ( h ( s ) , i ), and the statement follows.  It is straig htforward to c heck that the constructed assignments define a con- trav ar iant functor Λ n : Skew → BS . W e are now pr epared to for mulate a nd prove our adjunction theo rem. A DUALIZING OBJECT APPRO ACH TO NON-COMMUT A TIVE STONE DUALITY 11 Theorem 15. F or e ach n ≥ 0 the functor Λ n : Skew → BS op is t he left adjoint to the fun ctor λ n : BS op → Skew . F or a skew Bo ole an algebr a S the c omp onent η S of the u nit of the adjunction η is given by (6) η S ( a )( g ) = g ( a ) , a ∈ S, g ∈ Λ n ( S ) . Pr o of. It is immediate that η S is a morphism. Let S be a skew Bo olea n alg e br a, X a B o olean space and µ : S → λ n ( X ) a morphism of skew Bo olean alg ebras. Our aim is to show that there is a unique mor phism u : X → Λ n ( S ) of Bo olean space s such that µ = λ n ( u ) η S . F or each x ∈ X we put (7) u ( x )( s ) = µ ( s )( x ) , s ∈ S. Let us verify tha t u is a prop er contin uous map. F or this, we have to show that the inv erse image under u of a compa ct-op en subset of Λ n ( S ) is compa c t- op en in X . Since any compact-op en subs et is a finite unio n o f basic compact-o pe n sets, a nd any basic compact-op en s et is a finite intersection of the sets o f the form L ( s, i ), it is enough to verify that u − 1 ( L ( s, i )) is compact-op en in X for e a ch s ∈ S and each i ∈ { 1 , . . . , n + 1 } . W e have x ∈ u − 1 ( L ( s, i )) ⇔ u ( x ) ∈ L ( s, i ) ⇔ u ( x )( s ) = i ⇔ µ ( s )( x ) = i ⇔ x ∈ µ ( s ) − 1 ( i ) . Since µ ( s ) ∈ λ n ( X ) then µ ( s ) − 1 ( i ) is compact-o p en by the definition of λ n ( X ). So u − 1 ( L ( s, i )) is compac t-op en, to o . V erify the equality µ = λ n ( u ) η S . F or ea ch a ∈ S and x ∈ X we have ( λ n ( u ) η S ( a ))( x ) = η S ( a ) u ( x ) (by (2)) = u ( x )( a ) (b y (6)) = µ ( a )( x ) (b y (7)) , as r equired. The uniqueness of u is shown in a standar d wa y , we leav e the details to the reader.  Remark 1 6. L et S b e a skew Bo ole an algebr a and A , S funct ors fr om Subse ction 2.1. We set Ω = A Λ 1 : Skew → BA . It fol lows fr om the or em 15 that Ω is the left adjoint to the functor ω = λ 1 S : BA → Skew fr om [26] . The c onstruction of Λ 1 yields the ful l description of (the version with pr op er morphisms of ) Ω , that had b e en questione d in [26] . The statement, that follows, pr ovides a wa y to faithfully re present a skew Bo olean algebra S as a subalg ebra of a well-understoo d skew Bo olean algebra with int er sections λ n Λ n ( S ) who se dual space has ( n + 1)-e le men t stalks. If n = 1 we obtain a faithful repres ent a tion of S in a skew B o olean algebra λ 1 Λ 1 ( S ) w ith in- tersections whose dual space has has 2- e le ment stalks. This a lgebra has , roug hly sp eaking, a ‘lo w degree o f non-comm utativity’ (but note that its underlying Boo le an algebra (Λ 1 ( S )) ∗ is rather huge in compariso n with the underlying Bo olean algebr a S/ D of S ). In the terminology of Leech and Spinks [2 6], λ 1 Λ 1 ( S ) is a minimal skew Bo ole an c over of the Bo olean algebra (Λ 1 ( S )) ∗ . W e refer the rea de r to [2 6] for an int er esting discussio n of different approa ches to the definition of the notion of a minimal skew Bo olean c ov er. Theorem 17 . If n ≥ 1 then the map η S is inje ctive, and is ther efor e a faithful r epr esentation of S as a sub algebr a of λ n Λ n ( S ) . 12 GANNA KUDR Y A VTSEV A Pr o of. Let a, b ∈ S and a 6 = b . This implies tha t M ( a ) 6 = M ( b ). Then w e can assume that there is an ultra filter X a,F in S such that b 6∈ X a,F (note that a ∈ X a,F by de finitio n of X a,F ). Let x b e the only p oint of the s pace 2 ∗ . The ma p g : 2 ∗ → ( S/ D ) ∗ given by x 7→ F is a contin uous map. As follows from the discussion in the paragr aph a fter Theor em 1, w e can construct the cohomomorphism k : S ∗ → ( n + 2 ) ∗ ov er g given by k x ( X a,F ) = 1 a nd k x ( G ) = 2 for a ny G ∈ S ∗ F , G 6 = X a,F . Since b 6∈ X a,F then k ( b ) 6 = 1 . It follows that the morphism from S to n + 2 , that corres po nds to k , has different v alues at a a nd b . This implies that η S ( a )( k ) 6 = η S ( b )( k ).  5. The algebras of the monads of the adjunctions Λ n ⊣ λ n Let n ≥ 0 b e fixed throug hout this section. Prelimina r ies on monads can be found in, e.g., [1 , Cha pter 10 ] o r [2 7, Cha pter VI]. Let ( T , η , µ ) be the monad over the category Skew that arises from the adjunction Λ n ⊣ λ n . W e hav e T = λ n Λ n , η is the unit of the adjunction given b y (6), a nd µ = λ n ǫ Λ n is a natura l transforma tion fro m T 2 to T , where ǫ : 1 BS → Λ n λ n is the counit of the a djunction. The following sta tement is str aightforw ar d to verify . Lemma 18. L et X b e a Bo ole an sp ac e. Then (8) ǫ X ( x )( f ) = f ( x ) , x ∈ X , f ∈ λ n ( X ) . Let S b e a skew Bo o le a n algebr a. In this s ection we will need to work with skew Bo olean algebr as λ n Λ n λ n Λ n ( S ), λ n Λ n ( S ) and mor phis ms betw een them. These ob jects a re so mewhat complica ted, and to pr o ceed, we fir st pr op ose a convenien t wa y of working with them. Our insights will b e ba sed on the top ologica l r epresen- tation o f mo rphisms of skew Bo o lean algebra s by morphisms of their dual spa ces. W e start from a us eful description of the p oints of the space Λ n ( S ). Lemma 19. Ther e is a bije ctive c orr esp ondenc e b etwe en the p oints of the s p ac e Λ n ( S ) and elements of the set (9) { ( F, f ) : F ∈ ( S/ D ) ∗ , f ∈ { 1 , . . . , n + 1 } S ∗ F } . Pr o of. Let a b e the o nly po int of the Bo o lean space (( n + 2 ) / D ) ∗ = 2 ∗ . Let h ∈ Λ n ( S ). Then h − 1 induces a contin uous prop er map, tha t we denote also by h − 1 , from 2 ∗ to ( S/ D ) ∗ . Let F h ∈ ( S/ D ) ∗ be suc h that h − 1 ( a ) = F h . The n e h has the only one c o mpo nent e h a : S ∗ F h → ( n + 2 ) ∗ a = { 1 , . . . , n + 1 } . Show that the constr ucted cor resp ondence h 7→ ( F h , e h a ) is bijective. Assume that ( F h , e h a ) = ( F g , e g a ). This immediately implies that h = g a nd that h − 1 ( i ) = g − 1 ( i ) for all i = 1 , . . . , n + 1. This prov es that the constructed corr esp ondence is injective. F or the reverse directio n, let F ∈ ( S/ D ) ∗ and let f ∈ { 1 , . . . , n + 1 } S ∗ F . It is clea r that the map g from 2 ∗ to ( S/ D ) ∗ given by g ( a ) = F is pr op er and contin uous and so f is the o nly c omp onent o f a g -cohomomor phism from S ∗ to n + 2 ∗ . This and Theorem 1 imply that the constructed assignment is surjective.  W e now characteriz e the p oints of the spa ce Λ n λ n Λ n ( S ). Let T n +1 denote the set of a ll tra nsformations o f the set { 1 , . . . , n + 1 } . Lemma 20. Ther e is a bije ctive c orr esp ondenc e b etwe en the p oints of the s p ac e Λ n λ n Λ n ( S ) and elements of t he set (10) { ( F, f , g ) : F ∈ ( S/ D ) ∗ , f ∈ { 1 , . . . , n + 1 } S ∗ F , g ∈ T n +1 } . A DUALIZING OBJECT APPRO ACH TO NON-COMMUT A TIVE STONE DUALITY 13 Pr o of. Let h ∈ Λ n λ n Λ n ( S ). T ha t is, h is a skew Bo o lean a lgebra mo rphism from λ n Λ n ( S ) to n + 2 . Then h is a Bo o lean algebr a morphism fr o m ( λ n Λ n ( S )) / D to 2 . W e hav e that h − 1 (1) is an ultrafilter o f ( λ n Λ n ( S )) / D and so is equa l to some N G , where G ∈ Λ n ( S ), see (1). The only comp onent of the cohomo morphism ˜ h is a map from ( λ n Λ n ( S )) ∗ N G to { 1 , . . . , n + 1 } . This map defines g ∈ T n +1 where N G,i 7→ g ( i ) for all i ∈ { 1 , . . . , n + 1 } . By Lemma 1 9 we hav e that G corres p o nds to a pa ir ( F , f ). Hence h corr e sp onds to the triple ( F , f , g ). That this cor resp ondence is bijective by similar a r guments as in the pro o f of Lemma 19.  In what follows, we often identify the p oints of the spaces Λ n ( S ) a nd Λ n λ n Λ n ( S ) with the elements of the sets (9) and (10), r esp ectively , via the constructions given in the pro ofs of Le mma s 1 9 a nd 20. T o pro ceed, we need to establish how the action of maps of our interest lo oks like in this notation. W e do this in the following three lemmas. By id we deno te the ident ity tr ansformatio n of the set { 1 , . . . , n + 1 } . Lemma 21. ǫ Λ n ( S ) ( F, f ) = ( F , f , id ) . Pr o of. Let ϕ ∈ Λ n ( S ) and assume that ǫ Λ n ( S ) ( ϕ ) = ϕ ′ . By Lemma 18 ϕ ′ ( g ) = g ( ϕ ), g ∈ λ n Λ n ( S ). Thus for i = 1 , . . . , n + 1 we hav e g ∈ ϕ ′− 1 ( i ) ⇔ ϕ ∈ g − 1 ( i ) ⇔ g ∈ N ϕ,i . Therefore ϕ ′− 1 ( i ) = N ϕ,i . Thus also ϕ ′ − 1 (1) = N ϕ . This and the co nstructions in the pro ofs o f Le mma s 19 a nd 2 0 imply that if ϕ cor resp onds to ( F, f ) that ϕ ′ corres p o nds to ( F , f , id ), as required.  Lemma 22. η S − 1 ( N ( F, f ) ) = F and the c omp onent f η S N ( F,f ) of f η S is given by x 7→ N ( F, f ) ,f ( x ) , x ∈ S ∗ F . Pr o of. Let ϕ ∈ Λ n ( S ). Then fo r a ∈ S and i = 1 , . . . , n + 1 , applying the definition of N ϕ,i and (6), we have a ∈ η − 1 S ( N ϕ,i ) ⇔ η S ( a ) ∈ N ϕ,i ⇔ η S ( a )( ϕ ) = i ⇔ ϕ ( a ) = i ⇔ a ∈ ϕ − 1 ( i ) . Thu s η − 1 S ( N ϕ,i ) = ϕ − 1 ( i ). If ϕ corr esp onds to the pair ( F , f ) via the constr uction in the pro of o f Lemma 19 then the only comp onent o f e ϕ is f : S ∗ F → { 1 , . . . , n + 1 } . W e therefore hav e f η S − 1 ( N ( F, f ) ,i ) = f − 1 ( i ), and the sta tement follows.  Lemma 23 . µ S − 1 = ǫ ∗∗ Λ n ( S ) , that is, µ S − 1 ( N x ) = N ǫ Λ n ( S ) ( x ) and the c omp onent f µ S N ( F,f ) is given by N ( F, f ,id ) ,i 7→ N ( F, f ) ,i . Pr o of. The statement follows from µ S = λ n ( ǫ Λ n ( S ) ) applying Lemma 10 and Lemma 2 1.  W e are now prepa red to characterize the algebras for the mo nad ( T , η , µ ). By definition an algeb r a of t he m onad ( T , η , µ ) (or just a T -algebr a ) is a pair ( S, γ ) with S is a skew Bo olean a lgebra and γ : T ( S ) → S is a mo r phism such that the following diag rams co mm ute: T 2 ( S ) T ( S ) S T ( S ) T ( S ) S S T ( γ ) γ µ S γ η S 1 S γ 14 GANNA KUDR Y A VTSEV A A m orphism of T -algebr as h : ( S 1 , γ ) → ( S 2 , δ ) is a mo r phism h : S 1 → S 2 such that the following dia gram co mm utes: T ( S 1 ) T ( S 2 ) S 1 S 2 T ( h ) h γ γ Theorem 24. (1) A p air ( S, γ ) is an algebr a for the monad ( T , η , µ ) if and only if S = λ n ( X ) for some Bo ole an s p ac e X and γ = λ n ǫ X . (2) A map h : λ n ( X 1 ) → λ n ( X 2 ) is a morph ism of T -algebr as if and only if h = λ n ( f ) for some morphism f : X 1 → X 2 . Pr o of. The equality 1 S = γ η S implies the equality 1 ( S/D ) ∗ = η S − 1 γ − 1 . Applying Theorem 1 we have the following commuting diagr ams: S ∗ ( λ n Λ n ( S )) ∗ ( S/ D ) ∗ ( λ n Λ n ( S ) / D ) ∗ S ∗ ( S/ D ) ∗ f η S e γ 1 S ∗ η S − 1 1 ( S/ D ) ∗ γ − 1 Let F ∈ ( S/ D ) ∗ . Assume γ − 1 ( F ) = N ( G,f ) . By Lemma 2 2 w e hav e η S − 1 ( N ( G,f ) ) = G . Since the second diagram ab ove co mm utes we obtain that F = G . W e fix f = f F : S ∗ F → { 1 , . . . , n + 1 } such that γ − 1 ( F ) = N ( F, f ) . Now, since the fir st diagr am ab ove commutes and applying Lemma 2 2, we see tha t for x ∈ S ∗ F it ho lds (11) x f η S N ( F,f ) − − − − − → N ( F, f ) ,f ( x ) f γ F − − − − − → x. The la tter shows that the ma p f η S N ( F,f ) and also the restrictio n of the map map f γ F to the image of f η S N ( F,f ) are injective. Injectivit y of f η S N ( F,f ) implies that f must b e injectiv e, to o . By definition a nd applying Theorem 1 w e hav e the following diagr ams (wher e Λ n ( γ ) ∗∗ ( N x ) = N Λ n ( γ ) ( x ) ): ( λ n Λ n λ n Λ n ( S )) ∗ ( λ n Λ n ( S )) ∗ ( λ n Λ n λ n Λ n ( S ) / D ) ∗ ( λ n Λ n ( S ) / D ) ∗ ( λ n Λ n ( S )) ∗ S ∗ ( λ n Λ n ( S ) / D ) ∗ ( S/ D ) ∗ ^ λ n Λ n ( γ ) e γ f µ S e γ Λ n ( γ ) ∗∗ γ − 1 ǫ ∗∗ Λ n ( S ) γ − 1 Our goal now is to describ e the a ction o f Λ n ( γ ) and λ n Λ n ( γ ) if the elemen ts of the spaces ar e enco ded as is desc rib ed in the pro o fs of Lemmas 19 a nd 20. L e t g ∈ Λ n ( S ). Its co rresp onding pair is ( F , ˆ g ), where F = g − 1 (1) and ˆ g is the only comp onent of e g . Applying Λ n ( γ )( g ) = g γ and the co nstruction in the pro of of Lemma 2 0 it follows tha t we have (12) ( F, ˆ g ) Λ n ( γ ) − − − − − → ( F , f , ˆ g e γ F ) . A DUALIZING OBJECT APPRO ACH TO NON-COMMUT A TIVE STONE DUALITY 15 This and L e mma 10 yield (13) N ( F, f , ˆ g f γ F ) ,i ^ λ n Λ n ( γ ) ( F, ˆ g ) − − − − − − − − → N ( F, ˆ g ) ,i for e a ch i ∈ { 1 , . . . , n + 1 } . It follows that for F ∈ ( S/ D ) ∗ we have F γ − 1 − − − − − → N ( F, f ) Λ n ( γ ) ∗∗ − − − − − → N ( F, f ,f e γ F ) . On the o ther hand, observe that ( λ n Λ n ( S ) / D ) ∗ is isomorphic to Λ n ( S ) and ( λ n Λ n λ n Λ n ( S ) / D ) ∗ is isomo rphic to Λ n λ n Λ n ( S ) via the map N x 7→ x . Applying Lemma 2 1, for F ∈ ( S/ D ) ∗ we have F γ − 1 − − − − − → N ( F, f ) ǫ ∗∗ Λ n ( S ) − − − − − → N ( F, f ,id ) . Since the tw o expressions above must be equal, it follows that f is a bijection a nd e γ F = f − 1 . Hence | S ∗ F | = n + 1. Ther efore, all the stalks of S ∗ are ( n + 1)-element. Let F ∈ ( S/ D ) ∗ be fixed. Since | S ∗ F | = n + 1 and f is a bijection, we can enum er ate the germs o f S ∗ F so tha t S ∗ F = { F (1) , . . . , F ( n +1) } a nd f maps each F ( i ) to i . In this nota tio n we hav e (14) e γ F ( N ( F, f ) ,i ) = F ( i ) . Let i ∈ { 1 , . . . , n + 1 } . W e define the i -th layer of the spa ce S ∗ as the set S ∗ ( i ) = { F ( i ) : F ∈ ( S/ D ) ∗ } . W e pr o ceed to show that S ≃ λ n (( S/ D ) ∗ ). It is conv enient to work with the isomorphic copy S ∗∗ of S . Let s ∈ S . Observe that f η S ( M ( s ) ∩ S ∗ ( i ) ) is the i -th lay er of M ( η S ( s )). Since the pro jection of the latter set is co mpact-op en then its inv erse imag e under γ is co mpact-op en, to o. It follows that the map F ( i ) 7→ N F, i , where F ∈ ( S/ D ) ∗ and i ∈ { 1 , . . . , n + 1 } , induces an isomorphism be tw ee n S ∗∗ and λ n (( S/ D ) ∗ ) ∗∗ . Let X = ( S/ D ) ∗ . T o esta blis h the eq ua lity γ = λ n ǫ X we apply arguments similar to those in the pro ofs of Lemma 21 and Lemma 23 to observe that the actio n o f ] λ n ǫ X F coincides with the action of e γ F given in (14). W e are left to pr ov e the claim ab out morphisms. Assume first that f : X 2 → X 1 is a mor phism of Bo olea n spaces. It is straightforw ar d to verify that λ n ( f ) : λ n ( X 1 ) → λ n ( X 2 ) is a morphism of T -algebr as from ( T ( λ n ( X 1 )) , λ n ǫ X 1 ) to ( T ( λ n ( X 2 )) , λ n ǫ X 2 ). W e now as sume tha t h : λ n ( X 1 ) → λ n ( X 2 ) is a mor phism of T -a lgebras from ( T ( λ n ( X 1 )) , λ n ǫ X 1 ) to ( T ( λ n ( X 2 )) , λ n ǫ X 2 ). Since ( λ n ( X i ) / D ) ∗ ≃ X i , i = 1 , 2 , then h − 1 induces a mor phism, ˆ h , from X 2 to X 1 . W e show that h = λ n ( ˆ h ). Le t ( F, f ) ∈ Λ n ( S 2 ), w he r e F ∈ X 2 and f ∈ { 1 , . . . , n + 1 } ( λ n ( X 2 )) ∗ F . Then ( F, f ) Λ n ( h ) − − − − − → ( h − 1 ( F ) , f e h F ) and there fo re (15) N ( h − 1 ( F ) , f e h F ) ,i ^ λ n Λ n ( h ) N ( F,f ) − − − − − − − − → N ( F, f ) ,i for a ll i = 1 , . . . , n + 1 . Let X be a Bo olean s pace. Similar ly as it was done in Lemma 2 0 we establish a bijection betw een the p oints of Λ n λ n ( X ) and pa irs ( F , f ), F ∈ X , f ∈ T n +1 . If g ∈ Λ n λ n ( X ), then g : X ∗ → 2 determines F and the only comp o nent o f e g determines f . By Lemmas 10 in this notation we hav e (16) N ( F, id ) ,i ^ λ n ǫ X 2 F − − − − − → N ( F, i ) . 16 GANNA KUDR Y A VTSEV A F rom (15),(16) and the commutativ e diagram λ n Λ n λ n ( X 1 ) λ n Λ n λ n ( X 2 ) λ n ( X 1 ) λ n ( X 2 ) λ n Λ n ( h ) h λ n ǫ X 1 λ n ǫ X 2 we see that ^ λ n ǫ X 1 is defined on a ll N ( h − 1 ( F ) , e h F ) ,i , F ∈ X 2 , 1 ≤ i ≤ n + 1, and it m ust b e e h F = id . It fo llows that the action of e h F is g iven b y N h − 1 ( F ) , i 7→ N F, i and th us h = λ n ( ˆ h ). The pro of is complete.  Let λ n ( BS ) be the catego ry whose ob jects are λ n ( X ), where X is a Boo lean space, and who se a rrows ar e λ n ( f ), f is a mor phism of Bo olean spac e s. Corollary 25. The Eilenb er g-Mo or e c ate gory of t he monad ( T , η , µ ) is isomorphic to the c ate gory λ n ( BS ) . Conse qu ently, the adjunction Λ n ⊣ λ n is monadic for every n ≥ 0 . Pr o of. The first s ta tement fo llows from Theor em 24. The second statement holds bec ause the categor y λ n ( BS ) is obviously isomor phic to the catego ry BS op .  Corollary 26. The c ate gory λ n ( BS ) is a r efle ctive su b c ate gory of the the c ate gory Skew . The r efle ctor is given by t he fun ct or Λ n λ n . References [1] S. Awodey , Cate gory the ory , Oxf ord Uni v ers i t y Press, 2006. [2] A. Bauer, K. Cvetk o-V ah, Stone dualit y for ske w Bo olean algebras with i nt ersections, to appear i n Houston J. Math. [3] A. Bauer, K. Cve tko-V ah, M. Gehrke, S. v an Go ol, G. Kudrya vtsev a, A non-commutativ e Priestley duality , ar X iv:1206:5848v1. [4] R. J. Bignall, J. E. Leech, Sk ew Bo olean algebras and discriminator v arieties, Algebr a Uni- versalis 33 (1995), 387–398. [5] G. E. Bredon, She af The ory , Grad. T exts in Math., vol. 170, Springer-V erlag, 1997. [6] D. M. Clark, B. A. Da vey , Natur al dualities for the working algebr aist , Cambridge Univ ersi t y Press, 1998. [7] S. Burris, H. P . Sank appana v ar, A Course in Universal Algebr a , Grad. T exts in Math., v ol. 78, Springer-V erlag, 1981. [8] W. H. Cornis h, Bo olean skew algebras, A ct a M ath. Ac ad. Sci. Hungar. 36 (1980), 281–291. [9] H. P . Do ctor, The catego r ies of Bo olean l attices, Bo olean ri ngs and Bo olean spaces. Canad. Math. Bul l. 7 (1964), 245–252. [10] J. F unk, Semigroups and toposes, Semigr oup F orum 75 (2007), no. 3, 481-520. [11] J. F unk, P . Hofstra, T op os theoretic asp ects of semigroup actions, The ory Appl. Cate g. 24 (2010), no. 6, 117–147. [12] J. F unk, M. V. Lawson, B. Steinberg, Characterizations of Morita equiv alent inv erse semi- groups, J. Pur e Appl. Algebr a 215 (2011), no. 9, 2262–2279. [13] J. F unk, B. Steinberg, T he universal cov ering of an inv erse semigroup, Appl. Cate g. Structur es 18 (2010), no. 2, 135–163, [14] S. Giv an t, P . Halm os, Intr o duction to Bo ole an Algebr as , U ndergrad. T exts Math., Springer, 2009. [15] P . T. Johnstone, Stone sp ac es , Cambridge Unive r sity Press, 1986. [16] G. Kudrya vtsev a, A refinemen t of Stone duali ty to skew Boolean algebras, Alg. Uni versalis , 2012, D OI 10.1007/s0001 2-012-0192-1. [17] G. Kudry avtse v a, M. V. La wson, The structure of generalized inv erse semigroups, [18] G. Kudrya vtsev a, M. V. La ws on, Stone duality for generalized inv erse semigroups, in prepa- ration. A DUALIZING OBJECT APPRO ACH TO NON-COMMUT A TIVE STONE DUALITY 17 [19] M. V. Lawson, A non-commut ative generalization of Stone duali t y , J. Aus. Math. So c. 88 (2010), 385–404. [20] M. V. La wson, Non-commutativ e Stone duali ty: inv erse semigroups, top ological group oids and C ⋆ -algebras, accepte d by IJA C , arXiv1104.1054v1. [21] M. V. Lawson, D. H. Lenz, Ps eudogroups and their ´ etale group oids, [22] J. Leec h, Skew lattices in ri ngs, Alg. Unive rsalis 26 (1989), 48–72. [23] J. Leec h, Skew Bo olean Algebras, Alg. Universalis 27 (1990), 497–506. [24] J. Leec h, Normal Skew Lattices, Semigro up F oru m 44 (1992), 1–8. [25] J. Leech , Recen t developmen ts in the theory of skew l attices, Semigr oup F orum 52 (1996), 7–24. [26] J. Leec h, M . Spinks, Skew Bo olean algebras deri ve d f rom Bo olean algebras, Algebr a Univer- salis 5 8 (2008), no.3, 287–302. [27] S. Mac Lane, Categories for the working mathematician, Grad. T exts in Math., v ol. 5, Springer-V erlag, 1998. [28] S. M ac Lane, I. Mo erdij k, She aves in ge ometry and lo gic. A first intr o duction to top os the ory , Springer-V erlag, 1994. [29] H.-E. Porst, W. Tholen, Concrete dualities, Cate g ory the ory at work (Bremen, 1990), 111– 136. [30] P . Resende, ´ Etale groupoids and their quan tales, A dv. Math. 208 (2007) 147–209. [31] P . Resende, Le c t ur es on ´ etale gr oup oids, inve rse semigr oups and quantales , preprin t a v ailable at ww w . math.ist.utl.pt [32] B. Steinberg, Strong Mori ta equiv alence of i nv erse semigr oups, Houston J. Math. 37 (2011), 895-927. [33] M. H. Stone, Applications of the Theory of Boolean Ri ngs to General T opology . T r ans. A mer. Math. So c , 41 (1937), 375–481. G. K.: University of Ljubljana, F acul ty of Computer and In forma tion Science, Tr ˇ za ˇ ska cest a 25, SI-100 1, Ljubljana, S LO VENIA. E-mail addr ess : ganna.kud ryavtseva@fri.uni-lj.si