aa r X i v : . [ m a t h . GN ] S e p HOFMANN-LAWSON DUALITY FOR LOCALLY SMALLSPACES
ARTUR PIĘKOSZDepartment of Applied Mathematics, Cracow University of Technology,ul. Warszawska 24, 31-155 Kraków, Polandemail: [email protected]
Abstract.
We prove versions of the spectral adjunction, a Stone-typeduality and Hofmann-Lawson duality for locally small spaces with boundedcontinuous mappings. Introduction
Hofmann-Lawson duality belongs to the most important dualities in latticetheory. It was stated in [7] (with the reference to the proof in [6]). Its versionsfor various types of structures are numerous (see, for example, [3]).The concept of a locally small space comes from that of Grothendiecktopology through generalized topological spaces in the sense of Delfs andKnebusch (see [2, 11, 14]). Locally small spaces were used in o-minimal ho-motopy theory ([2, 13]) as underlying structures of locally definable spaces.A simple language for locally small spaces was introduced and used in [12]and [14], compare also [16]. It is analogical to the language of Lugojan’s gen-eralized topology ([9]) or Császár’s generalized topology ([1]) where a familyof subsets of the underlying set is satisfying some, but not all, conditions fora topology. However treating locally small spaces as topological spaces withadditional structure seems to be more useful.While Stone duality for locally small spaces was discussed in [15], weconsider in this paper the spectral adjunction, a Stone-type duality andHofmann-Lawson duality.The set-theoretic axiomatics of this paper is what Saunders Mac Lane callsthe standard Zermelo-Fraenkel axioms (actually with the axiom of choice)for set theory plus the existence of a set which is a universe, see [8], page 23.
Notation.
We shall use a special notation for family intersection U ∩ V = { U ∩ V : U ∈ U , V ∈ V } . Date : September 8, 2020.2010
Mathematics Subject Classification.
Primary: 06D22, 06D50. Secondary: 54A05,18F10.
Key words and phrases.
Hofmann-Lawson duality, Stone-type duality, locally smallspace, spatial frame, continuous frame. Preliminaries on frames and their spectra.
First, we set the notation and collect basic material on frame theory thatcan be found in [4, 5, 7, 10, 18].
Definition 1. A frame is a complete distributive lattice L satisfying the(possibly infinite) distributive law a ∧ W i ∈ I b i = W i ∈ I ( a ∧ b i ) for a, b i ∈ L and i ∈ I . Definition 2. A frame homomorphism is a lattice homomorphism betweenframes preserving all joins. The category of frames and frame homomor-phisms will be denoted by Frm . The category of frames and right Galoisadjoints of frame homomorphisms will be denoted by
Loc . Definition 3.
For a frame L , its spectrum Spec ( L ) is the set of non-unitprimes of L : Spec ( L ) = { p ∈ L \ { } : ( p = a ∧ b ) = ⇒ ( p = a or p = b ) } . For a ∈ L , we define ∆ L ( a ) = { p ∈ Spec ( L ) : a p } . For a subset S ⊆ L , we set ∆ L ( S ) = { ∆ L ( a ) : a ∈ S } . Fact 4 ([4, Prop. V-4.2]) . For any frame L and a, b, a i ∈ L for i ∈ I , wehave ∆ L (0) = ∅ , ∆ L (1) = Spec ( L ) , ∆ L ( _ i ∈ I a i ) = [ i ∈ I ∆ L ( a i ) , ∆ L ( a ∧ b ) = ∆ L ( a ) ∩ ∆ L ( b ) . Consequently, the mapping ∆ L : L ∋ a ∆ L ( a ) ∈ ∆ L ( L ) is a surjectiveframe homomorphism. Definition 5.
The set ∆ L ( L ) is a topology on Spec ( L ) , called the hull-kerneltopology . Remark 6.
In this paper (as opposed to [15]), being sober implies being T (Kolmogorov). Fact 7 ([10, Ch. II, Prop. 6.1], [4, V-4.4]) . For any frame L , the topologicalspace ( Spec ( L ) , ∆ L ( L )) is sober. Definition 8.
A frame L is spatial if Spec ( L ) order generates (or: inf-generates ) L , which means that each element of L is a meet of primes. Fact 9 ([10, Ch. II, Prop. 5.1]) . If L is a spatial frame, then ∆ L is anisomorphism of frames. Definition 10 ([7, p. 286]) . For a, b ∈ L , we say that b is well-below a (or: b is way below a ) and write b ≪ a if for each (up-)directed set D ⊆ L suchthat a ≤ sup D there exists d ∈ D such that b ≤ d . Definition 11.
A frame is called continuous if for each element a ∈ L wehave a = _ { b ∈ L : b ≪ a } . OFMANN-LAWSON DUALITY FOR LOCALLY SMALL SPACES 3
Fact 12 ([10, Ch. VII, Prop. 6.3.3], [4, I-3.10]) . Every continuous frame isspatial.
The following two facts lead to Hofmann-Lawson duality.
Fact 13 ([4, Thm. V-5.5], [10, Ch. VII, 6.4.2]) . If L is a continuous frame,then ( Spec ( L ) , ∆ L ( L )) is a sober locally compact topological space and ∆ L isan isomorphism of frames. Fact 14 ([4, Thm. V-5.6], [10, Ch. VII, 5.1.1]) . If ( X, τ X ) is a sober locallycompact topological space, then τ X is a continuous frame. Finally, we recall the classical Hofmann-Lawson duality theorem.
Theorem 15 ([18, Thm. 2-7.9], [7, Thm. 9.6]) . The category
ContFrm ofcontinuous frames with frame homomorphisms and the category
LKSob oflocally compact sober spaces and continuous maps are dually equivalent. Categories of consideration.
Now we present the basic facts on the theory of locally small spaces.
Definition 16 ([14, Definition 2.1]) . A locally small space is a pair ( X, L X ) ,where X is any set and L X ⊆ P ( X ) satisfies the following conditions:(LS1) ∅ ∈ L X ,(LS2) if L, M ∈ L X , then L ∩ M, L ∪ M ∈ L X ,(LS3) ∀ x ∈ X ∃ L x ∈ L X x ∈ L x (i. e., S L X = X ).The family L X will be called a smopology on X and elements of L X will becalled small open subsets (or smops ) in X . Remark 17.
Each topological space ( X, τ X ) may be expanded in many waysto a locally small space by choosing a suitable basis L X of the topology τ X such that L X is a sublattice in τ X containg the empty set. Definition 18 ([14, Def. 2.9]) . For a locally small space ( X, L X ) we definethe family L woX = the unions of subfamilies of L X of weakly open sets. Then L woX is the smallest topology containing L X . Example 19.
The nine families of subsets of R from Example 2.14 in [14](compare Definition 1.2 in [16]) are smopologies and share the same familyof weakly open sets (the natural topology on R ). Analogically, Definition 4.3in [17] shows many generalized topological spaces induced by smopologieson R with the same family of weakly open sets (the Sorgenfrey topology). Definition 20 ([15, Def. 31]) . A locally small space ( X, L X ) is called T (or Kolmogorov ) if the family L X separates points , which means ∀ x, y ∈ X ( x = y ) = ⇒ ∃ V ∈ L X | V ∩ { x, y } | = 1 . Definition 21.
Assume ( X, L X ) and ( Y, L Y ) are locally small spaces. Thena mapping f : X → Y is: ( a ) bounded ([14, Definition 2.40]) if L X is a refinement of f − ( L Y ) , ( b ) continuous ([14, Definition 2.40]) if f − ( L Y ) ∩ L X ⊆ L X , ( c ) weakly continuous if f − ( L woY ) ⊆ L woX . A. PIĘKOSZ
The category of locally small spaces and their bounded continuous mappingsis denoted by
LSS ([14, Remark 2.46]). The full subcategory of T locallysmall spaces is denoted by LSS ([15, Def. 33]). Definition 22.
We have the following full subcategories of
LSS :(1) the category SobLSS generated by the topologically sober objects ( X, L X ) , i.e., such object that the topological space ( X, L woX ) is sober(compare [16, Def. 3.1]).(2) the category LKSobLSS generated by the topologically locally com-pact sober objects.
Example 23.
For X = R , we take L lin = the smallest smopology containing all straight lines, L alg = the smallest smopology containing all proper algebraic subsets.Then ( R , L lin ) and ( R , L alg ) are two different topologically locally compactsober locally small spaces with the same topology of weakly open sets (thediscrete topology).Now we introduce some categories constructed from Frm . Definition 24.
The category
FrmS consists of pairs ( L, L s ) with L a frameand L s a sublattice with zero sup-generating L (this means: every member of L is the supremum of a subset of L s ) as objects and dominating compatibleframe homomorphisms h : ( L, L s ) → ( M, M s ) as morphisms. Here a framehomomorphism h : L → M is called(1) dominating if ∀ m ∈ M s ∃ l ∈ L s h ( l ) ∧ m = m (then we shall also say that h ( L s ) dominates M s ).(2) compatible if ∀ m ∈ M s ∀ l ∈ L s h ( l ) ∧ m ∈ M s (then we shall also say that h ( L s ) is compatible with M s ). Remark 25. If h : L → M is a frame homomorphism satisfying h ( L s ) = M s ,then h : ( L, L s ) → ( M, M s ) is dominating compatible. Definition 26.
The category
LocS consists of pairs ( L, L s ) with L a frameand L s a sublattice with zero sup-generating L as objects and special lo-calic maps (i.e., right Galois adjoints h ∗ : ( M, M s ) → ( L, L s ) of dominatingcompatible frame homomorphisms h : ( L, L s ) → ( M, M s ) ) as morphisms. Remark 27.
The categories
FrmS op and LocS are isomorphic.
Definition 28.
We introduce the following categories:(1) the full subcategory
SpFrmS in FrmS generated by objects ( L, L s ) where L is a spatial frame,(2) the full subcategory SpLocS in LocS generated by objects ( L, L s ) where L is a spatial frame,(3) the full subcategory ContFrmS in FrmS generated by objects ( L, L s ) where L is a continuous frame,(4) the full subcategory ContLocS in LocS generated by objects ( L, L s ) where L is a continuous frame. OFMANN-LAWSON DUALITY FOR LOCALLY SMALL SPACES 5 The spectral adjunction.
Theorem 29 (the spectral adjunction) . The categories
LSS and
LocS areadjoint.Proof.
Step 1:
Defining functor
Ω :
LSS → LocS .Functor
Ω :
LSS → LocS is defined by Ω( X, L X ) = ( L woX , L X ) , Ω( f ) = ( L wo f ) ∗ , where L wo f : L woY → L woX is given by ( L wo f )( W ) = f − ( W ) for a boundedcontinuous f : ( X, L X ) → ( Y, L Y ) .For any locally small space ( X, L X ) , the pair ( L woX , L X ) consists of a frameand a sublattice with zero that sup-generates the frame.For a bounded continuous map f : ( X, L X ) → ( Y, L Y ) the frame homo-morphism L wo f : L woY → L woX is always:(1) dominating (because f is bounded): ∀ W ∈ L X ∃ V ∈ L Y ( L wo f )( V ) ∩ W = W, (2) compatible (because f is continuous): ∀ W ∈ L X ∀ V ∈ L Y ( L wo f )( V ) ∩ W ∈ L X . The mapping ( L wo f ) ∗ : L woX → L woY , defined by the condition ( L wo f ) ∗ ( W ) = [ { V ∈ L woY : f − ( V ) ⊆ W } = [ ( L wo f ) − ( ↓ W ) , is a special localic map. Clearly, Ω preserves identities and compositions. Step 2:
Defining functor
Σ :
LocS → LSS .Functor
Σ :
LocS → LSS is defined by Σ( L, L s ) = ( Spec ( L ) , ∆ L ( L s )) , Σ( h ∗ ) = h ∗ | Spec ( M ) : Spec ( M ) → Spec ( L ) . (Notice that Σ( h ∗ ) is always well defined by [4, Prop.4.5]).The pair ( Spec ( L ) , { ∆ L ( a ) } a ∈ L s ) is always a topologically sober locallysmall space by Facts 4 and 7.For h : ( L, L s ) → ( M, M s ) a dominating compatible frame homomor-phism, Σ( h ∗ ) = h ∗ | Spec ( M ) is always a bounded continuous mapping betweenlocally small spaces from ( Spec ( M ) , ∆ M ( M s )) to ( Spec ( L ) , ∆ L ( L s )) :(1) Take any a ∈ M s . Since h is dominating, for some b ∈ L s we have h ∗ | Spec ( L ) (∆ M ( a )) ⊆ h ∗ | Spec ( L ) (∆ M ( h ( b ))) ⊆ ∆ L ( b ) . This is why h ∗ | Spec ( L ) (∆ M ( M s )) refines ∆ L ( L s ) .(2) For any d ∈ L s , f ∈ M s , we have ( h ∗ | Spec ( L ) ) − (∆ L ( d )) ∩ ∆ M ( f ) = ∆ M ( h ( d ) ∧ f ) ∈ ∆ M ( M s ) . This is why ( h ∗ | Spec ( L ) ) − (∆ L ( L s )) ∩ ∆ M ( M s ) ⊆ ∆ M ( M s ) . Clearly, Σ preserves identities and compositions. Step 3:
There exists a natural transformation σ from ΩΣ to Id LocS .We define the mapping σ L : (∆ L ( L ) , ∆ L ( L s )) → ( L, L s ) by the formula σ L = (∆ L ) ∗ : ∆ L ( L ) = τ (∆ L ( L s )) ∋ A → _ ∆ − L ( ↓ A ) ∈ L. This σ L is an (injective) morphism in LocS since ∆ L is a (surjective) dom-inating compatible frame homomorphism: A. PIĘKOSZ (1) ∆ L ( L s ) obviously dominates ∆ L ( L s ) .(2) Take any D ∈ ∆ L ( L s ) and f ∈ L s . Choose d ∈ L s such that ∆ L ( d ) = D . Then D ∩ ∆ L ( f ) = ∆ L ( d ∧ f ) ∈ ∆ L ( L s ) , so ∆ L is compatible.For a special localic map h ∗ : ( M, M s ) → ( L, L s ) and a ∈ M , we have σ L ◦ ΩΣ( h ∗ )(∆ M ( a )) = σ L ◦ ( L wo h ∗ | Spec ( M ) ) ∗ (∆ M ( a )) = σ L ( _ { Z ∈ ∆ L ( L ) : ( h ∗ | Spec ( M ) ) − ( Z ) ⊆ ∆ M ( a ) } ) = h ∗ ( a ) = h ∗ ◦ σ M (∆ M ( a )) , so σ is a natural transformation. Step 4:
There exists a natural transformation λ from Id LSS to ΣΩ .W define λ X : ( X, L X ) → ( Spec ( L woX ) , ∆( L X )) , where ∆ = ∆ L woX , by λ X : X ∋ x ext X { x } ∈ Spec ( L woX ) , which is a bounded continuous map:(1) Take any W ∈ L X . Then λ X ( W ) = { ext { x } : x ∈ W } , which iscontained in ∆( W ) = { V ∈ Spec ( L woX ) : W V } . This is why λ X ( L X ) refines ∆( L X ) .(2) Take any W ∈ L X . Since x ∈ W iff ext { x } ∈ ∆( W ) , we have W = λ − X (∆( W )) . This is why λ − X (∆( L X )) ∩ L X ⊆ L X .For a bounded continuous map f : ( X, L X ) → ( Y, L Y ) , we have (ΣΩ( f ) ◦ λ X )( x ) = ( L wo f ) ∗ (ext X { x } ) = [ { Z ∈ Spec ( L woX ) : x / ∈ f − ( Z ) } = ext Y { f ( x ) } = ( λ Y ◦ f )( x ) . This means λ is a natural transformation. Step 5:
Functor Ω is a left adjoint of functor Σ .We are to prove that σ L | Spec (∆( L )) ◦ λ Spec ( L ) = id Spec ( L ) , σ L woX ◦ ( L wo λ X ) ∗ = id L woX . For p ∈ Spec ( L ) , we have σ L | Spec (∆( L )) ◦ λ Spec ( L ) ( p ) = σ L | Spec (∆( L )) (ext Spec ( L ) { p } ) = id Spec ( L ) ( p ) . For W ∈ L woX , we have σ L woX ◦ ( L wo λ X ) ∗ ( W ) = σ L woX (∆ L woX ( W )) = id L woX ( W ) . (cid:3) A Stone-type duality.
Theorem 30.
The categories
SobLSS , SpLocS and
SpFrmS op are equiv-alent.Proof. Assume ( X, L X ) is an object of SobLSS . Then λ X : ( X, L X ) → ( Spec ( L woX ) , ∆( L X )) is a homeomorphism by [10, Ch. II, Prop. 6.2] and λ X ( L X ) = ∆( L X ) . Hence λ X is an isomorphism in SobLSS .Assume ( L, L s ) is an object of SpFrmS . Then, by Fact 9, ∆ L is anisomorphism of frames and ∆ − L (∆ L ( L s )) = L s , so, by Remark 25, both ∆ L : ( L, L s ) → (∆ L ( L ) , ∆ L ( L s )) and ∆ − L are dominating compatible framehomomorphisms. Hence σ L = (∆ L ) ∗ is an isomorphism in SpLocS .Restricted σ : ΩΣ Id SpLocS and λ : Id SobLSS ΣΩ are naturalisomorphisms. Hence SobLSS and
SpLocS are equivalent. Similarly toRemark 27, categories
SpLocS and
SpFrmS op are isomorphic. (cid:3) OFMANN-LAWSON DUALITY FOR LOCALLY SMALL SPACES 7 Hofmann-Lawson duality.
In this section we give a new version of Theorem 15.
Lemma 31.
Let ( L, L s ) be an object of ContFrmS . Then ( Spec ( L ) , ∆( L s )) is an object of LKSobLSS .Proof.
By the proof of Theorem 29, ( Spec ( L ) , ∆( L s )) is a locally small space.By Facts 7 and 13, this space is topologically sober locally compact. (cid:3) Lemma 32.
For an object ( X, L X ) of LKSobLSS , the pair ( L woX , L X ) isan object of ContFrmS (so also of
ContLocS ).Proof.
By the proof of Theorem 29, ( L woX , L X ) is an object of SpFrmS . ByFact 14, L woX is a continuous frame. (cid:3) Theorem 33 (Hofmann-Lawson duality for locally small spaces) . The cat-egories
LKSobLSS , ContLocS and
ContFrmS op are equivalent.Proof. By lemmas 31, 32, the restricted functors
Σ :
ContLocS → LKSobLSS , Ω :
LKSobLSS → ContLocS are well defined. The restrictions λ : Id LKSobLSS ΣΩ , σ : ΩΣ Id ContLocS of natural isomorphisms from Theorem 30 are natural isomorphisms. Obvi-ously,
ContLocS and
ContFrmS op are isomorphic. (cid:3) Remark 34.
Further equivalences of categories may be obtained using Ex-ercise V-5.24 in [4] (or Exercise V-5.27 in [5]).
Example 35.
Consider the space ( R , L l + om ) from [14, Ex. 2.14(4)] where L l + om = the finite unions of bounded from above open intervalsand the following functions:(1) the function − id : R ∋ x
7→ − x ∈ R is continuous but is notbounded, so L wo ( − id) : ( τ nat , L l + om ) → ( τ nat , L l + om ) is a compati-ble frame homomorphism but is not dominating,(2) the function sin : R ∋ x sin( x ) ∈ R is bounded an weakly contin-uous but not continuous, so L wo sin : ( τ nat , L l + om ) → ( τ nat , L l + om ) is a dominating frame homomorphism but is not compatible,(3) the function arctan : R ∋ x arctan( x ) ∈ R is bounded contin-uous, so L wo arctan : ( τ nat , L l + om ) → ( τ nat , L l + om ) is a dominatingcompatible frame homomorphism,(4) the function : R ∋ x exp( − x ) ∈ R is continuous but notbounded, so the mapping L wo : ( τ nat , L l + om ) → ( τ nat , L l + om ) is acompatible frame homomorphism but is not dominating. References [1] Császár, Á.: Generalized topology, generalized continuity. Acta Math. Hungar. (4), 351–357 (2002)[2] Delfs, H., Knebusch, M.: Locally Semialgebraic Spaces. Lecture Notes in Math.,Vol. 1173. Springer, Berlin-Heidelberg (1985) A. PIĘKOSZ [3] Erné, M.: Choiceless, Pointless, but not Useless: Dualities for Preframes. Appl.Categor. Struct. , 541–572 (2007)[4] Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M., Scott D.S.: ACompendium of Continuous Lattices. Springer, Berlin-Heidelberg (1980)[5] Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M., Scott D.S.:Continuous Lattices and Domains. Cambridge University Press, Cambride (2003)[6] Hofmann, K.H., Keimel, K.: A General Character Theory for Partially OrderedSets and Lattices. Mem. Amer. Math. Soc. No. 122. Amer. Math. Soc., Providence(1972)[7] Hofmann, K.H., Lawson, J.D.: The spectral theory of distributive continuous lat-tices. Trans. Amer. Math. Soc. , 285–310 (1978)[8] Mac Lane, S.: Categories for the Working Mathematician. Grad. Texts in Math.,vol. 5. Springer, New York (1998)[9] Lugojan, S.: Generalized topology. Stud. Cerc. Mat. , 348–360 (1982)[10] Picado, J., Pultr, A.: Frames and Locales. Birkhäuser, Basel (1992)[11] Piękosz, A.: On generalized topological spaces I. Ann. Polon. Math. (3), 217–241 (2013)[12] Piękosz, A.: On generalized topological spaces II. Ann. Polon. Math. (2), 185–214 (2013)[13] Piękosz, A.: O-minimal homotopy and generalized (co)homology. Rocky MountainJ. Math. (2), 573–617 (2013)[14] Piękosz, A.: Locally small spaces with an application. Acta Math. Hungar. (1),197–216 (2020)[15] Piękosz, A.: Stone duality for Kolmogorov locally small spaces. arXiv:1912.12490v3(2020)[16] Piękosz, A., Wajch, E.: Compactness and compactifications in generalized topol-ogy. Topology Appl. , 241–268 (2015)[17] Piękosz, A., Wajch, E.: Bornological quasi-metrizability in generalized topology.Hacet. J. Math. Stat.48