Algebras of the extended probabilistic powerdomain monad
aa r X i v : . [ m a t h . GN ] M a r Algebras of the extended probabilisticpowerdomain monad
Jean Goubault-Larrecq [email protected]
Xiaodong Jia ∗ [email protected] LSV, ENS Paris-Saclay, CNRS, Universit´e Paris-Saclay, FranceMarch 25, 2019
Abstract
We investigate the Eilenberg-Moore algebras of the extended proba-bilistic powerdomain monad V w over the category TOP of T topologicalspaces and continuous maps. We prove that every V w -algebra in our set-ting is a weakly locally convex sober topological cone, and that a mapis the structure map of a V w -algebra if and only if it is continuous andsends every continuous valuation to its unique barycentre. Conversely, forlocally linear sober cones—a strong form of local convexity—, the mereexistence of barycentres entails that the barycentre map is the structuremap of a V w -algebra; moreover the algebra morphisms are exactly thelinear continuous maps in that case.We also examine the algebras of two related monads, the simple valu-ation monad V f and the point-continuous valuation monad V p . In TOP their algebras are fully characterised as weakly locally convex topologicalcones and weakly locally convex sober topological cones, respectively. Inboth cases, the algebra morphisms are continuous linear maps betweenthe corresponding algebras. The probabilistic powerdomain construction on directed complete partially or-dered sets (dcpo for short) was introduced by Jones and Plotkin and employedto give semantics to programming languages with probabilistic features [14, 13].The probabilistic powerdomain of a dcpo consists of continuous valuations de-fined on the Scott-opens of the dcpos, where a valuation is a function assigning ∗ This research was partially supported by Labex DigiCosme (project ANR-11-LABEX-0045-DIGICOSME) operated by ANR as part of the program “Investissement d’Avenir” IdexParis-Saclay (ANR-11-IDEX-0003-02). continuous abstract probabilistic domains , and the algebrahomomorphisms are continuous linear maps. Kirch [20] generalised Jones andPlotkin’s probabilistic powerdomain by stipulating that a valuation might takevalues that are not finite. He showed that this construction is again a monadthat can be restricted to the category of continuous domains, and the algebrasof this monad in the category of continuous domains are the continuous d-cones ,a notion well investigated in [25].A topological counterpart of the probability powerdomain construction wasconsidered by Heckmann in [11] and then by Alvarez-Manilla, Jung and Keimelin [15, 2]. They considered the weak topology on the set of continuous valua-tions instead of the Scott topology. In [11, Proposition 5.1], Heckmann provedthat the resulting space is sober for any topological spaces; and in [15, 2], theauthors proved that the resulting topological space is stably compact if the un-derlying space is stably compact. This topological construction is consistentwith earlier work [20], where Kirch proved that the weak topology and Scotttopology coincide on the set of continuous valuations if one starts with a contin-uous domain. Cohen, Escard´o and Keimel further developed this constructionin [4], where they employed the theory of topological cones to retrieve the defi-nition and called the construction the extended probabilistic powerdomain over T spaces. They showed that the extended probabilistic powerdomain construc-tion is a monad over the category of T topological spaces and considered itsalgebras in related categories in the same paper, leaving a conjecture that thealgebras of this monad on the category of stably compact spaces and continuousfunctions are the stably compact locally convex topological cones. Restrictingthis monad to the category of compact ordered spaces (compact pospaces) andcontinuous monotone maps, Keimel located the algebras of this monad to be thecompact convex ordered sets embeddable in locally convex ordered topologicalvector spaces [16]. Outline.
We are concerned about the algebras of the extended probabilistic powerdo-main in the category of T topological spaces and continuous functions. Werecall some known facts about the extended probabilistic powerdomain monadin Section 2, and on cones in Section 3. We prove in Section 4 that every algebraof this monad in the category of T spaces is a weakly locally convex sober topo-logical cone , and algebra morphisms must be continuous linear maps . We thenshow the tight connection that there is between algebras of the extended prob-abilistic powerdomain monad and barycentres in a sense inspired from Choquet[3], and already used by Cohen, Escard´o and Keimel in [4]: the structure mapsof algebras map every continuous valuation to one of its barycentres, and con-versely, if barycentres are unique and the barycentre map is continuous, then it2s the structure map of an algebra. Moreover, on so-called locally linear cones,the mere existence of barycentres defines an algebra, and on convex- T cones, allcontinuous linear maps are algebra morphisms. Compared to [4], we do not needany stable compactness assumption, and this is due to the Schr¨oder-Simpsontheorem (see Section 3.2). We also isolate the new notion of local linearity,which seems to have been overlooked in ibid.In Section 5, we consider two related probabilistic powerdomain construc-tions, the simple valuation monad V f and the point-continuous valuationmonad V p . Those were initially considered by Heckmann in [11]. We fullycharacterise the algebras of this two monads as weakly locally convex topo-logical cones and weakly locally convex sober topological cones, respectively.In both cases, the algebra morphisms are shown to be continuous linear mapsbetween the corresponding algebras. Those are simple consequences of Heck-mann’s results. Preliminaries.
We use standard notions and notations in domain theory [7, 1] and in non-Hausdorff topology [8]. The category of topological spaces and continuous func-tions is denoted by
TOP . For the convenience of our discussion, we restrictourselves to its full subcategory
TOP of T topological spaces. The categoryof dcpos and Scott-continuous functions is denoted by DCPO . We use R + todenote the set of positive reals, and R + to denote the positive reals extendedwith ∞ . The extended positive reals R + will play a vital role in our discussion.Whenever R + is treated as a topological space, we mean that it is equippedwith the Scott topology until stated otherwise. Definition 2.1 A valuation on a topological space ( X, O X ) is a function µ from O X to the extended positive reals R + satisfying for any U, V ∈ O X : • (strictness) µ ( ∅ ) = 0 ; • (monotoncity) µ ( U ) ≤ µ ( V ) if U ⊆ V ; • (modularity) µ ( U ) + µ ( V ) = µ ( U ∪ V ) + µ ( U ∩ V ) .A continuous valuation µ on ( X, O X ) is a valuation that is Scott-continuousfrom O X to R + , that is, for every directed family of open subsets U i , i ∈ I , itholds: • (Scott-continuity) µ ( S i ∈ I U i ) = sup i ∈ I µ ( U i ) . aluations on the same topological space X are ordered by µ ≤ ν if andonly if µ ( U ) ≤ ν ( U ) for all U ∈ O X . The order is sometimes referred as the stochastic order .The set of continuous valuations on X with the stochastic order is denotedby V X . Example 2.2
Let X be a topological space, the Dirac mass δ x at x ∈ X isdefined by δ x ( U ) = 1 if x ∈ U and otherwise. The Dirac mass δ x is acontinuous valuation on X for every x ∈ X . Example 2.3
Let X be a topological space, The linear combinations P ni =1 r i δ x i of Dirac masses are also continuous valuations, where r i ∈ R + , x i ∈ X . Thesevaluations are called simple valuations . The set of all simple valuations on X is denoted as V f X . Example 2.4
Let X be a topological space, µ, ν be continuous valuations on X and r, s ∈ R + . The linear combinations rµ + sν , defined as ( rµ + sν )( U ) = r · µ ( U ) + s · ν ( U ) for every open subset U , are again continuous valuations. Proposition 2.5 V X is a dcpo in the stochastic order.Proof. For a directed family of continuous valuations µ i , i ∈ I , and any opensubset U ⊆ X , define (sup i ∈ I µ i )( U ) = sup i ∈ I µ i ( U ). One verifies that sup i ∈ I µ i is another continuous valuation. See [7, Lemma IV-9.8] and [11, Section 3.2.(5)]for details. ✷ We can extend V to a functor from the category of topological spaces andcontinuous functions to the category of dcpos and Scott-continuous functionsby using the following proposition. Proposition 2.6
Let f : X → Y be a continuous function between topologicalspaces X and Y , and µ be any continuous valuation on X . Then the map V f : µ ( U ∈ O Y µ ( f − ( U ))) is Scott-continuous from V X to V Y .Proof. Straightforward. ✷ Corollary 2.7 V is a functor from the category TOP to the category
DCPO .Proof.
Straightforward. ✷ There is a canonical functor Σ from the category
DCPO to TOP , namely,the Scott-space construction. For any dcpo L , Σ L is the topological space( L, σL ), where σL is the Scott topology on L ; and for any Scott-continuousfunction f : L → M , Σ f = f is continuous from Σ L to Σ M .Post-composing the functor Σ with V , one obtains an endofunctor Σ ◦ V overthe category TOP , we denote it by V s . Pre-composing the functor Σ with V ,however, yields an endofunctor V ◦
Σ over the category
DCPO , we denote itby V d .In her PhD thesis [13], Jones showed that V d is a monad over the cate-gory DCPO , and moreover, V d can be restricted to the full subcategory of4ontinuous domains. She then used this monad to model probabilistic side ef-fects of programming languages.Naturally, one wonders whether V s is a monad on TOP . Unfortunately, thisis not the case in general. The problem is that the topology O X is, in general,too sparse to sufficiently restrict the Scott topology on V X . Alternatively, oneconsiders the weak topology on V X , and we will see this is the right topologyon V X that gives rise to a monad structure. Definition 2.8 [20, Satz 8.5]
For a topological space X , the weak topology on V X is generated by a subbasis of sets of the form [ U > r ] , U ∈ O X, r ∈ R + ,where [ U > r ] denotes the set of continuous valuations µ such that µ ( U ) > r .We use V w X to denote the space V X equipped with the weak topology and call V w X the extended probabilistic powerdomain or the valuation powerdomain over X . Analogously, we can extend V w into a functor on TOP by defining its actions V w f on continuous maps f : X → Y by V w f ( µ )( V ) = µ ( f − ( V )). Proposition 2.9 V w is an endofunctor on the category TOP .Proof.
The main thing is to check that V w f is continuous for every continuousmap f : X → Y . For every open subset V of Y , for every r ∈ R + \ { } ,( V w f ) − ([ V > r ]) = { µ ∈ V X | µ ( f − ( V )) > r } = [ f − ( V ) > r ]. ✷ Continuous valuations are variations on the idea of measure. While measuresallow one to integrate measurable functions, continuous valuations allow oneto integrate lower semi-continuous functions. A lower semi-continuous function from a topological space to R + is the same thing as a continuous function from X to R + , where the latter is equipped with the Scott topology. We write L X for the set of lower semi-continuous functions from X to R + .For any topological space X , every lower semi-continuous function h : X → R + has a Choquet type integral with respect to a continuous valuation µ on X defined by: Z x ∈ X h ( x ) dµ = Z ∞ µ ( h − ( r, ∞ ]) dr, where the right side of the equation is a Riemann integral. If no risk of confusionoccurs, we usually write R x ∈ X h ( x ) dµ as R h dµ . For the discussion that follows,we collect some properties of this integral, and readers are referred to [20, 24, 21]for more details. Lemma 2.10
1. For every simple valuation µ = P ni =1 r i δ x i , R h dµ = P ni =1 h ( x i ) . In particular, for the Dirac mass δ x , R h dδ x = h ( x ) .2. For all lower semi-continuous functions h, k : X → R + and r, s ∈ R + , R ( rh + sk ) dµ = r R h dµ + s R k dµ . . For every directed family (in the pointwise order) of lower semi-continuousfunctions h a : X → R + , a ∈ A , we have that R (sup a ∈ A h a ) dµ =sup a ∈ A R h a dµ .4. For every open set U , R χ U dµ = µ ( U ) , here χ U is the characteristicfunction of U defined as χ U ( x ) = 1 when x ∈ U and otherwise.5. For all continuous valuations µ, ν ∈ V w X and r, s ∈ R + , for every lowersemi-continuous function f : X → R + , R f d ( rµ + sν ) = r R f dµ + s R f dν .6. Let f : X → Y be a continuous map, µ be a continuous valuationon X , and g : Y → R + be a lower semi-continuous function. Then R y ∈ Y g ( y ) d V w f ( µ ) = R x ∈ X ( g ◦ f )( x ) dµ . Those properties imply a form of the Riesz representation theorem for con-tinuous valuations [20]. It states that integrating with respect to a continuousvaluation ν defines a lower semi-continuous linear functional f R f dν on L X and that, conversely, for every lower semi-continuous linear functional φ on L X , there is a unique continuous valuation ν representing φ , in the sense that φ ( f ) = R f dν for every f ∈ L X , and ν is given by ν ( U ) = φ ( χ U ) for everyopen set U .For all h ∈ L X and r ∈ R + , we define [ h > r ] = { µ ∈ V w X | R h dµ > r } .It is routine to check that [ h > r ] are open in the weak topology of V w X . Theyalso form a subbase of the weak topology, as [ U > r ] = [ χ U > r ]. Using integration, we now argue that V w defines a monad on the category TOP .Recall that a monad on a category C consists of an endofunctor T : C → C together with two natural transformations: η : 1 C → T (where 1 C denotes theidentity functor on C ) and m : T → T , satisfying the equalities m ◦ T m = m ◦ mT and m ◦ T η = m ◦ ηT = 1 T . The natural transformations η and m arecalled the unit and the multiplication of the monad, respectively. Alternatively,one can use the following equivalent description, due to Manes. Definition 2.11 [22] A monad on a category C is a triple ( T, η, † ) consistingof a map T from objects X of C to objects T X of C , a collection η = ( η X ) X ofmorphisms η X : X → T X , one for each object X of C , and a so-called extensionoperation † that maps every morphism f : X → T Y to f † : T X → T Y such that:1. η † X = id T X ;2. for every morphism f : X → T Y , f † ◦ η X = f ;3. for all morphisms f : X → T Y and g : Y → T Z , g † ◦ f † = ( g † ◦ f ) † . T is a functor before checking it is a monad. In fact every monad definedin this sense gives rise to an endofunctor, by defining its action on morphisms f : X → Y as T f = ( η Y ◦ f ) † . The unit of the monad η is given by ( η X ) X andthe multiplication m is given by m X = id † T X for every object X in C .The following is folklore, and is implicit in [4], for example. Proposition 2.12
The functor V w is a monad on the category TOP . The unit η is given by η X : x δ x for every X , and for continuous function f : X → V w Y the extension operation is given by f † ( µ )( U ) = Z x ∈ X f ( x )( U ) dµ. For every lower semi-continuous function h : Y → R + , the following disintegra-tion formula holds: Z y ∈ Y h ( y ) df † ( µ ) = Z x ∈ X (cid:18)Z y ∈ Y h ( y ) df ( x ) (cid:19) dµ. (1) In particular, the function x R y ∈ Y h ( y ) df ( x ) is lower semi-continuous.Proof. The map x ∈ X f ( x )( U ) is continuous for every open set U by thedefinition of the weak topology, hence the formula makes sense. We directlyprove the last claim. Let us assume that x ∈ X satisfies R y ∈ Y h ( y ) df ( x ) > r ,where r ∈ R + \ { } . The function h is the supremum of the countable chain ofmaps h N , defined as N P N N k =1 χ h − ( k/ N , ∞ ] , so R y ∈ Y h N ( y ) df ( x ) > r for some N ∈ N . Let us write h N as ǫ P nk =1 χ U k (a so-called step function ), where ǫ > U k is open, to avoid irrelevant details. Then ǫ P nk =1 f ( x )( U k ) > r ,so there are numbers r k ∈ R + \ { } such that f ( x )( U k ) > r k for each k and ǫ P nk =1 r k ≥ r . Then T nk =1 f − ([ U k > r k ]) is an open neighbourhood of x , and R y ∈ Y h ( y ) df ( x ′ ) ≥ R y ∈ Y h N ( y ) df ( x ′ ) = ǫ P nk =1 f ( x ′ )( U k ) > r for every x ′ inthat neighbourhood.Let us define Λ( h ) as R x ∈ X (cid:16)R y ∈ Y h ( y ) df ( x ) (cid:17) dµ for every h ∈ L X , whichnow makes sense. It is easy to see that Λ is linear and lower semi-continuous,hence there is a unique continuous valuation ν such that Λ( h ) = R h dν forevery h ∈ L Y . We have ν ( U ) = Λ( χ U ), and this gives us back the definition of f † ( µ )( U ).It remains to check the monad equations. That could be done as in [20], butManes’ formulation makes it easier. Equations (i) and (ii) are immediate. For(iii), we have:( g † ◦ f † )( µ )( U ) = Z y ∈ Y g ( y )( U ) df † ( µ )= Z x ∈ X (cid:18)Z y ∈ Y g ( y )( U ) df ( x ) (cid:19) dµ by (1) , g † ◦ f ) † ( µ )( U ) = Z x ∈ X g † ( f ( x ))( U ) dµ = Z x ∈ X (cid:18)Z y ∈ Y g ( y )( U ) df ( x ) (cid:19) dµ by definition. ✷ Remark 2.13
For a topological space X , the multiplication m X of the monadat V w X sends every continuous valuation ̟ ∈ V w ( V w X ) to id †V w X ( ̟ ) =( U R µ ∈V w X µ ( U ) d̟ ) . In particular, for any continuous valuation µ ∈ V w X , m X ( δ µ ) = µ . In this paper, we are mainly interested in the Eilenberg-Moore algebras ofthe valuation powerdomain monad over the category
TOP . Recall that an algebra of a monad T over category C is a pair ( A, α ), where A is an object in C and α A : T A → A is a morphism of C , called the structure map , such that α A ◦ η A = id A and α A ◦ m A = α A ◦ T α A . A morphism f : A → B in C is calleda T -algebra morphism if f ◦ α A = T f ◦ α B .From the basic theory of algebras of monads, we know that in particularthe pair ( T A, m A ) is an algebra of T , where m is the multiplication of T . Inour case, ( V w X, m X ) is a V w -algebra for every topological space X . In order tolocate all the algebras, let us first examine the structure of V w X for an arbitrarytopological space X . We will see that V w X is a topological cone as defined below. The following notions are from [17].
Definition 3.1 A cone is a commutative monoid C together with a scalar mul-tiplication by nonnegative real numbers satisfying the same axioms as for vectorspaces; that is, C is endowed with an addition ( x, y ) x + y : C × C → C whichis associative, commutative and admits a neutral element , and with a scalarmultiplication ( r, x ) r · x : R + × C → C satisfying the following axioms for all x, y ∈ C and all r, s ∈ R + : r · ( x + y ) = r · x + r · y ( rs ) · x = r · ( s · x ) 0 · x = 0( r + s ) · x = r · x + s · x · x = x r · We shall often write rx instead of r · x for r ∈ R + and x ∈ C .A semitopological cone is a cone with a T topology that makes + and · separately continuous.A topological cone is a cone with a T topology that makes + and · jointlycontinuous. emark 3.2 If · is separately continuous, it is automatically jointly continuous[17, Corollary 6.9 (a)]. This is a consequence of a theorem due to Ershov [6,Proposition 2], which states that every separately continuous map from X × Y to Z where X is a c-space (in particular, a continuous poset in its Scott topology)and Y and Z are arbitrary spaces is jointly continuous. Definition 3.3
A function f : C → D from cone C to D is linear if and onlyif for all r, s ∈ R + and x, y ∈ C , f ( rx + sy ) = rf ( x ) + sf ( y ) . Example 3.4
The extended reals R + is a topological cone in the Scott topology,with the usual addition and multiplication extended with r + ∞ = ∞ + r = ∞ for all r ∈ R + , · ∞ = ∞ · , and r · ∞ = ∞ · r = ∞ for r = 0 . Example 3.5
For any topological space X , L X is a cone with the pointwiseaddition and multiplication. It is a semitopological cone with the Scott topologyinduced by the pointwise order. It is a topological cone if X is core-compact (i.e.,if O X is a continuous lattice). Indeed, in that case L X is also a continuouslattice; this can be obtained from [7, Proposition II-4.6] and the fact that R + isa continuous lattice. Every continuous dcpo is a c-space in its Scott topology,then we use [17, Corollary 6.9 (c)], which says that every semitopological conewith a c-space topology is topological. Example 3.6
For every bounded sup-semi-lattice ( L, ≤ , ⊤ , ∨ ) , we can define x + y as x ∨ y , r · x as x if r > , ⊥ otherwise. This is a cone. With the Scotttopology, it is a semitopological cone, and a topological cone if L is continuous[11, Section 6.1]. This illustrates that how far from vector spaces cones can be. Example 3.7
1. For any cone C , the set of all linear maps from C to R + is a cone with pointwise addition and scalar multiplication.2. For any semitopological cone C , the set of all lower semi-continuous linearmaps from C to R + is a cone with pointwise addition and multiplication.We denote it as C ∗ and call it the dual cone of C . We endow C ∗ with the upper weak ∗ topology , that is, the coarsest topology making the functions η C ( x ) = ( φ φ ( x )) : C ∗ → R + continuous for all x ∈ C . The cone C ∗ with the upper weak ∗ topology is atopological cone, as is every subcone of any power R + I with the subspacetopology of the product topology, see the discussion after Definition 5.1 in[17], or [4, Section 3] for example. Proposition 3.8
For any topological space X , V w X is a T topological cone. V w X can be identified with the dual cone ( L X ) ∗ , by a form of the Riesz repre-sentation theorem [20]; see also Section 3.2. This is the path taken in [4]. Wegive an explicit proof. Showing that C ∗ , for a general semitopological cone C ,is a T topological cone is done similarly.9 roof. For all continuous valuations µ, ν ∈ V w X and r ∈ R + , we define( r · µ )( U ) = r ( µ ( U )) and ( µ + ν )( U ) = µ ( U ) + ν ( U ). It is easy to see that V w X with + and · form a cone structure. We proceed to check that + and · are jointlycontinuous. To this end, we assume that µ + ν ∈ [ U > r ] for some U open in X and r ∈ R + \ { } . By definition, that means that µ ( U ) + ν ( U ) > r . If either µ ( U ) or ν ( U ) is equal to ∞ , say µ ( U ) = ∞ , then we know that µ ∈ [ U > r ], andwe pick the whole V w X as an open neighbourhood of ν . Obviously, for any µ ′ ∈ [ U > r ] and ν ′ ∈ V w X , µ ′ + ν ′ ∈ [ U > r ]. If µ ( U )+ ν ( U ) is finite, we choose some s ∈ R + such that µ ( U ) + ν ( U ) > s > r , we let ε = s − r , r µ = max { µ ( U ) − ε, } and r ν = max { ν ( U ) − ε, } . Then µ ∈ [ U > r µ ] and ν ∈ [ U > r ν ], and for all µ ′ ∈ [ U > r µ ] and ν ′ ∈ [ U > r ν ], ( µ ′ + ν ′ )( U ) = µ ′ ( U ) + ν ′ ( U ) > r µ + r ν > r .So we have proved that + is jointly continuous. The joint continuity of scalarmultiplication can be verified similarly.For T -ness, let µ and µ be two different continuous valuations. Then thereexists an open set U such that µ ( U ) = µ ( U ). Without loss of generality weassume that µ ( U ) < µ ( U ). Choose s such that µ ( U ) < s < µ ( U ). Then[ U > s ] is an open subset of V w X containing µ but not µ . ✷ The cone structure on V w X also has additional properties. Definition 3.9 • A subset A of a cone C is called convex if and only if,for any two points a, b ∈ A , the linear combination ra + (1 − r ) b is in A for any r ∈ [0 , . • A subset A of a cone C is called a half-space if and only if both A and itscomplement are convex. • A cone C with a T topology is called weakly locally convex if and only iffor every point x ∈ C , every open neighbourhood U of x contains a convex(not necessarily open) neighbourhood of x . • A cone C with a T topology is called locally convex if and only if eachpoint has a neighbourhood basis of open convex neighbourhoods. • A cone C with a T topology is called locally linear if and only if C has asubbase of open half-spaces. Weak local convexity was introduced in [11], where it is simply called localconvexity. Our notion of local convexity is that of [17, 4]. The notion of locallinearity is new. Note that all those notions would be equivalent in the contextof topological vector spaces.
Proposition 3.10
Every locally linear topological cone is locally convex, andevery locally convex topological cone is weakly locally convex. ✷ Example 3.11
The dual cone C ∗ of any semitopological cone C (defined inExample 3.7) is locally linear. One verifies that the sets ( η C ( x )) − (( r, ∞ ]) arehalf-spaces for all x ∈ X and r ∈ R + , and they form a subbase for the upperweak ∗ topology on C ∗ . xample 3.12 Specializing the previous construction to V w X ∼ = ( L X ) ∗ , thesubbasic open subsets [ U > r ] of V w X are all half-spaces, so V w X is locallylinear. The topology on V w X is more than T , it is actually sober by [11, Proposition5.1]. Hence: Proposition 3.13 V w X is a locally linear sober topological cone for anyspace X . ✷ Example 3.14
For every core-compact space X , L X is a continuous lattice.It follows that L X (with its Scott topology) is a locally convex topological cone,using an argument that Keimel [17, Lemma 6.12] attributes to Lawson. Weargue that L X is in fact locally linear. More generally, L X is a locally linearsemitopological cone for every space X whose sobrification X s is ⊙ -consonant[5, Definition 13.1]. (If X is core-compact, then X s is locally compact sober [8,Theorem 8.3.10], every locally compact sober space is LCS-complete, and everyLCS-complete space is ⊙ -consonant [5, Lemma 13.2].) First, L X is homeo-morphic to L X s , where X s is the sobrification of X [10, Lemma 2.1]. This isbecause R + is sober, and therefore every continuous map from X to R + has aunique continuous extension to X s . This homeomorphism is also an isomor-phism of cones. If X s is ⊙ -consonant, then the Scott topology on L X s coincideswith the compact-open topology [5, Corollary 13.5]. The subbasic open subsets { f ∈ L X s | f ( Q ) ⊆ ( r, ∞ ] } ( Q compact saturated in X s , r ∈ R + ) are easilyseen to be open half-spaces. Example 3.15
Here is an example of a locally convex, non-locally linear topo-logical cone. Consider any complete lattice L , and equip it with the Scott topologyand with the cone structure of Example 3.6. Its non-empty convex subsets areits directed subsets. In particular, every open subset is convex, which impliesthat L is trivially locally convex. For every non-empty convex closed subset C , C is directed and closed, so x = sup C is in C , and therefore C is the downwardclosure ↓ x of x . Hence the proper open half-spaces are exactly the complementsof downward closures of points. It follows that the topology generated by theopen half-spaces is the upper topology . In particular, L is locally linear if andonly if the upper and Scott topologies coincide. In particular, for a continuous(complete) lattice L , L is locally linear if and only if L is hypercontinuous [7,Proposition VII-3.4]. The distributive hypercontinuous lattices are the Stoneduals of quasi-continuous dcpos [7, Propositions VII-3.7, VII-3.8]. Hence anylattice of the form O X , where X is core-compact but not a quasi-continuous dcpo(or does not have the Scott topology), is a locally convex topological cone thatis not locally linear. For example O R , where R comes with its metric topology,fits. Remark 3.16
We have already mentioned that local linearity was not used in[4], and one may think that this is due to the author’s reliance on stable compact-ness. However, there are stably compact, locally convex but non-locally linear opological cones: any continuous, non-hypercontinuous lattice L will serve asan example (Example 3.15), since every continuous lattice is stably compact inits Scott topology [8, Fact 9.1.6]. Recall that a retraction r : X → Y is a continuous map between topologicalspaces such that there is a continuous map s : Y → X with r ◦ s = id Y . Y is the retract of X . A linear retraction is any retraction r : C → D betweensemitopological cones that is also linear. Then D is a linear retract of C . Bewarethat we do not require the associated section s to be linear in any way.Heckmann showed that every linear retract of a weakly locally convex coneis weakly locally convex [11, Proposition 6.6]. It follows: Proposition 3.17
Let C be a locally linear topological cone, D be a topologicalcone, and r : C → D be a linear retraction. Then D is a weakly locally convexcone. ✷ We will see in Section 5 that, conversely, every weakly locally convex coneis a linear retract of some locally linear topological cone.
Keimel’s Separation Theorem , which we reproduce below, is an analogue ofthe Hahn-Banach separation theorem on semitopological cones, and provides uswith a rich collection of lower semi-continuous linear maps.
Theorem 3.18 [17, Theorem 9.1]
In a semitopological cone C consider anonempty convex subset A and an open convex subset U . If A and U are dis-joint, then there exists a lower semi-continuous linear functional Λ : C → R + such that Λ( x ) ≤ < Λ( y ) for all x ∈ A and y ∈ U . ✷ Following Keimel, we call a semitopological cone C convex- T if and only iffor every pair of distinct points a , b of C , there is a lower semi-continuous linearfunction Λ : C → R + such that Λ( a ) = Λ( b ) [17, Definition 4.7]. The followingis an immediate consequence of [17, Corollary 9.3]. We give the explicit, shortproof. Corollary 3.19
Every locally convex semitopological cone is convex- T .Proof. Since C is T , we may assume that there exists an open open U containing a but not b . Since C is locally convex, we can find an open convex subset V suchthat a ∈ V ⊆ U . Realising that the singleton set { b } is a convex set and b / ∈ V ,we apply Theorem 3.18 and we find a lower semi-continuous linear functional Λsuch that Λ( b ) ≤ < Λ( y ) for all y ∈ V . Hence Λ( b ) < Λ( a ), since a ∈ V . ✷ Linear maps on cones such as R + , L X , V w X follow our intuition. Let usexplore the stranger cones from Example 3.6. Example 3.20
Consider any complete lattice L with its Scott topology and thecone structure of Example 3.6. For every lower semi-continuous linear map Λ : L → R + , Λ − ((1 , ∞ ]) is a proper open half-space, hence of the form L \ ↓ x for some point x ∈ L (see Example 3.15). Then x ≤ x if and only if Λ( x ) < for every x ∈ L , and the equality Λ( rx ) = r Λ( x ) implies that Λ( x ) can only be qual to or to ∞ . It follows that the semi-continuous linear maps on L areexactly the maps ∞ · χ L \↓ x , where x ∈ X .As a consequence, the dual cone L ∗ can be equated with the opposite lattice L op with the upper topology. The cone structure is that of L op : addition isinfimum in L , r · x is equal to x if r = 0 , to the top element of L otherwise. We have already mentioned a Riesz-type representation theorem for continuousvaluations [20]. That states that ν ( f R f dν ) and φ ( U φ ( χ U ))define mutually inverse maps between continuous valuations on X and lowersemi-continuous linear functions on L X . Additionally, those define a homeo-morphism between V w X and the dual cone ( L X ) ∗ , namely, the weak topologyon the former is in one-to-one correspondence with the upper weak ∗ topologyon the latter under this bijection.There is yet another representation theorem, the so-called Schr¨oder-SimpsonTheorem, stating that any linear lower semi-continuous functional φ from V w X to R + is uniquely determined by a semi-continuous function h ∈ L X in thesense that φ ( ν ) = R h dν for all ν ∈ V w X . The theorem was originally provedby Schr¨oder and Simpson [23], Keimel gave a conceptual proof of it in [18], andthe first author gave an elementary proof in [9]. Theorem 3.21 (The Schr¨oder-Simpson Theorem)
Let X be a topological space,and Λ be a lower semi-continuous linear map from V w X to R + . There is aunique lower semi-continuous map h ∈ L X such that Λ( ν ) = R h dν for every ν ∈ V w X , and h ( x ) = Λ( δ x ) . V w In order to describe the structure maps of the V w -algebras, let us first definebarycentres of continuous valuations by imitating a definition due to [3, Chap-ter 6, 26.2], and following [4]. Definition 4.1
Let C be a semitopological cone, and ν be a continuous valua-tion on C . A barycentre of ν is any point b ν ∈ C such that, for every linearlower semi-continuous map Λ : C → R + , Λ( b ν ) = R Λ dν . Remark 4.2
Given a probability measure ν , Choquet called its barycenters its resultants . One can also encounter the name centre of gravity , or centre ofmass , of ν . Choquet’s definition applies to the case where C is a Hausdorfflocally convex vector space, not a semitopological cone, and uses continuousmaps Λ from C to R with its standard topology, not its Scott topology. xample 4.3 Let C be a semitopological cone, and P ni =1 r i δ x i be a simplevaluation on C . Then P ni =1 r i x i is a barycentre of P ni =1 r i δ x i . In par-ticular, for any x ∈ C , x is a barycentre of the Dirac mass δ x . Indeed,for every lower semi-continuous linear function f : C → R + , we have that f ( P ni =1 r i x i ) = P ni =1 r i f ( x i ) = R f d ( P ni =1 r i δ x i ) . Example 4.4
Let L be a complete lattice with its Scott topology and the conestructure of Example 3.6. For every ν ∈ V w L , the support supp ν is the com-plement of the largest open set U such that ν ( U ) = 0 . (The family of thoseopen sets is directed, by the modularity law, and its supremum must be in it,by Scott-continuity.) We claim that the barycentre of ν is W supp L . Indeed,using the definition of barycentres and the fact that the lower semi-continuouslinear maps Λ are the maps of the form ∞ · χ L \↓ x , x ∈ X , we obtain that x is a barycentre of ν if and only if the following holds: ( ∗ ) for every x ∈ X , x ≤ x if and only if ∞ · ν ( L \ ↓ x ) = 0 . Since ∞ · ν ( L \ ↓ x ) = 0 is equivalentto supp ν ⊆ ↓ x , hence to the fact that x is an upper bound of supp ν , ( ∗ ) isequivalent to stating that x is the least upper bound of supp ν . Lemma 4.5
Barycentres on a convex- T semitopological cone C are uniquewhen they exist.Proof. If x and x are two barycentres of the same continuous valuation ν ,then Λ( x ) = Λ( x ) for every lower semi-continuous linear map Λ : C → R + .Since C is convex- T , x = x . ✷ We now show that the structure maps of the V w -algebras are nothing butmaps that send valuations to their barycentres. Lemma 4.6
Let ( X, α ) be an algebra of the monad V w on the category TOP .Then X is a topological cone with + defined by x + y = α ( δ x + δ y ) , and scalarmultiplication defined by r · x = α ( rδ x ) for r ∈ R + and x, y ∈ X . Moreover, thestructure map α is linear and sends each µ ∈ V w X to a barycentre of µ . We say that the cone structure obtained this way is induced by the algebra(
X, α ). The fact that α is linear and α ( µ ) is a barycentre of µ has to beunderstood with respect to that induced cone structure. Proof.
We first notice that every extension map f † , as given in Proposi-tion 2.12, is linear, so m X = id †V w X and V w α = ( η Y ◦ α ) † are linear.Let us show that X with the addition and scalar multiplication defined aboveis a cone. We only verify the associativity of addition and scalar multiplication.For any x, y, z ∈ X and r, s ∈ R + , we do the following computation:( x + y ) + z = α ( δ α ( δ x + δ y ) + δ z ) definition of addition on X = α ( δ α ( δ x + δ y ) + δ α ( δ z ) ) definition of structure map= α ( V w α ( δ δ x + δ y ) + V w α ( δ δ z )) naturality of the unit= α ( V w α ( δ δ x + δ u + δ δ z )) linearity of V w α = αm X ( δ ( δ x + δ y ) + δ δ z ) definition of structure map= α (( δ x + δ y ) + δ z ) definition of m X . x + ( y + z ) = α ( δ x + ( δ y + δ z )), so ( x + y ) + z = x + ( y + z ). Moreover, r · ( s · x ) = r · ( α ( sδ x )) definition of scalar multiplication on X = α ( rδ α ( sδ x ) ) definition of scalar multiplication on X = α ( r V w α ( δ sδ x )) naturality of the unit= α ( V w α ( rδ sδ x )) linearity of V w α = αm X ( rδ sδ x ) definition of structure map= α ( rsδ x ) linearity of m X = ( rs ) · x definition of scalar multiplication on X. To see that X is a topological cone, we assume that U is an open set in X and x + y ∈ U . This means that α ( δ x + δ y ) ∈ U , hence δ x + δ y ∈ α − ( U ).Since V w X is a topological cone and the unit map η X : x δ x : X → V w X is continuous, we can find open sets U x , U y such that x ∈ U x , y ∈ U y and forany x ′ ∈ U x , y ′ ∈ U y , δ x ′ + δ y ′ ∈ α − ( U ), which means that x ′ + y ′ ∈ U forall x ′ ∈ U x and y ′ ∈ U y . This proves that + is jointly continuous. The jointcontinuity of scalar multiplication can be proved similarly.We proceed to prove that α is linear. Let r ∈ R + and µ, ν ∈ V w X . We havethe following: α ( µ + ν ) = α ( m X ( δ µ ) + m X ( δ ν )) monad law= α ( m X ( δ µ + δ ν )) linearity of m X = α ( V w α ( δ µ + δ ν )) definition of structure map= α ( V w α ( δ µ ) + V w α ( δ ν )) linearity of V w α = α ( δ α ( µ ) + δ α ( ν ) ) naturality of the unit= α ( µ ) + α ( ν ) definition of addition on X. Similarly, we can prove that α ( rµ ) = r · α ( µ ).Finally, we prove that α ( µ ) is a barycentre of µ for all µ ∈ V w X . Assume thatΛ : X → R + is a lower semi-continuous linear map. Notice that the compositionΛ ◦ α is then a linear map from V w X to R + . Hence by the Schr¨oder-SimpsonTheorem there exists a unique lower semi-continuous map h : X → R + such thatΛ ◦ α ( ν ) = R h dν for all ν ∈ V w X . In particular Λ( x ) = Λ ◦ α ( δ x ) = R h dδ x = h ( x ) for all x ∈ X . This implies that h = Λ, and hence Λ( α ( µ )) = R Λ dµ forall µ ∈ V w X . So α ( µ ) is a barycentre of µ by definition. ✷ Corollary 4.7
Let X be a topological space. For every ̟ ∈ V w V w X , m X ( ̟ ) is the barycentre ( U R ν ∈V w X ν ( U ) d̟ ) of ̟ in V w X .Proof. By general category theory, ( V w X, m X ) is an algebra of V w . ✷ Corollary 4.8
For every convex- T semitopological cone C , there is at mostone map α : V w C → C that makes ( C, α ) a V w -algebra and induces the originalcone structure on C . roof. By Lemma 4.6, and since the induced cone structure is the original one, α must map every ν to one of its barycentres, and barycentres are unique byLemma 4.5. ✷ Proposition 4.9
Let ( X, α ) be an algebra of the monad V w on the category TOP . Then X is a weakly locally convex sober topological cone with the inducedcone structure.Proof. By Lemma 4.6, X is a topological cone, and α is linear. It is also acontinuous retraction by definition of algebras, since α ◦ η X = id X . Hence X islinear retract of V w X , which is locally linear and sober. Since sobriety is pre-served by continuous retractions, X is a weakly locally convex sober topologicalcone by Proposition 3.17. ✷ We may guess that the V w -algebras are the sober, weakly locally convex,topological cones, or maybe those on which, additionally, every continuous val-uation has a barycentre. This is not quite enough. The function α mapping ν to its barycentre must be continuous as well, and barycentres should be unique .The latter happens in all convex- T cones, but we do not know whether thecone structure induced by a V w -algebra (Lemma 4.6) is convex- T . Proposition 4.10
Let C be a semitopological cone, and α be a continuous mapfrom V w C to C . If α ( ν ) is the unique barycentre of ν for every ν ∈ V w C , then ( C, α ) is an algebra of the monad V w on the category TOP .In that case, the cone structure on C induced by the algebra ( C, α ) coincideswith the original cone structure on C . C is a sober, weakly locally convex,topological cone.Proof. For every x ∈ C , α ( δ x ) = x by uniqueness of barycentres, and since x is abarycentre of δ x (Example 4.3). In order to show that α ( V w α ( ̟ )) = α ( m C ( ̟ ))for every ̟ ∈ V w V w C , we consider any lower semi-continuous linear functionΛ : C → R + , and we observe that:Λ( α ( m C ( ̟ ))) = Z x ∈ C Λ( x ) dm C ( ̟ ) α ( m C ( ̟ )) is a barycentre of m C ( ̟ )= Z x ∈ C Λ( x ) d (id V w C ) † ( ̟ ) m C = (id V w C ) † = Z ν ∈V w C (cid:18)Z x ∈ C Λ( x ) dν (cid:19) d̟ disintegration formula (1)= Z ν ∈V w C Λ( α ( ν )) d̟ α ( ν ) is a barycentre of ν = Z x ∈ C Λ( x ) d V w α ( ̟ ) item (vi) in Lemma 2.10.This shows that α ( m C ( ̟ )) is also a barycentre of V w α ( ̟ ). Since barycentresare unique, α ( m C ( ̟ )) = α ( V w α ( ̟ )). 16inally, for all x, y ∈ C we observe that α ( δ x + δ y ) and x + y are bothbarycentres of δ x + δ y , hence they are equal. Similarly, for every r ∈ R + , α ( rδ x ) = r · x . Hence the induced cone structure coincides with the originalcone structure on C . We conclude by Proposition 4.9. ✷ This simplifies in the case of locally linear cones, where the uniqueness ofbarycentres and the continuity of the barycentre map are automatic.
Proposition 4.11
Let C be a locally linear semitopological cone such that ev-ery continuous valuation ν on C has a barycentre b ν . The barycentre map β : V w C → C , defined by β ( ν ) = b ν , is the structure map of a V w -algebra (andin particular, C is sober and a topological cone).Proof. Since C is locally linear, it is locally convex hence convex- T (Corol-lary 3.19). Therefore Lemma 4.5 applies, showing that the barycentre b ν isunique for every ν ∈ V w X , hence that β is well-defined.We now prove that that β is continuous. Let H be an open half-space of C ; since C is locally linear, it suffices to show that β − ( H ) is open. If H = C ,then β − ( H ) = V w X is open. Otherwise, by Theorem 3.18, there exists alinear lower semi-continuous function h : C → R + such that h ( a ) ≤ < h ( b )for all a ∈ C \ H and b ∈ H . Then H = h − ((1 , ∞ ]), and β − ( H ) is thenthe set of continuous valuations ν such that h ( β ( ν )) >
1. By the definition ofbarycentres, h ( β ( ν )) = R x ∈ C h ( x ) dν , so β − ( H ) is equal to the open set [ h > β is the structure map of a V w -algebra. It follows that C is sober, and a topological cone, by Proposition 4.9. ✷ Example 4.12
The extended real numbers R + with the map µ R x ∈ R + x dµ is a V w -algebra, since R + with the Scott topology is a locally linear topologicalcone. Example 4.13
Let L be a complete lattice with its Scott topology and the conestructure of Example 3.6.1. If L is a continuous, non-hyper-continuous dcpo (see Example 3.15), then β : ν supp ν is the structure map of a V w -algebra on L , as a consequenceof the following proposition, although L is not locally linear.2. If L is not weakly Hausdorff (see below), then β ( ν ) is the unique barycentreof ν for every ν ∈ V L , but the cone structure of L is induced by no V w -algebra, again by the following proposition. We will also see that everyweakly Hausdorff complete lattice is sober, hence Isbell’s example of a non-sober complete lattice [12] is not weakly Hausdorff. A weakly Hausdorff space is a topological space X such that for all x, y ∈ X , forevery open subset U of X that contains ↑ x ∩ ↑ y , there are open neighborhoods V of x and W of y such that V ∩ W ⊆ U [19, Lemma 6.6]. Proposition 4.14
Let L be a complete lattice with its Scott topology and thecone structure of Example 3.6. For every ν ∈ V L , let β ( ν ) = W supp ν . Thefollowing are equivalent: . there is a V w -algebra structure on L that induces its cone structure;2. β is the structure map of a V w -algebra on L ;3. β is continuous;4. ∨ : L × L → L is jointly continuous;5. L is weakly Hausdorff.In particular, (i)–(v) hold if L is core-compact, and (i)–(v) imply that L is sober.Proof. We have seen in Example 4.4 that β ( ν ) is the barycentre of ν . Thisbarycentre is unique since L is locally convex (Example 3.15). The implication(iii) ⇒ (ii) then follows from Proposition 4.10. The converse implication is trivial.The equivalence of (i) with (ii) follows from Corollary 4.8.Since every V w -algebra is a topological cone (Proposition 4.9), (i) implies(iv). We now assume (iv), and aim to show (iii). We will repeatedly use thefollowing fact: an open set U intersects supp ν if and only if ν ( U ) >
0. Indeed, U intersects supp ν if and only if U is not included in the largest open set with ν -measure zero.Let ν ∈ V L and V be an open neighborhood of W supp ν . We will exhibitan open neighborhood U of ν such that W supp µ ∈ V for every µ ∈ U .Since V is Scott-open, there are finitely many points x , · · · , x n ∈ supp ν whose supremum W ni =1 x i is in V . By (iv), there are open neighborhoods U i of x i , for each i , such that for all y ∈ U , . . . , y n ∈ U n , W ni =1 y i is in V . Let usdefine U as T ni =1 [ U i > i , U i intersects supp ν at x i , so ν ( U i ) > ν is in U . For every µ ∈ U , we have µ ( U i ) > i , so U i must meet supp µ , say at y i . Then W supp µ ≥ W ni =1 y i ∈ V , so W supp µ is in V . This shows (iii).The equivalence between (iv) and (v) is immediate, since ↑ x ∩ ↑ y = ↑ ( x ∨ y ).As for the last part, the binary supremum operation is jointly continuouson any core-compact complete lattice by [7, Corollary II-4.15], so if L is core-compact then (iv) holds. If (i) holds, then L is sober by Proposition 4.9. ✷ Here is a final example. In that case, L X is locally linear (see Example 3.14). Proposition 4.15
For every core-compact space X , for every continuous valu-ation ν on L X , the map β ( ν ) : x ∈ X R f ∈L X f ( x ) dν is the unique barycenterof ν on L X with its Scott topology. The map β is the structure map of a V w -algebra on L X , and the cone structure it induces on L X is the usual one.Proof. Kirch characterised the algebras of the V monad on the category CONT of continuous dcpos: the Eilenberg-Moore category of V on CONT is equivalentto the category of continuous d-cones [20, Satz 7.1]. A d-cone is a dcpo witha cone structure whose addition and scalar multiplication are Scott-continuous.Every continuous d-cone is a topological cone, as a consequence of Ershov’stheorem (see Remark 3.2, and [17, Corollary 6.9 (c)]). Since X is core-compact, L X is a continuous d-cone, hence there is a map β : V ( L X ) → L X that turns18 L X, β ) into a V -algebra. Using another result of Kirch [20, Satz 8.6], whichstates that for every continuous dcpo Y in its Scott topology, the Scott andweak topologies agree on V Y , β is a V w -algebra map. The value β ( ν ) is givenas a directed supremum of barycentres of simple valuations way-below ν , andone can check that this is equal to R f ∈L X f ( x ) dν . Here is an alternative proof,which the reader may find interesting.We will use Jones’ version of Fubini’s theorem [13, Theorem 3.17]. Thisstates that for every (jointly) continuous map f : X × Y → R + , where R + hasthe Scott topology, for all continuous valuations µ on X and ν on Y , Z x ∈ X (cid:18)Z y ∈ Y f ( x, y ) dν (cid:19) dµ = Z ( x,y ) ∈ X × Y f ( x, y ) d ( µ × ν ) = Z y ∈ Y (cid:18)Z x ∈ X f ( x, y ) dµ (cid:19) dν, where µ × ν is the uniquely determined product valuation . Implicit in thattheorem is the following fact: ( ∗ ) the maps y ∈ Y R x ∈ X f ( x, y ) dµ and x ∈ X R y ∈ Y f ( x, y ) dν are lower semi-continuous. The latter can be shown usingstep functions as in the proof of Proposition 2.12.Let g ( x ) = R f ∈L X f ( x ) dν . This makes sense because the map f ∈ L X f ( x ) is Scott-continuous, hence lower semi-continuous, for every x ∈ X . Inorder to show that g is continuous, we first notice that App : L X × X → R + ,which maps ( f, x ) to f ( x ), is (jointly) continuous. Indeed, L X is a continuouslattice (see Example 3.5), hence a c-space. App is clearly separately continuous,and then jointly continuous by Ershov’s theorem (see Remark 3.2). By ( ∗ ), themap x ∈ X R f ∈L X App ( f, x ) dν is lower semi-continuous. But that map issimply g .Now that g is in L X , we check that it is a barycenter of ν . Let Λ be any lowersemi-continuous function from L X to R + . Λ is integration with respect to someuniquely defined continuous valuation µ on X , by the Riesz-type representationtheorem mentioned earlier. Then:Λ( g ) = Z x ∈ X g ( x ) dµ = Z x ∈ X (cid:18)Z f ∈L X App ( f, x ) dν (cid:19) dµ = Z f ∈L X (cid:18)Z x ∈ X App ( f, x ) dµ (cid:19) dν by Jones’s version of Fubini’s theorem= Z f ∈L X Λ( f ) dν, showing that, indeed, g is a barycenter of ν . Note that Jones’ version of Fubini’stheorem applies, crucially, because App is jointly continuous.There is nothing more to prove: we merely apply Proposition 4.11, using thefact that L X is locally linear. ✷ .2 The morphisms of algebras of V w Adapting the definition of morphisms of algebras in our setting, a morphism f between two V w -algebras ( X, α ) and (
Y, β ) is a continuous function f : X → Y such that β ◦ V w f = f ◦ α . Considering the cone structure of X and Y inducedby α and β (Proposition 4.9), respectively, we will see that f is a linear mapbetween X and Y . Indeed, for any a, b ∈ X and r ∈ R + , f ( a + b ) = f ( α ( δ a + δ b )) definition of addition= β ( V w f ( δ a + δ b )) f is a morphism of algebras= β ( δ f ( a ) + δ f ( b ) ) naturality of the unit= f ( a ) + f ( b ) definition of addition.Similarly, we have f ( r · a ) = r · f ( a ). Conversely, we want to know whethercontinuous linear maps are exactly the V w -algebra morphisms. To prove this,however, we need to assume that Y is convex- T , and not just weakly locallyconvex. Proposition 4.16
Let ( X, α ) , ( Y, β ) be two V w -algebras, viewed as topologicalcones in the sense of Proposition 4.9. If Y is convex- T , then the V w -algebramorphisms from ( X, α ) to ( Y, β ) are precisely the continuous linear maps be-tween them.Proof. Let f : X → Y be a linear map. For every lower semi-continuous linearmap Λ : Y → R + , Λ ◦ f : X → R + is lower semi-continuous and linear. Sincestructure maps send valuations to their barycentres by Lemma 4.6, for anycontinuous valuation µ ∈ V w X we have,Λ( f ( α ( µ ))) = (Λ ◦ f )( α ( µ ))= Z Λ ◦ f dµ α ( µ ) is a barycentre of µ = Z Λ d ( V w f ( µ )) item (vi) in Lemma 2.10= Λ( β ( V w f ( µ ))) β ( V w f ( µ )) is a barycentre of V w f ( µ ) . Since Y is convex- T , we use Corollary 3.19 to conclude that f ( α ( µ )) = β ( V w f ( µ )). ✷ V f and V pBesides the space V w X of continuous valuations on any topological space X ,Heckmann also considered its subspaces V f X of simple valuations and V p X ofpoint-continuous valuations on X [11]. In the same paper he showed that V p X is the sobrification of V f X [11, Theorem 5.5]. We will see that V f and V p canalso be extended to monads on the category TOP .20e have seen simple valuations in Example 2.3. We proceed to define point-continuous valuations.For a topological space X , according to Heckmann [11], one considers, in-stead of the Scott topology, the point topology on O X determined by the sub-basic open sets O ( x ), x ∈ X , where O ( x ) = { U ∈ O X | x ∈ U } for each x ∈ X .We denote O X with the point topology by O p X . One can equate O X with theset of continuous maps from X to Sierpi´nski space S = { , } (with the Scotttopology of ≤ ), and then the point topology is the subspace topology inducedby the inclusion into S X . Definition 5.1 A point-continuous valuation µ on ( X, O X ) is a valuation thatis continuous from O p X to R + . The set of all point-continuous valuations on X is denoted by V p X . One easily sees that every simple valuation is point-continuous, and everypoint-continuous valuation is a continuous valuation, since the Scott topologyon O X is finer than the point topology.In what follows, we consider V f X and V p X as subspaces of V w X , that is,the topologies considered are the subspace topologies induced from the weaktopology on V w X . Proposition 5.2
Let f : X → Y be a continuous function between topologi-cal spaces X and Y . Then the map V ∗ f : µ ( U ∈ O Y µ ( f − ( U ))) iscontinuous from V ∗ X to V ∗ Y , where µ ∈ V ∗ X and ∗ is p or f .Proof. The only difficult point is to show that V p f sends point-continuous valua-tions to point-continuous valuations. To this end, let µ be any point-continuousvaluation and U be any open subset in Y , and r be any positive number in R + with V p f ( µ )( U ) > r . By definition, µ ( f − ( U )) > r . Since µ is point-continuous,we can find a finite subset F of points such that f − ( U ) ∈ T x ∈ F O ( x ) and forevery open subset V ∈ T x ∈ F O ( x ), µ ( V ) > r . We claim that T y ∈ f ( F ) O ( y )is an open set containing U and such that, for every W ∈ T y ∈ f ( F ) O ( y ), V p f ( µ )( W ) > r . The former is obvious since f − ( U ) ∈ T x ∈ F O ( x ) means that F ⊆ f − ( U ), i.e., f ( F ) ⊆ U . For the latter claim, we know that f ( F ) ⊆ W ,so we have f − ( W ) ∈ T x ∈ F O ( x ). From the point-continuity of µ , we have µ ( f − ( W )) > r , hence V p f ( µ )( W ) > r . V p f is continuous since V w f is (Proposition 2.9). ✷ Remark 5.3
The following formula holds: V f f ( P ni =1 a i δ x i ) = P ni =1 a i δ f ( x i ) . Proposition 5.4
For all topological spaces X and Y , and for every continuousfunction f : X → V ∗ Y , the map f †∗ : µ (cid:18) U Z x ∈ X f ( x )( U ) dµ (cid:19) : V ∗ X → V ∗ Y is well-defined and continuous, where ∗ is p or f . roof. If f †∗ indeed takes its values in V ∗ Y , then it is continuous, because f † is—that is part of Proposition 2.12.We proceed to prove that f †∗ takes its values in V ∗ Y . When ∗ = f, we assumethat µ = P ni =1 r i δ x i . Then f † f ( µ )( U ) = P ni =1 r i f ( x i )( U ). Since for each i ∈ I , r i f ( x i ) is a simple valuation, f † f ( µ ), as a finite sum of simple valuations, is againa simple valuation.We now show that f † p takes its values in V p Y . In order to see this, we firstnotice that f † f is also a continuous map from V f X to V p Y , considering V f Y as asubspace of V p Y . Since V p X is the sobrification of V f X [11, Theorem 5.5], thefunction f † f has a unique continuous extension e from V p X to V p Y . Considering V p Y as a subspace of V w Y , then from Proposition 2.12 we know that both e and f † p are continuous functions from V p X to V w Y . Since V w Y is T , and e and f † p coincide on V f X , they coincide on the sobrification V p X as well. Thus f † p sends point-continuous valuations to point-continuous valuations since e does. ✷ With all the ingredients listed above, we conclude the following:
Proposition 5.5 V ∗ ( ∗ is p or f ) is a monad on the category TOP , with theunit η ∗ : x → δ x : X → V ∗ X and extension f †∗ : µ ( U Z x ∈ X f ( x )( U ) dµ ) : V ∗ X → V ∗ Y for continuous map f : X → V ∗ Y . The multiplication m ∗ X of V ∗ at X is (id V ∗ X ) †∗ . ✷ Similarly to V w X , V f X and V p X are also locally linear topological coneswith the canonical operations of addition and scalar multiplication. Moreover,we have the following: Theorem 5.6 [11, Theorem 6.7, Theorem 6.8] V f X is the free weakly locally convex cone over X in the category TOP .2. V p X is the free weakly locally convex sober cone over X in the category TOP . This means that for a T space X , V f X (resp., V p X ) is a weakly locally convex(resp., weakly locally convex sober) topological cone, and for every continuousfunction f : X → C from X to a weakly locally convex (resp., weakly locallyconvex sober) topological cone C , there is a unique continuous linear function f : V ∗ X → M such that f ◦ η ∗ = f , where ∗ is f or p.The following is a straightforward consequence of the above theorem. Corollary 5.7
A topological cone C is weakly locally convex (resp., weakly lo-cally convex and sober) if and only if it is a continuous linear retract of a locallylinear (resp., locally linear and sober) topological cone. roof. We have seen the “if” direction in Proposition 3.17.For the “only if” direction, since id C , the identity map over C , is continuousand linear, there exists a unique continuous linear map id C : V f C → C such thatid C ◦ η f = id C . This exhibits that C is a linear retract of V f C which is a locallylinear topological cone.To show that every weakly locally convex sober topological cone is a linearretract of some locally linear sober topological cone, we just change V f into V p in the above and the same argument applies. ✷ The following results from [11, Theorem 6.1] are needed for our furtherdiscussion.
Theorem 5.8
Let X be a topological space.1. Every linear function from V f X to some cone is uniquely determined byits values on Dirac masses.2. Every continuous linear function from V p X to some topological cone isuniquely determined by its values on Dirac masses. We now have enough ingredients to prove the main results of this section.
Theorem 5.9
Let X be a T topological space, and α : V f X → X be a contin-uous map. If ( X, α ) is a V f -algebra, then X is a weakly locally convex topo-logical cone, and α is the standard barycentre map P ni =1 r i δ x i P ni =1 r i x i .Conversely, for every weakly locally convex topological cone C , there exists a(unique) continuous linear map α from V f C to C , sending each simple valua-tion to its barycentre, and the pair ( C, α ) is a V f -algebra.Proof. A similar argument as in the proof of Lemma 4.6 shows that X is atopological cone with + defined by x + y = α ( δ x + δ y ), and scalar multiplicationdefined by r · x = α ( rδ x ) for all r ∈ R + and x, y ∈ X . X is weakly locally convexsince V f X is locally linear and the structure map α is a linear retraction. Weshow that α sends each simple valuation to its barycentre. This is easy since α ( δ x ) = x by the definition of structure map, hence by linearity α sends everysimple valuation P ni =1 r i δ x i to P ni =1 r i x i , which is a barycentre of P ni =1 r i δ x i (Example 4.3).Conversely, assume that C is a weakly locally convex topological cone. Since V f C is the free weakly locally convex topological cone over C , there exists aunique map α such that α ◦ η f = id C . Hence α ( δ x ) = x for every x ∈ C . Then α sends each simple valuation to its barycentre, since α is linear. Finally, tosee that ( C, α ) is a V f -algebra, we only need to verify that α ◦ m f C = α ◦ V f α C .Notice that both sides of the equals sign are continuous linear functions from V f V f C to C . From Theorem 5.8 we only need to show they are equal on Diracmasses. To this end, let us assume that µ is a simple valuation on X , and wecompute the following, α ◦ m f C ( δ µ ) = α ( µ ) monad law= α ( δ α ( µ ) ) α ◦ η = id= α ◦ V f α ( δ µ ) naturality of the unit,23nd this concludes the proof. ✷ The V f -algebra morphisms are precisely the continuous linear maps betweenthem. Theorem 5.10
Let
X, Y be two weakly locally convex topological cones, and α : V f X → X, β : V f Y → Y be the corresponding barycentre maps. Then acontinuous map f : X → Y is a V f -algebra morphism if and only if f is linear.Proof. Assume first that f a V f -algebra morphism. For any a, b ∈ X , r ∈ R + ,we have f ( a + b ) = f ( α ( δ a + δ b )) α is the barycentre map= β ( V f f ( δ a + δ b )) f is a morphism of algebras= β ( δ f ( a ) + δ f ( b )) Remark 5.3= f ( a ) + f ( b ) α is the barycentre mapSimilarly, we can prove that f ( ra ) = rf ( a ).Conversely, assume f is linear. Then for any simple valuation P ni =1 r i δ x i ,we have: f ( α ( n X i =1 r i δ x i )) = f ( n X i =1 r i x i ) α is the barycentre map= n X i =1 r i f ( x i ) f is linear , and β ( V f f ( X i ∈ I r i δ x i )) = β ( X i ∈ I r i δ f ( x i ) ) Remark 5.3= X i ∈ I r i f ( x i ) β is the barycentre map.Hence f ◦ α = β ◦ V f f , and therefore f is a V f -algebra morphism. ✷ In weakly locally convex topological cones, even simple valuations may haveseveral barycentres. As in Theorem 5.9, we call x the standard barycentre of δ x .Note that this is well-defined: if δ x = δ y , then for every open set U , δ x ( U ) = 1if and only if δ x ( U ) = 1, hence x and y have the same open neighbourhoods,which implies x = y since the cone is T . Theorem 5.11
Let X be a T topological space, and α : V p X → X be a func-tion. If ( X, α ) is a V p -algebra, then X is a weakly locally convex sober topologicalcone, and α is a linear continuous function that maps every point-continuousvaluation to one of its (Choquet) barycentres, and every Dirac mass to its stan-dard barycentre.Conversely, for every weakly locally convex sober topological cone C , thereexists a (unique) continuous linear map α from V p C to C that sends each Diracmass to its standard barycentre. Then the pair ( C, α ) is a V p -algebra. roof. The same reasoning as in the proof of Theorem 5.9 will show that X is acontinuous linear retract of V p X . Since V p X is locally linear and sober, then X ,as a continuous linear retract, is weakly locally convex sober. The proof that α is linear is as in Lemma 4.6.To see that α maps every point-continuous valuation µ to one of its barycen-tres, for any continuous linear map Λ : X → R + we consider two maps from V p X to R + . They are µ Λ ◦ α ( µ ) and µ R Λ dµ . Note that these two mapsare continuous linear maps and coincide on Dirac masses on X , hence they areequal from Item (ii) of Theorem 5.8. Hence α is indeed a barycentre map.Conversely, for any weakly locally convex sober cone C , let f be the identitymap from C to C . By Item (ii) of Theorem 5.6, there is a unique continuouslinear map f such that f ◦ η p = f , and this is the desired α . That ( C, α ) is a V p -algebra can be verified similarly as in the proof of Theorem 5.9. ✷ Theorem 5.12
Let
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