On geometric properties of the functors of positively homogenous and semiadditive functionals

solute retract, soft mapping, monad. 0. Introduction. V.Fedorchuk posed a general problem concerning geometric properties of functors, that is, how functors affect certain geometric properties of spaces and mappings between them [11]. Under geometric properties we understand the property of being an AR for a space, the properties of being soft or a Tychonov fibering for a mapping etc. There were many investigations in this direction involving such functors as the hyperspace functor exp, the probability measures functor P , the superextension functor λ, the inclusion hyperspace functor G and others (see, e.g. [10] or [11]). Let us now consider as an example the functors of probability measures P and superextension λ. There is a natural structure of linear convexity on P (X). As for λ, de Groot constructed some abstract convexity (not linear) on any space of the form λ(X) (see [12]), and this convexity is binary, whereas the linear convexity on P (X) is not. The functors λ and P differ in their geometric properties as well. Consider the property of being an AR, for instance. In the metrizable case, λ(X) ∈ AR if and only if X is a continuum, and P (X) is an absolute retract for each compactum X. When X is not metrizable, the space P (X) can be AR only in case X is openly generated and of weight ≤ ω 1 . As for the superextension functor, λ(X) ∈ AR whenever X is an openly generated continuum, without limitations on weight. The algebraic aspects of functors are formalized by the notion of a monad in the sense of Eilenberg and Moore [13]. The notion of convexity considered in this paper is considerably broader than the classic one: specifically, it is not restricted to the context of linear spaces. Such convexities appeared in the process of studying different structures like partially ordered sets, semilattices, lattices, superextensions etc. We base our approach on the notion of topological convexity from [14] where the general convexity theory is covered from axioms to application in different areas. T.Radul assigned to each monad F some abstract convexity structure on every space F X, where F is the functorial part of the monad F. Some additional conditions on these monads (that they are L-monads which weakly preserve preimages) guarantee that the considered convexities generate the topology of the space F X for the functor F included in an L-monad. It was shown that L-monads which weakly preserve preimages and with binary convexities can give absolute retracts in all weights [3]. Also, the morphisms of their algebras can be soft in nonmetrizable case under certain conditions. Note that the property of binarity of the convexity generated by monad F is equivalent to the superextension monad being the submonad of F (again [3]). In this article we consider functors OS and OH (introduced in [5], [6]), which both generate L-monads. The monad OS does not generate binary convexities, in turn OH does, and this as well appears to be the reason for the difference in their geometric properties: the properties of OS are close to that of P , and OH is closer to λ. 1. Definitions and facts. In the present paper we shall deal with objects and morphisms of the category Comp, that is, with compact Hausdorff spaces and continuous mappings. By C(X), where X ∈ Comp, we denote the Banach space of all continuous real-valued functions on X with the sup-norm ϕ = sup{|ϕ(x)| | x ∈ X}. By c X , where c ∈ R, we denote the constant function: c X (x) = c for all x ∈ X. Let X ⊂ Y . We say that a space X is a retract of Y if there exists a map r : Y → X such that r| X = id X . The space X is an absolute retract (shortly X ∈ AR), if for any embedding i : X ֒→ Y the subspace i(X) is a retract of Y . Recall that a τ -system, where τ is any cardinal number, is a continuous inverse system consisting of compacta of weight ≤ τ and epimorphisms over a τ -complete indexing set. As usual, ω stands for the countable cardinal number. A compactum X is called openly generated, if it can be represented as the limit of some ω-system with open bonding mappings [1]. The mapping f : X → Y is called soft if for any space Z and its closed subset A, any functions We say that a commutative diagram A triple (F, η, µ), where F is an endofunctor in category Comp, η : [13]. Suppose that F = (F, η, µ) is a monad. A pair (X, ξ), where ξ : F (X) → X, is called an Let ν : C(X) → R be a functional. We say that ν is: 1) normed, if ν(1 X ) = 1; 2) weakly additive, if for any φ ∈ C(X) and c ∈ R we have ν(φ + c X ) = ν(φ) + c; 3) order-preserving, whenever for any ϕ, ψ ∈ C(X) such that ϕ(x) ≤ ψ(x) for all x ∈ X (i.e. ϕ ≤ ψ) the inequality ν(ϕ) ≤ ν(ψ) holds; 4) positively homogeneous, if for any ϕ ∈ C(X) and any real t ≥ 0 we have in that way, V forms a covariant functor in the category Comp. For any space X by O(X) denote the set of functionals satisfying 1)-3) (order-preserving functionals), by OH(X) the set of all functionals on C(X) which satisfy properties 1)-4) (positively homogenous functionals), and by OS(X) we denote the set of functionals on C(X) which satisfy properties 1)-5) (semiadditive functionals). Also recall that P (X) stands for the set of all functionals on C(X) which are normed ( µ = 1), positive (µ(ϕ) ≥ 0 for all ϕ ≥ 0) and linear. Let F stand for one of O, OH, OS, P . The space F (X) is considered as the subspace of V (X). For any function f : X → Y , the map F (f ) : F (X) → F (Y ) is the restriction of V (f ) on the corresponding space F (X). Then F forms a covariant functor in Comp, which is a subfunctor of V . It was shown in [5] and [6] that the functor OS is normal, and OH is weakly normal, both OH(X) and OS(X) being convex compacta for any space X. Each of the abovementioned functors generates a monad. If F is one of V, O, OH, OS, P , the identity and multiplication maps are defined as follows. The natural transformation η : Id Comp → F is given by ηX(x)(ϕ) = ϕ(x) for any x ∈ X and ϕ ∈ C(X), and the natural transformation µ : Later by µ F X we shall denote the multiplication map for the corresponding functor F . According to the characterization given in [15], by L-monad we mean any submonad of V. Hence, OH and OS, being submonads of V are both L-monads. We say that an L-monad F = (F, η, µ) weakly preserves preimages ([3]) if for any mapping ) and all ϕ ∈ C(X). Let us recall the notion of convexities introduced in [3]. Let (F, η, µ) be a monad, and (X, ξ) be an F-algebra. Let A be a closed subset of X. By f A denote the quotient map f The family C F (X, ξ) forms a convexity on X. Also, any F-algebras morphism preserves convexities defined above [3]. Later we'll restrict ourselves with the binary monads. A monad F is binary if C F (X, ξ) is binary, i.e. the intersection of each linked subsystem of C F (X, ξ) is not empty (we call a family of subsets of a space linked if the intersection of the finite number of any of its elements is not empty). Theorem A.([3, Theorem 3.3]) Let F be a binary L-monad which weakly preserves preimages, and let X be such that F X is an openly generated (connected) compactum. Then each By exp X, for any compact X, we denote the space of all nonempty closed subsets of X equipped with the Vietoris topology (see, e.g., [10]). In what follows we shall need the characterization of OS(X), given in [5]. In particular, the following facts take place: • For any A ∈ exp P (X) the functional ν A given by ν A (ϕ) = sup{µ(ϕ)|µ ∈ A}, where ϕ ∈ C(X), exists and belongs to OS(X). Also ν A = ν conv(A) for any A ∈ exp P (X) (Proposition 3.2); • Any ν ∈ OS(X) coincides with a functional of the form ν A , where A = {µ ∈ P (X)|µ(ϕ) ≤ ν(ϕ) ∀ϕ ∈ C(X)} is a convex compactum in P (X), in addition, for each ϕ ∈ C(X) there is µ ∈ A such that µ(ϕ) = ν(ϕ) (Theorem 3.3); • The correspondence between functionals from OS(X) and closed convex subsets of P (X) is one-to-one (Theorem 3.4); • For any f : X → Y and ν A ∈ OS(X) we have OS(f )(ν A ) = ν P (f )(A) . For any subset A ⊂ OH(X), we see that sup A, inf A also belong to OH(X). Thus, OH(X) The following statement can be obtained by applying the same arguments as in [ From the remarks on OS made in the first section one can see that OS is in fact isomorphic to the composition of the functors cc and P . Some properties of the functor cc were studied in [8]. For any convex compact X, ccX is defined to be the set of all nonempty closed convex subsets of X, ccX is considered as the subspace of exp X. For any affine mapping f : It was shown in [3] that the monad O generated by the functor of weakly additive functionals weakly preserves preimages (Theorem 4.2). Since OH and OS are submonads of O, they weakly preserve preimages as well. Recall that the notation L stands for the superextension monad generated by the superextension functor λ (see [10] for details). For any compact X, the space λX has a functional representation which can be defined by the embedding iX : λX → ϕ∈CX [min ϕ, max ϕ] such that iX(A)(ϕ) = sup{inf ϕ(A)|A ∈ A}, where A is from λX and ϕ ∈ C(X). It is easy to see that the image iX(λX) lies in OH(X). Actually, the natural transformation i = {iX} is a monad morphism which embeds the superextension monad in OH. Therefore, by [3, Theorem 3.2], OH is binary. Now take any openly generated compactum X. Whereas the functor OH is open and the space OH(X) is convex, OH(X) is an openly generated continuum. From Theorem A we see that whenever F is a binary L-monad that weakly preserves preimages, then F (X) ∈ AR for some compact X provided F (X) is an openly generated connected compactum. Applying this fact in our case we see that OH(X) ∈ AR. Conversely, if we suppose that OH(X) ∈ AR for some compact X, then an argumentation similar to that of [4,Theorem 2] provides that X is an openly generated compactum. We therefore obtain the following fact: Theorem 1. OH(X) is an absolute retract if and only if X is an openly generated com- So what we get is that OH(X) can be an AR even when the weight of X exceeds ω 1 . The same could be said on some other functors which generate L-monads and contain L as submonad, for instance G, O, λ by itself. The functor OS seems to be closer to P . It does not give an AR in weights higher than ω 1 : Proposition 3. OS(X) is an absolute retract if and only if X is openly generated with w(X)≤ ω 1 . Proof. Follows from the results of [8], namely [8, Theorem 4.1] combined with results of [7] providing that a statement analogous to that of the proposition holds for the functors cc and P . Corollary 1. There is no monad embedding i : L ֒→ OS. Indeed, assuming the contrary, we would obtain that OS is binary. Therefore, according to [3,Theorem 3.3], the space OS(D ω 2 ), for example, must be an absolute retract, a contradiction. 3. The softness of multiplication maps for OH and OS. Theorem 2. If the multiplication map µ OS X for OS is soft then X is metrizable. Proof. Suppose that X is not metrizable and µ OS X is soft. Use [9,Theorem 3] to obtain that X is openly generated. Represent X as the limit of an ω-system S = {X α , p β α , A} with open bonding maps. Whereas µX is soft, we can assume that all limit diagrams OS 2 (X) are soft [9,Theorem 2], hence open. Now our aim is to obtain α 0 ∈ A and an accumulation point x ∈ X α 0 such that p -1 α 0 (x) contains more than one point. The weight of X is uncountable, so its character is uncountable too, since w(X) = χ(X) for any openly generated compactum [4]. Choose x 0 ∈ X with χ(x 0 , X) > ω and some α ∈ A, put x α = p α (x 0 ). Then p -1 α (x α ) contains more than one point, otherwise x 0 would have the countable character. If x α is not isolated, Then x α is the required point. Suppose that x α is isolated. Consider x 1 ∈ p -1 α (x α ) distinct from x 0 . We can choose α 1 > α with p α 1 (x 1 ) = p α 1 (x 0 ). Again p -1 α 1 (x 0 ) is not a singleton, and if p α 1 (x 0 ) is an accumulation point, we are done. Assume the opposite. Take any x 2 ∈ p -1 α 1 (p α 1 (x 0 )) with x 2 = x 0 and α 2 > α 1 such that p α 2 (x 2 ) = p α 2 (x 0 ) and continue the process as described above. If on any step i the point p α i (x 0 ) is not an accumulation point, we obtain the sequence {x i } i∈N of points in X and the up-directed chain of elements {α i } i∈N of A which has the least upper bound α 0 ∈ A. Then the space X α 0 is the limit of the inverse system {X α i , p α j α i , i ≤ j} and lim i→∞ p α 0 (x i ) = p α 0 (x 0 ). Indeed, the family {(p α 0 α i ) -1 (p α i (x 0 ))} forms a base of neighborhoods at p α (x 0 ), and for any such (p α 0 α i ) -1 (p α i (x 0 )) we see that p α 0 (x j ) is contained in it for all j ≥ i. Therefore, α 0 ∈ A and x = p α 0 (x 0 ) chosen above are as required. According to our assumption, the diagram Proof. Necessity. Let X = lim S, where S = {X α , p α , A} is an ω-system consisting of metrizable compacta and epimorphisms. The mapping µ OH X is soft, hence we can assume that all the limit diagrams of the form are open. Assume that X is not openly generated, so that there exists α ∈ A such that p α is not open. Then by Proposition 2 the mapping OH(p α ) is not open. Therefore, there is a functional ν ∈ OH(X α ) and a net {ν i } i∈I converging to ν such that the net OH(p α ) -1 (ν i ) converges to some A = OH(p α ) -1 (ν). We have that A ⊂ OH(p α ) -1 (ν). Choose two comparable elements θ 1 ∈ A and θ 2 ∈ OH(p α ) -1 (ν)\A. Let, for example, θ 1 ≤ θ 2 . Let {θ i } be a net converging to θ 2 such that θ i ∈ OH(p α ) -1 (ν i ) for all i ∈ I. We see that the net {(θ i , ηOH(X α )(ν i ))} converges to (θ 2 , ηOH(X α )(ν)). Now let V ∈ OH 2 (X) be a functional such that V(Φ) = max{Φ(θ 1 ), Φ(θ 2 )}. Then χ(V) = (θ 2 , ηOH(X α )(ν)), where χ is the characteristic map of the diagram. Choose Φ ∈ C(OH(X)) with Φ(θ 2 ) = 1 and Φ(θ) = 0 for any θ ∈ A. Then we may take that Φ(θ) ≤ f rm[o]--/2 for any OH(p α ) -1 (ν i ), hence, using [4, Lemma 2], we get that Θ(Φ) ≤ 1/2 for all Θ ∈ (OH 2 (p α )) -1 (ηOH(X α )(ν i )). Thus we obtained an open neighborhood of V of the form V = {Θ ∈ OH 2 (X) |Θ(Φ) > 1/2} with V ∩ χ -1 (θ i , ηOH(X α )(ν i )) = Ø, a contradiction which shows that X must be openly generated. Sufficiency. The monad OH is a binary monad which weakly preserves preimages. Since µ OH X : OH 2 (X) → OH(X) is an open OH-algebras morphism and OH 2 (X) is openly generated (by Theorem 3), the softness of µ OH X follows from Theorem A. The statement is proved.