@ Applied General Topology c© Universidad Politécnica de Valencia Volume 13, no. 1, 2012 pp. 1-10 On cofree S-spaces and cofree S-flows Behnam Khosravi Abstract Let S-Tych be the category of Tychonoff S-spaces for a topological monoid S. We study the cofree S-spaces and cofree S-flows over topo- logical spaces and we prove that for any topological space X and a topological monoid S, the function space C(S, X) with the compact- open topology and the action s · f = (t �→ f(st)) is the cofree S-space over X if and only if the compact-open topology is admissible and Ty- chonoff. Finally we study injective S-spaces and we characterize injec- tive cofree S-spaces, when the compact-open topology is admissible and Tychonoff. As a consequence of this result, we characterize the cofree S-spaces and cofree S-flows, when S is a locally compact topological monoid. 2010 MSC: 54C35, 22A30, 20M30, 54D45, 54H10. Keywords: S-space, S-flow, Cofree S-space, Cofree S-flow Compact-open topology, Injective S-space. 1. Introduction and Preliminaries There are many works about S-spaces or more specially G-spaces and their applications, and some authors study the free and projective S-spaces (G- spaces) and their applications [1, 7, 10, 11, 13, 14, 15, 16, 18]. Also, there are some results about injective and cofree Boolean S-spaces (see [1]). Recall that, for a monoid S, a set A is a left S-set (or S-act) if there is, so called, an action μ : S × A → A such that, denoting μ(s,a) := sa, (st)a = s(ta) and 1a = a. The definitions of an S-subset A of B and an S- homomorphism (also called S-map) between S-sets are clear. In fact S-maps are action-preserving maps: f : A → B with f(sa) = sf(a), for s ∈ S, a ∈ A. Each monoid S can be considered as an S-set with the action given by its mul- tiplication. Let S be a monoid and A be an S-set. Recall that for s ∈ S, the 2 B. Khosravi S-homomorphism λs : A → A is defined by y �→ sy for any y ∈ A. Similarly, for a ∈ A, the S-map ρa : S → A is defined by t �→ ta for any t ∈ S. Let C be a concrete category over D and U : C → D be the forgetful functor. An object K in C with a morphism ψ : K → D in D, where D ∈ D, is the cofree object over D, if for every morphism f : C → D in D there exists a unique morphism f̃ : C → K in C such that ψ ◦ f̃ = f in D. For any two topological spaces X and Y , we denote the set of all continuous maps from X to Y by C(X,Y ). If τ is a topology on the set C(X,Y ), then the corresponding space is denoted by Cτ (X,Y ). The category of all Tychonoff spaces is denoted by Tych. Note that all of the spaces in this note are Tychonoff (completely regular and Hausdorff). A monoid S with a Hausdorff topology τS such that the multiplication · : S × S → S is (jointly) continuous, is called a topological monoid. For a topological monoid S, an S-space is an S-set A with a topology τA such that the action S × A → A is (jointly) continuous. The category of all Tychonoff S-spaces with continuous S-maps is denoted by S-Tych (see [14, 15, 16, 18]). A compact Hausdorff S-space is called an S-flow (see [2, 13]). Let Y and Z be two topological spaces. A topology on the set C(Y,Z) is called splitting if for every space X, the continuity of a map g : X × Y → Z implies that of the map g̃ : X → Cτ (Y,Z) defined by g̃(x)(y) = g(x,y), for every x ∈ X and y ∈ Y (this topology is also called proper [3, 8] or weak [6]). A topology τ on C(Y,Z) is admissible if the mapping ω(y,f) := f(y) from Y × C(Y,Z) into Z is continuous in y and f. Equivalently, a topology τ on C(Y,Z) is admissible if for every topological space X, the continuity of an f : X → Cτ (Y,Z) implies the continuity of ̂f : X × Y → Z, where ̂f(x,y) := f(x)(y) for every (x,y) ∈ X × Y (see [8]) (the latter definition is usually used as the definition of admissible topology, but we use the former). A topological space Y is said to be exponential if for every space X there is an admissible and splitting topology on C(Y,X) (see [6]). For any topological spaces X and Y , we denote the set C(X,Y ) with the compact-open topology by Cco(X,Y ). For any compact subset K of X and an open set U in Y , by (K,U) we mean the set {f ∈ C(X,Y )|f(K) ⊆ U}. For topological topics and facts about Stone-Cech compactification, we refer to [3, 9, 17]. In this note, we study the cofree and injective S-spaces and S-flows. Recall that for a set E and a monoid S, ES, the set of all functions from S to E with the action sf := (t �→ f(st)), for any function f : S → E and s ∈ S, is the cofree S-set over E (see [12]). In Section 2, we study the cofree S-spaces and S-flows over topological spaces. As a consequence of these results, we characterize the cofree S-spaces and the cofree S-flows over topological spaces, when the compact-open topology is admissible and Tychonoff (more specially, when S is locally compact). Finally, in Section 3, we study injective cofree S-spaces and S-flows over topological spaces, when the compact-open topology is admissible and Tychonoff. Note that we state and prove our results for On cofree S-spaces and cofree S-flows 3 topological monoids and plainly all of our results hold for topological groups and G-spaces. 2. The Cofree S-spaces and S-flows over a topological space One of the main steps in the study of injective objects in a category is the study of cofree objects. These objects can be used for presenting injective cover for objects in a category1. In this section, first we study the cofree Tychonoff S- spaces over a Tychonoff space, then we study the cofree S-flow over a compact space2. Finally in this section, as a consequence of these results, we will show that if S is a locally compact topological monoid, then the cofree S-space and the cofree S-flow exist and we present them explicitly. Remark 2.1. It is a known fact that the cofree S-set over a set E, is the set ES of all functions from S to E with the action defined by s · f := (t �→ f(st)) for all f ∈ ES, s ∈ S and t ∈ S. Let E and D be two sets. Recall that for an S-set A and any function h : A → E, the S-homomorphism h : A → ES defined by h(a) := h ◦ ρa is the unique S-map such that ψ ◦ h = h, where ψ : ES → E is defined by ψ(f) := f(1), for any f ∈ ES (see [12]). Remark 2.2. (i) Let S be a topological monoid and X be a topological space. The S-space S×X with the product topology and the action λ1 : S×(S×X) → S × X, t(s,x) = (ts,x), is denoted by L(X). For a topological space X, the S-space X with the trivial action, sx := x, is denoted by T(X). (ii) Note that for any topological space X and a non-empty topological space Y , if we define cx(y) := x for every y ∈ Y , and C := {cx|x ∈ X}, then C as a subspace of Cco(Y,X) is homeomorphic to X. So the function j : X → Cco(Y,X) defined by j(x) := cx is an embedding from X to Cco(Y,X). From now on, we denote this embedding by jX for any topological space X. By Theorem 2.9 in [6], we have Remark 2.3. Let X be a topological space. Then the following are equivalent (a) X is exponential; (a) For every space Y , there exists a splitting and admissible topology on C(X,Y ); (c) X is core compact. Note that for Hausdorff spaces (and more generally for sober spaces) core com- pactness is the same as local compactness (see [5]). Furthermore, it is a known fact that if X is locally compact, then the compact-open topology on C(X,Y ) is admissible and splitting. 1Note that the cofree objects in an arbitrary category are not injective in general. 2One can easily see that if the cofree S-flow exists over a space X, then X is compact. So this assumption is not an extra assumption. 4 B. Khosravi Theorem 2.4. Let S be a topological monoid. Then the following are equiva- lent (a) For every Tychonoff space X, the compact-open topology on C(S,X) is admissible and Tychonoff; (b) For every space X, Cco(S,X) is Tychonoff and Cco(S,X) with the ac- tion defined by sf = (s′ �→ f(ss′)) is the cofree S-space over X. Proof. (a) ⇒ (b) Let X be a topological space. First we show that Cco(S,X) with its introduced action is an S-space, then we show that it has the cofree universal property. First, note that since S is a topological monoid, the action is well-defined. Let f ∈ C(S,X), s ∈ S and (K,U) be subbasis element for Cco(S,X) containing sf. Therefore for any k ∈ K, f(sk) = (sf)(k) ∈ U. Since by the assumption, the compact-open topology on C(S,X) is admissible and since for any k ∈ K, we have ω(sk,f) ∈ U, there exist open neighborhoods Of and Wsk of f and sk, in Cco(S,X) and S, respectively such that ω(Wsk,Of ) ⊆ U. On the other hand, since S is a topological monoid, for sk ∈ Wsk, there exist open sets Wks and Wk in S which contain {s} and {k}, respectively and Wks · Wk ⊆ Wsk. Since K is compact and obviously {Wk}k∈K forms an open cover for K, there exist k1, · · · ,kn in K such that K ⊆ ∪ni=1Wki . Define Ws := ∩ni=1Wkis . Clearly for Ws ∈ τS we have ω(Ws · K,Of ) ⊆ ω(Ws · (∪ni=1Wki),Of ) ⊆ U ⇒ sf ∈ WsOf ⊆ (K,U). Hence Cco(S,X) with its introduced action is an S-space. Now we prove the universal property. First note that the function ψ : Cco(S,X) → X is continuous. Let (A,τA) be an S-space and h : (A,τA) → X be continuous. We show that for any S-space (A,τA) and a continuous function h : (A,τA) → X, the function h : (A,τA) → Cτ (S,X) defined by h(a) = (s �→ h(sa)) is continuous and ψ ◦ h = h. First, note that for every a ∈ A and s ∈ S, h(a)(s) = h(sa) = h ◦ ρa(s), and since (A,τA) is an S-space, for every a ∈ A, h(a) ∈ C(S,X), so h is a well defined function. Consider the continuous function h ◦ λ : S × A → A → X (s,a) �→ sa �→ h(sa) Since τ is splitting, the function ˜(h ◦ λ) : A → Cτ (S,X), where ˜(h ◦ λ)(a) := (s �→ (h ◦ λ)(s,a)), is continuous. Therefore, h is continuous. Hence Cco(S,X) with its introduced action is the cofree S-space over X. (b) ⇒ (a) Let λ denote the action of the cofree S-space over X. It is a known fact that the compact-open topology is splitting. On the other hand, since Cco(S,X) with λ is an S-space and since ψ : Cco(S,X) → X is continuous, ω = ψ ◦ λ is continuous. Therefore the compact-open topology is admissible. Therefore, the compact-open topology is admissible and Tychonoff. � As a quick consequence of the above theorem, we have On cofree S-spaces and cofree S-flows 5 Corollary 2.5. Let S be a locally compact topological monoid. Then for any Tychonoff space X, Cco(S,X) with the action defined by sf = f ◦ λs is the cofree S-space over X. Proof. Since S is locally compact, by Remark 2.3, the compact-open topology on C(S,X) is admissible. On the other hand, by [5, Corollary 3.8], Cco(S,X) is Tychonoff. So by the above theorem we have the result. � Theorem 2.6. Let S be a completely regular topological monoid3. Then the following are equivalent: (a) for every compact space X, the compact-open topology on C(S,X) is admissible and Tychonoff; (b) for every compact space X, Cco(S,X) is completely regular and there exists an action ˜λ : S×β(Cco(S,X)) → β(Cco(S,X)) such that ˜λ|Cco(S,X) coincides with the action of Cco(S,X) and β(Cco(S,X)) is the cofree S-flow over the space X. Proof. (a)⇒ (b) Let S be a topological monoid such that the compact-open topology on C(S,X) is admissible, for every compact space X. Let X be a compact space and let λ denote the action of the cofree S-space Cco(S,X). First, we introduce ˜λ and we show that it is continuous, then we prove the universal property. Since S × Cco(S,X) is Tychonoff, β(S × Cco(S,X)) exists. By the char- acteristic of the Stone-Cech compactification, for the continuous action λ : S × Cco(S,X) → Cco(S,X), there exists a continuous function λ : β(S × Cco(S,X)) → β(Cco(S,X)) such that λ|S×Cco(S,X) = λ. Fix an arbitrary t ∈ S and define k : Cco(S,X) → S × Cco(S,X) as follows k(f) := (t,f), for ev- ery f ∈ Cco(S,X). Consider the closure of k(Cco(S,X)) in β(S × Cco(S,X)). It is obvious that there exists a compact space B such that the closure of k(Cco(S,X)) in β(S × Cco(S,X)) is equal to {t} × B. Again by the charac- teristic of the Stone-Cech compactification, there exists a continuous function h : β(Cco(S,X)) → B such that h ◦ i = k, where i is the natural inclusion map from Cco(S,X) to β(Cco(S,X)). Define λ ′ := λ|S×B. Now we define ˜λ := λ′ ◦ (idS × h) : S × β(Cco(S,X)) → S × B → β(Cco(S,X)) and we show that ˜λ is an action. Let b ∈ β(Cco(S,X)) and s,s′ ∈ S. Then since g ∈ β(Cco(S,X)), there exists a net (fα) ⊆ Cco(S,X) such that fα → g. ˜λ(ss′,g) = ˜λ(ss′, limαfα) = limαλ ′(ss′,k(fα)) = limαλ(ss ′,fα) = limαλ(s,λ(s ′,fα)) = ˜λ(s, ˜λ(s ′,g)). Therefore ˜λ is a continuous action and β(Cco(S,X)) with action ˜λ is an S-flow. Now we prove the universal property. First, let ˜ψ : β(Cco(S,X)) → X be the continuous extension of ψ : Cco(S,X) → X which exists by the characteristic of the Stone-Cech compactification. To prove the universal property, we show that for any S-flow (F,τF ) and a continuous function l : (F,τF ) → X, there 3Clearly for a topological group, this assumption is not necessary. 6 B. Khosravi exists a continuous S-map l : (F,τF ) → β(Cco(S,X)) such that ˜ψ ◦ l = l. Let (F,τF ) be an S-flow and let l : (F,τF ) → X be a continuous function. Since by Theorem 2.4, Cco(S,X) with action λ is the cofree S-space over X, there exists a continuous S-map l : (F,τF ) → Cco(S,X) such that ψ ◦ l = l. Since ˜ψ|Cco(S,X) = ψ, we have clearly ˜ψ ◦ l = l. Therefore β(Cco(S,X)) with action ˜λ is the cofree S-flow over the space X. (b) ⇐ (a) Suppose that there exists a continuous action ˜λ on β(Cco(S,X)) such that β(Cco(S,X)) with this action is an S-flow and ˜λ|Cco(S,X) = λ, where λ is the action of Cco(S,X). Therefore Cco(S,X) is an S-space. On the other hand, since ψ : Cco(S,X) → X is continuous, ω = ψ ◦ λ : S × Cco(S,X) → Cco(S,X) → X is continuous. Therefore the compact-open topol- ogy on C(S,X) is admissible and Tychonoff. � Recall that for a topological group G, a G-space (A.τA) is called G-compactifi- cable or G-Tychonoff, if (A,τA) is a G-subspace of a G-flow (compact G-space) (see [14, 15]). Since S is locally compact, it is a Tychonoff space. Therefore by the above theorem, we have Corollary 2.7. (a) Let S be a locally compact topological monoid. Then for any compact space X, there exists an action ˜λ on β(Cco(S,X)) such that β(Cco(S,X)) with this action is the cofree S-flow over the space X. (b) Let G be a locally compact topological group and Xbe a topological space. Then the cofree G-space over X is G-compactificable. 3. Injective cofree S-spaces and S-flows Recall that by an embedding of topological spaces (S-spaces) we mean a homeomorphism (homeomorphism S-map) onto a subspace (an S-subset). A topological space (S-space) Z is called injective over an embedding of topo- logical spaces (of S-spaces) j : X ↪→ Y if any continuous map (continuous S-homomorphism) f : X → Z extends to a continuous map (continuous S- homomorphism) f : Y ↪→ Z along j. A space (an S-space) is injective if it is injective over embeddings (see [5]). Proposition 3.1. Let S be a completely regular topological monoid and (E,τ) be an S-space. If (E,τ) is an injective S-space, then (|E|,τ) is injective in Tych. Proof. Suppose that we are given the following diagram in Tych i X ↪→ Y f ↓ (E,τ) On cofree S-spaces and cofree S-flows 7 where X and Y are topological spaces and f is a continuous function. Now consider the following diagram id × i L(X) ↪→ L(Y ) id × f ↓ S × E ↙ λ ↓ h E Since (E,τ) is an injective S-space, there exists a continuous S-map h from L(Y ) to (E,τ) such that h(id × i) = λ(id × f). (Note that λ(id × f) is a continuous S-homomorphism.) Note that for any topological space Z, the spaces Z and Z × {1} with the product topology are homeomorphic. Furthermore, we have (id × i)|{1}×X : {1} × X ↪→ {1} × Y , and λ ◦ (id × i)(1,x) = h ◦ (id × i)(1,x) = f(x). Now, define h′ := h|{1}×Y : {1} × Y → E and consider the following diagram (id × i)|{1}×X {1} × X −→ {1} × Y ⊆ S × Y g1 ↓ ↓ g2 X −→ Y f ↓ i E where g1 and g2 are the following homeomorphisms g1 : {1} × X → X, and g2 : {1} × Y → Y (1,x) �→ x (1,y) �→ y. Since g1 and g2 are homeomorphisms and, h ′((id×i)|{1}×X) = λ((id×f)|{1}×X) = fg1, we have h′ ◦ (id × i)|{1}×X ◦ g−11 = f. (I) On the other hand, since the rectangular in the above diagram is commutative, we have (id × i)|{1}×X ◦ g−11 = g−12 ◦ i. (II) Now, define f◦ := h′ ◦ g−12 . Clearly, by the Relations (I) and (II), f◦ is a continuous function from Y to E such that f◦i = h′g−12 i = h ′ ◦ (id × i)|{1}×X ◦ g−11 = f. � similarly, we can prove that Proposition 3.2. Let S be a compact topological monoid and (F,τF ) be an S- flow. If (F,τF ) is an injective S-flow, then (|F |,τF ) is injective in the category of compact Hausdorff spaces. 8 B. Khosravi It is known that the cofree S-spaces are not injective in general. In the next proposition, we characterize the injective cofree S-spaces when S is a locally compact topological monoid. Proposition 3.3. For a locally compact monoid S and a topological space X, the cofree S-space over X, Cco(S,X) is injective in S-Tych if and only if X is injective in Tych. Proof. (⇒) Suppose that we are given the following diagram in Tych Z f ↓ X ↪→ Y Consider the following diagram in S-Tych. T(Z) jX ◦f ↓ Cco(S,X) ↪→ T(Y ) Since Cco(S,X) is injective, there exists a continuous S-homomorphism h : T(Y ) → Cco(S,X) such that h ◦ i = jX ◦ f. Therefore, f = ψ ◦ h ◦ i. Take k := ψ ◦ h. Hence f = k ◦ i and X is injective. (⇐) Suppose that we are given i : (A,τA) ↪→ (B,τB) and f : (A,τA) → Cco(S,X) for two S-spaces (A,τA) and (B,τB). Consider the following diagram i (A,τA) ↪→ (B,τB) f ↓ Cco(S,X) ψ ↓ X Since X is injective in Tych, there exists g : (B,τB) → X such that g ◦ i = ψ ◦ f. Since Cco(S,X) is the cofree S-space over X, there exists h : (B,τB) → Cco(S,X) such that ψ ◦ h = g. We claim that h ◦ i = f. Clearly we have ψ ◦ h ◦ i = ψ ◦ f. So for every a ∈ A and s ∈ S, we have h ◦ i(a)(s) = h ◦ i(a) ◦ λs(1) = ψ(h ◦ i(a) ◦ λs) = ψ(f(a) ◦ λs) = f(a) ◦ λs(1) = f(a)(s). Hence, h ◦ i = f, as we wanted. So Cco(S,X) is an injective S-space. Hence Cco(S,X) is injective in Tych. � Similarly we have Proposition 3.4. For a completely regular monoid S and a compact Hausdorff space X, the cofree S-flow over X, β(Cco(S,X)) is injective in the category of S-flows if and only if X is injective in the category of compact Hausdorff spaces. As an immediate result of Propositions 3.1 and 3.3, we have Proposition 3.5. Let S be a locally compact monoid. Cco(S,X) is injective in Tych if and only if Cco(S,X) is injective in S-Tych. On cofree S-spaces and cofree S-flows 9 Proof. (⇐) Since for any space Z, T(Z) is an S-space, the result is obvious. (⇒) Suppose that Cco(S,X) is injective in Tych and we are given i : (A,τA) ↪→ (B,τB) and f : (A,τA) → Cco(S,X) for two S-spaces (A,τA) and (B,τB). Since Cco(S,X) is injective in Tych, there exists a continuous function g : (B,τB) → Cco(S,X) such that g ◦ i = f. i (A,τA) ↪→ (B,τB) f ↓ ↙g Cco(S,X) ψ ↓ X Since Cco(S,X) is the cofree S-space over X and ψ ◦ g : (B,τB) → X is con- tinuous, there exists a continuous S-homomorphism h : (B,τB) → Cco(S,X) such that ψ ◦ h = ψ ◦ g. Clearly we have ψ ◦ h ◦ i = ψ ◦ g ◦ i. So, by the same argument as the proof of Proposition 3.3, we have h ◦ i = f. Hence Cco(S,X) is injective in S-Tych. � Acknowledgements. The author would like to imply his gratitude to the referee and Professor Sanchis for their kindness and helps. 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