On k-groups and Tychonoff k R -spaces (Category theory for topologists, topology for group theorists, and group theory for categorical topologists)

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1 On k-groups and Tychonoff k R -spaces 0 On k-groups and Tychonoff k R -spaces (Category theory for topologists, topology for group theorists, and group theory for categorical topologists) Gábor Lukács Universität Bremen, Germany The financial support of the Humboldt Foundation is gratefully acknowledged

2 On k-groups and Tychonoff k R -spaces 1 H (G, L) is the space of continuous homomorphisms between the groups G and L, equipped with the compact-open topology. If A is abelian, one puts  = H (A, T), where T is the unit circle;  is a topological group again. Important questions: What is the relationship between A and Â? What is the relationship between A and Â? Is H (A, T) the right dual object? Every g A gives rise to a (continuous) character ω g of Â, by evaluation ω g (ζ) = ζ(g). This defines an algebraic homomorphism ω A : A  (g ω g ); it is natural in A, but it is not necessarily continuous! A is Pontryagin-reflexive if ω A is an isomorphism of topological groups.

3 On k-groups and Tychonoff k R -spaces 2 When A is LCA (locally compact and abelian), then the famous Pontryagin duality holds: ω A is an isomorphism of topological groups; Compact groups have discrete duals, and vice versa; If B A is a closed subgroup, then ˆB = Â/B and Â/B = B, where B is the annihilator of B in Â; For c(a) the connected component of A, and B(A) the subgroup of elements g such that g is compact, c(a) = Â/B(Â) and B(A) = Â/c(Â). Trouble: Infinite products of LCA groups are not LC anymore.

4 On k-groups and Tychonoff k R -spaces 3 Recall that a category C is cartesian closed if for every a C, the functor a : C C has a right adjoint (which we denote by [a, ]). One approach to overcoming the problem of the continuity of ω A is working in a (complete) cartesian closed category C. It can be immediately seen that in such categories, for every x, y C, the natural map x [[x, y], y] given by evaluation at x is a morphism. If a and b are abelian group objects in C, then it is possible to define the internal group homomorphism-functor {a, b} (which is going to be an abelian group object again!), and one obtains that a {{a, b}, b} is a morphism of group objects in C. Another interesting feature one gets for free from cartesian closure is that if d is a fixed abelian group object (that we think of as the dualizing object), then {a, {b, d}} {b, {a, d}} is an isomorphism in C (natural in a and b); in other words, we get a dual adjunction for free.

5 On k-groups and Tychonoff k R -spaces 4 Candidate I: k-spaces A test function for X Top is a continuous map t: K X with K compact Hausdorff. A map f : X Y is k-continuous if for every test function t for X, f t is continuous. (When X is Hausdorff, it suffices to require that f K is continuous for every compact subset K of X.) X is a k-space if every k-continuous map of X is continuous. Basic properties of ktop and khaus (due to Vogt and Brown, respectively): ktop and khaus are complete, cocomplete, and cartesian closed; ktop and khaus are coreflective in Top and Haus, respectively (the coreflector is the k-ification, k); The product of X, Y in ktop or khaus is given by k(x Y ); Internal hom-functor is given by the coreflection kc (X, Y ) of the compact-open topology.

6 On k-groups and Tychonoff k R -spaces 5 Abelian group objects in khaus behave very nicely in terms of ω A (in fact, the continuity of ω A for LCA groups can be deduced from here). Drawbacks: 1. The addition is only k-continuous, and it is not necessarily continuous in both variables; 2. As a result, group objects in khaus need not be Tychonoff; 3. If B is a closed subgroup of A, A/B need not be Hausdorff! Recall that a space X is weakly Hausdorff (t 2 -space) if for every test function t for X, the image of t t is closed in X X; such spaces are clearly T 1. Although by switching to the category of weakly Hausdorff k-spaces, as Lamartin (1977) did, difficulty no. 3 can be eliminated, the two other troubles nevertheless remain. Another problem: T can capture only the Hausdorff part.

7 On k-groups and Tychonoff k R -spaces 6 Remark. The success of Dubuc and Porta (1971) in describing topological algebras in the category khaus (i.e., the operations are k-continuous) was due to a dual adjunction similar to what is described above. Often the source of such dual adjunctions is cartesian closure. The incompatibility of k-spaces with the Tychonoff property is worse than one would imagine: Example(s). For topological spaces X and Y, put P for the topology of separate continuity on X Y, and set Q to be the cross-topology: V Q if the intersection of V with every fiber {x} Y and X {y} is an open subset of the fiber. Henriksen and Woods (1999) proved that k(x Y, P) = (X Y, Q) and τ(x Y, Q) = (X Y, P) for every Tychonoff k-space X and Y. (Here τ : Top Tych is the Tychonoff reflection.)

8 On k-groups and Tychonoff k R -spaces 7 Candidate II: k R -spaces X is a k R -space if every k-continuous function from X to a Tychonoff space is continuous. The k R -ification k R X always exists. Features (GL): If X is a k R -space, then so is τx; if X is Tychonoff, then so is k R X; k R Haus is coreflective in Haus; k R Tych is coreflective in Tych; k R Tych is reflective in k R Haus; Tych τ Haus (The dashed arrows are right adjoints.) k R k R k R Tych k τ τ k R Haus khaus k

9 On k-groups and Tychonoff k R -spaces 8 k R Tych is equivalent to a proper epireflective subcategory of khaus; k R Tych is cartesian closed, and the internal hom-functor is k R C (X, Y ); If P is a Tychonoff space that contains a path (i.e., a homeomorphic image of I), then η X : X k R C (k R C (X, P ), P ) is an embedding. Pros and cons: 1. Nice dual adjunction (+); 2. The trouble with forming quotients remains ( ); 3. Tychonoff k R -spaces turn out to be useful (+).

10 On k-groups and Tychonoff k R -spaces 9 Candidate III: Convergence groups & Binz-Butzmann duality A convergence space is a set X together with a relation between filters on X and points of X such that: (Conv1) F x and G x = F G x; (Conv2) F x and F G = G x; (Conv3) ẋ x, where ẋ = {A X x A}. Remark. Every convergence structure on X defines also a topology, but the convergence with respect to that topology need not be the same as the original convergence. Conv is cartesian closed, and the internal hom-functor is given by the continuous convergence structure: It is the coarsest convergence structure that makes the evaluation map e : Conv(X, Y ) X Y continuous: F ζ Conv(X, Y ) if e(f H) ζ(x) for every filter H x X.

11 On k-groups and Tychonoff k R -spaces 10 A convergence group is a group object in the category of convergence spaces and their continuous maps (Conv). If A is an abelian convergence group, one puts Γ c A to be subspace of Conv(A, T) consisting of the homomorphisms. A is BB-reflexive if the continuous homomorphism κ A : A Γ c Γ c A is an isomorphism of convergence groups. Main features (due to Butzmann): If {A i } i I are BB-reflexive, then so are A i and A i ; i I i I If A is a topological group, then Γ c A is locally compact (as a convergence group), and Γ c Γ c A is a topological group again; κ A is an embedding if and only if A is locally quasi-convex ; If ω A is continuous, then Â Γ c Γ c A; in this special case, BB-reflexivity implies P -reflexivity (Chasco & Martín-Peinador).

12 On k-groups and Tychonoff k R -spaces 11 k-groups (of Noble) A group G is a k-group if every k-continuous homomorphism ϕ: G H is continuous. Features: kgrp is a coreflective subcategory of Grp(Haus), with coreflector k G ; Quotients of k-groups are k-groups; If H is an open subgroup of G, then H is a k-group iff G is so; If {G i } i I is a family of k-groups, then G i, G i, Σ G i, and i I i I i I G i are k-groups [ G i is G i equipped with the final topology]; i I i I i I Groups that are k-spaces are also k-groups; in particular, LC and metrizable groups are k-groups; An arbitrary product M i of metrizable groups is a k-group. i I

13 On k-groups and Tychonoff k R -spaces 12 Examples and remarks: 1. Let H be a non-lc k-group such that C (H, R) is metrizable, and hence a k-group (Warner, 1958). Put G = H C (H, R); G is a k-group. The evaluation e: H C (H, R) R is k-continuous, but it can be continuous only if H is LC (Arens, 1946). Hence, G is not a k R -space. 2. A group G admits a quasi-invariant basis if for every nbhd U of e there exists a countable family V of nbhds of e such that for any g G there exists V V such that gv g 1 U. Groups with this property are precisely the subgroups of products M i of metrizable groups. i I Since not every complete group admitting a quasi-invariant basis is a k-group, such groups provide a large number of examples for closed subgroups of k-groups that are not k-groups themselves.

14 On k-groups and Tychonoff k R -spaces 13 Examples and remarks (continued): 3. The only compact subgroup of R is the trivial one, and since R has many non-continuous homomorphisms, this shows that kgrp is not the coreflective hull of the compact Hausdorff groups. 4. kgrp is the coreflective hull of the groups that are generated by a compact subset. In fact, it is also the coreflective hull of the class of free groups on compact Hausdorff spaces. By free group on a Tychonoff space X we mean the group F X, where F : Tych Grp(Haus) is the left adjoint to the forgetful functor U : Grp(Haus) Tych. We note that the unit ι X : X F X of this adjunction is a closed embedding for every X Tych and that F X is algebraically generated by X.

15 On k-groups and Tychonoff k R -spaces 14 Examples and remarks (continued): 5. The projective limit of k-groups need not be a k-group: Put A = Z ω 1 2 (only finite number of non-zero coordinates). Set B α = {(g β ) g β = 0 for β < α} for α < ω 1. Topologize A such that each B α is an open subgroup. A is complete, with base {B α } at 0; Thus, A = lim A/B α ; Each A/B α, being countable and discrete, is a k-group; A is a P -space (G δ -sets are open), so its compact subspaces are finite; Therefore, k G A is discrete. Since A is not discrete, this shows that A is not a k-group.

16 On k-groups and Tychonoff k R -spaces 15 k R Tych comes to the rescue: Example no. 5 shows that the forgetful functor U 0 : kgrp Tych does not preserve limits, because lim kgrpa/b α = k G A A = lim A/B α. In particular, it is hopeless to construct a free k-group functor on Tych. For every Tychonoff k R -space X, the free group F X is a k-group. Since the k R -ification absorbs k G in the sense that k R k G = k R, our conclusion is that the right forgetful functor is k R U 0 : kgrp k R Tych. Indeed, the restricted free group functor F 0 : k R Tych kgrp has a right adjoint, namely k R U 0. Remark. k R is an embedding of kgrp into Grp(k R Tych). Furthermore, k R H (k R G, k R L) = k R H (G, L) for G, L kgrp.

17 On k-groups and Tychonoff k R -spaces 16 Abelian groups: Let A be an abelian k-group. Noble proved: ω A is continuous; A is complete & admits a base of open subgroups A is P -reflexive; A is an open subgroup of a P -reflexive k-group A is P -reflexive; A is a kk-group if every k-continuous homomorphism of A into a compact group is continuous. The next result is due to Deaconu. Let Σ be a subgroup of Hom(A, T). There exists an LCA topology T on A such that Σ = Ĝ (algebraically) if and only if both conditions are fulfilled: (i) Σ is dense in Hom(G, T) in the topology of pointwise-convergence; (ii) Σ is a kk-group and kσ is LC.

18 On k-groups and Tychonoff k R -spaces 17 kab is monoidal closed: Let A, B, and C be abelian k-groups. A H (B, C) (1) A k G H (B, C) (2) k R A k G H (B, C) (3) k R A k R H (B, C) = k R H (k R B, k R C) (4) k R (A B) bil k R C (5) k R (A B) bil C (6) F 0 k R (A B) bil C (7) F 0 k R (A B)/R(A, B) C (8) Here R(A, B) stands for the closed subgroup generated by the commutator and the usual bilinear relations.

19 On k-groups and Tychonoff k R -spaces 18 Put A k B = F 0 k R (A B)/R(A, B); it is clearly an abelian k-group, and A k : kgrp kgrp is the left adjoint to k G H (B, ): kgrp kgrp. Since this definition is very inconvenient for computations, all computations have to be done using the right adjoints. In particular, that is the way to show that (i) Z is the neutral object with respect to k ; (ii) k satisfies the pentagon condition (with correctly chosen maps). For (ii), one uses the cartesian closure of k R Tych. Consequently, k G H (A k B, C) = k G H (A, k G H (B, C)); in particular k G A k B = k G H (A, k G ˆB) = k G H (B, k G Â). For à = kgâ, it follows from the above that γ A : A à is continuous. (Notice that à = k G Â.)

20 On k-groups and Tychonoff k R -spaces 19 Open questions: 1. How to characterize the image U 0 (kgrp) of U 0 in Tych? 2. Is à the right dual object for k-groups? For gourmands: 3. Is there a totally minimal h-complete group that is not a k-group?