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
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.
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.
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.
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.
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.
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.)
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
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 (+).
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.
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).
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
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.
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.
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.
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.
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.
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.
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.)
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?