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Duality theory of locally precompact groups Gábor Lukács [email protected] University of Manitoba Winnipeg, Manitoba, Canada Financial support of the Killam Trust is gratefully acknowledged Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.0/12 Ethical issues This research (including this presentation) was prepared Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.0/12 Ethical issues This research (including this presentation) was prepared without animal experiments; Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.0/12 Ethical issues This research (including this presentation) was prepared without animal experiments; without child labour; Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.0/12 Ethical issues This research (including this presentation) was prepared without animal experiments; without child labour; without using Microsoft products. Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.0/12 Pontryagin-van Kampen duality Let L be a locally compact abelian (LCA) group. Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.1/12 Pontryagin-van Kampen duality Let L be a locally compact abelian (LCA) group. L̂ is the group of continuous characters χ : L → T, equipped with the compact open topology. (T = R/Z.) Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.1/12 Pontryagin-van Kampen duality Let L be a locally compact abelian (LCA) group. L̂ = Hco (L, T). (T = R/Z.) L̂ is a locally compact abelian group. Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.1/12 Pontryagin-van Kampen duality Let L be a locally compact abelian (LCA) group. L̂ = Hco (L, T). (T = R/Z.) L̂ is LCA. ˆ The evaluation αL : L → L̂ is a topological isomorphism. Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.1/12 Pontryagin-van Kampen duality Let L be a locally compact abelian (LCA) group. L̂ = Hco (L, T). (T = R/Z.) L̂ is LCA. ˆ αL : L → L̂ is a topological isomorphism. [ ∼ M̂ ∼ = L̂/M ⊥ and L/M = M ⊥ for every closed subgroup M ≤ L. (M ⊥ = {χ ∈ L̂ | χ(M ) = 0}.) Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.1/12 Pontryagin-van Kampen duality Let L be a locally compact abelian (LCA) group. L̂ = Hco (L, T). (T = R/Z.) L̂ is LCA. ˆ αL : L → L̂ is a topological isomorphism. [ ∼ M̂ ∼ = L̂/M ⊥ and L/M = M ⊥ for every closed subgroup M ≤ L. (M ⊥ = {χ ∈ L̂ | χ(M ) = 0}.) L is discrete ⇐⇒ L̂ is compact. Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.1/12 How to extend or generalize the Pontryagin-van Kampen duality? Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.2/12 Precompact subset Unless otherwise stated, all topological groups and spaces are Tychonoff. Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.3/12 Precompact subset Unless otherwise stated, all topological groups and spaces are Tychonoff. N (G) = neighborhoods of identity in the group G. Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.3/12 Precompact subset Unless otherwise stated, all topological groups and spaces are Tychonoff. N (G) = neighborhoods of identity in the group G. B ⊆ G is precompact if for every U ∈ N (G) there is F ⊆ G finite such that B ⊆ F U . Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.3/12 Precompact subset Unless otherwise stated, all topological groups and spaces are Tychonoff. N (G) = neighborhoods of identity in the group G. B ⊆ G is precompact if ∀U∈ N (G) ∃g1 , . . . , gl ∈ G such that B ⊆ g1 U ∪ . . . ∪ gl U . B Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.3/12 Precompact subset Unless otherwise stated, all topological groups and spaces are Tychonoff. N (G) = neighborhoods of identity in the group G. B ⊆ G is precompact if ∀U∈ N (G) ∃g1 , . . . , gl ∈ G such that B ⊆ g1 U ∪ . . . ∪ gl U . g1 U B Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.3/12 Precompact subset Unless otherwise stated, all topological groups and spaces are Tychonoff. N (G) = neighborhoods of identity in the group G. B ⊆ G is precompact if ∀U∈ N (G) ∃g1 , . . . , gl ∈ G such that B ⊆ g1 U ∪ . . . ∪ gl U . g1 U g2 U B Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.3/12 Precompact subset Unless otherwise stated, all topological groups and spaces are Tychonoff. N (G) = neighborhoods of identity in the group G. B ⊆ G is precompact if ∀U∈ N (G) ∃g1 , . . . , gl ∈ G such that B ⊆ g1 U ∪ . . . ∪ gl U . g1 U g2 U ··· ··· ··· B Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.3/12 Precompact subset Unless otherwise stated, all topological groups and spaces are Tychonoff. N (G) = neighborhoods of identity in the group G. B ⊆ G is precompact if ∀U∈ N (G) ∃g1 , . . . , gl ∈ G such that B ⊆ g1 U ∪ . . . ∪ gl U . g1 U g2 U ··· ··· ··· ··· gl U B ··· ··· ··· Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.3/12 Completion {gα } ⊆ G is a Cauchy net (two-sided uniformity) if gα gβ−1 −→ e, and gα−1 gβ −→ e. Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.4/12 Completion {gα } ⊆ G is a Cauchy net (two-sided uniformity) if gα gβ−1 −→ e, and gα−1 gβ −→ e. G is complete if every Cauchy net converges. Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.4/12 Completion {gα } ⊆ G is a Cauchy net (two-sided uniformity) if gα gβ−1 −→ e, and gα−1 gβ −→ e. G is complete if every Cauchy net converges. (Raı̆kov, 1946) Every topological group G admits a group completion Ḡ. Ḡ is unique up to topological isomorphism. Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.4/12 Completion {gα } ⊆ G is a Cauchy net (two-sided uniformity) if gα gβ−1 −→ e, and gα−1 gβ −→ e. G is complete if every Cauchy net converges. (Raı̆kov, 1946) Every topological group G admits a group completion Ḡ. Ḡ is unique up to topological isomorphism. (Weil, 1937) G is precompact if and only if Ḡ is compact. Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.4/12 Comfort-Ross duality A a discrete abelian group; K := Â its (compact) dual. Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.5/12 Comfort-Ross duality A a discrete abelian group; K := Â its (compact) dual. For H ≤ K, τH is the initial group topology with respect to ιH : A −→ TH a 7−→ (χ(a))χ∈H . Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.5/12 Comfort-Ross duality A a discrete abelian group; K := Â its (compact) dual. For H ≤ K, τH is the initial group topology with respect to ιH : A −→ TH . Comfort and Ross (1964): (A, τH ) is precompact. Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.5/12 Comfort-Ross duality A a discrete abelian group; K := Â its (compact) dual. For H ≤ K, τH is the initial group topology with respect to ιH : A −→ TH . Comfort and Ross (1964): (A, τH ) is precompact. \ If (A, τ ) is precompact, then τ = τH for H = (A, τ ). That is, every precompact group topology on A has the form τH . Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.5/12 Comfort-Ross duality A a discrete abelian group; K := Â its (compact) dual. For H ≤ K, τH is the initial group topology with respect to ιH : A −→ TH . Comfort and Ross (1964): (A, τH ) is precompact. \ If (A, τ ) is precompact, then τ = τH for H = (A, τ ). (A, τH ) is Hausdorff ⇐⇒ H is dense in K. Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.5/12 Locally precompact groups G is locally precompact if N (G) contains a precompact set. Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.6/12 Locally precompact groups G is locally precompact if N (G) contains a precompact set. (Weil, 1937) G is locally precompact if and only if Ḡ is locally compact. Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.6/12 Locally precompact groups G is locally precompact if N (G) contains a precompact set. (Weil, 1937) G is locally precompact if and only if Ḡ is locally compact. A locally precompact group G can be encoded as a discrete group Gd , a locally compact group Ḡ, and a dense injection iG : Gd → Ḡ. Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.6/12 Locally precompact groups G is locally precompact if N (G) contains a precompact set. (Weil, 1937) G is locally precompact if and only if Ḡ is locally compact. G locally precompact, G 7−→ (iG : Gd → Ḡ). Let D be discrete, L be locally compact. Given a dense injection i : D → L, G = i(L) is locally precompact, and i = iG . Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.6/12 Locally precompact groups G is locally precompact if N (G) contains a precompact set. (Weil, 1937) G is locally precompact if and only if Ḡ is locally compact. Equivalence of categories: G locally precompact, G 7−→ (iG : Gd → Ḡ). D discrete, L is LC, i : D → L is a dense injection, i(L) ←−[ (i : D → L). Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.6/12 Duality of morphisms Set M = LCA·→· , the category of morphisms in LCA. Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.7/12 Duality of morphisms Set M = LCA·→· , the category of morphisms in LCA. ˆ: M −→ Mop is an equivalence of categories. Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.7/12 Duality of morphisms Set M = LCA·→· , the category of morphisms in LCA. ˆ: M −→ Mop is an equivalence of categories. f ∈ ob M is injective ⇐⇒ fˆ is dense. Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.7/12 Duality of morphisms Set M = LCA·→· , the category of morphisms in LCA. ˆ: M −→ Mop is an equivalence of categories. f ∈ ob M is injective ⇐⇒ fˆ is dense. ⇓ The full subcategory DI of dense injections in M is self-dual. Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.7/12 Duality of morphisms Set M = LCA·→· , the category of morphisms in LCA. ˆ: M −→ Mop is an equivalence of categories. f ∈ ob M is injective ⇐⇒ fˆ is dense. DI (= dense injections in M) is self-dual. Consider the following full subcategories of DI: ob D = {i : D → L | D is discrete}. ob K = {i : L → K | K compact}. Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.7/12 Duality of morphisms Set M = LCA·→· , the category of morphisms in LCA. ˆ: M −→ Mop is an equivalence of categories. DI (= dense injections in M) is self-dual. Consider the following full subcategories of DI: ob D = {i : D → L | D is discrete}. ob K = {i : L → K | K compact}. ˆ: D −→ Kop is an equivalence of categories. Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.7/12 Duality of morphisms Set M = LCA·→· , the category of morphisms in LCA. ˆ: M −→ Mop is an equivalence of categories. DI (= dense injections in M) is self-dual. Consider the following full subcategories of DI: ob D = {i : D → L | D is discrete}. ob K = {i : L → K | K compact}. ˆ: D −→ Kop is an equivalence of categories. D is equivalent to the category LPA. Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.7/12 Duality of morphisms Set M = LCA·→· , the category of morphisms in LCA. ˆ: M −→ Mop is an equivalence of categories. DI (= dense injections in M) is self-dual. Consider the following full subcategories of DI: ob D = {i : D → L | D is discrete}. ob K = {i : L → K | K compact}. ˆ: D −→ Kop is an equivalence of categories. D is equivalent to the category LPA. K is the category of group compactifications of LCA groups. Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.7/12 Duality of LPA groups Let G be an LPA group. Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.8/12 Duality of LPA groups Let G be an LPA group. b̄ → G cd is its dual; ibG : G Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.8/12 Duality of LPA groups Let G be an LPA group. b̄ → G cd is its dual; ibG : G b̄ , τ ) of the group cd is the completion ((G) G d G b̄ equipped with the precompact topology τ . (G) d G Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.8/12 Duality of LPA groups Let G be an LPA group. b̄ → G cd is its dual; ibG : G b̄ , τ ) of the group cd is the completion ((G) G d G b̄ equipped with the precompact topology τ . (G) d G Let i : L → K be a compactification (L is LCA). Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.8/12 Duality of LPA groups Let G be an LPA group. b̄ → G cd is its dual; ibG : G b̄ , τ ) of the group cd is the completion ((G) G d G b̄ equipped with the precompact topology τ . (G) d G Let i : L → K be a compactification (L is LCA). G = î(K̂) is its dual (î : K̂ → L̂) ; Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.8/12 Duality of LPA groups Let G be an LPA group. b̄ → G cd is its dual; ibG : G b̄ , τ ) of the group cd is the completion ((G) G d G b̄ equipped with the precompact topology τ . (G) d G Let i : L → K be a compactification (L is LCA). G = î(K̂) is its dual; G carries the subgroup topology induced by L̂. Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.8/12 Duality of LPA groups (GL, 2006) The functors b̄ → G cd ) G 7−→ (ibG : G î(K̂) ←−[ (i : L → K) form a duality between LPA and K. Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.8/12 Duality of LPA groups (GL, 2006) The functors b̄ → G cd ) G 7−→ (ibG : G î(K̂) ←−[ (i : L → K) form a duality between LPA and K. Special cases: Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.8/12 Duality of LPA groups (GL, 2006) The functors b̄ → G cd ) G 7−→ (ibG : G î(K̂) ←−[ (i : L → K) form a duality between LPA and K. Special cases: G = L: Its dual is L̂ → bL̂ (Bohr compactification), so one obtains the Pontryagin duality. Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.8/12 Duality of LPA groups (GL, 2006) The functors b̄ → G cd ) G 7−→ (ibG : G î(K̂) ←−[ (i : L → K) form a duality between LPA and K. Special cases: G = L: Its dual is L̂ → bL̂ (Bohr compactification), so one obtains the Pontryagin duality. b̄ is discrete. This G precompact: Ḡ is compact, so G is precisely the Comfort-Ross duality. Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.8/12 Dual properties Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.9/12 Gδ -topology in LCA Gδ -topology: ∞ T Un is a Gδ -neighborhood of x0 if n=1 each Un is a neighborhood of x0 . Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.10/12 Gδ -topology in LCA Gδ -topology: N (δG) = { ∞ T Un | Un ∈ N (G)}. n=1 Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.10/12 Gδ -topology in LCA Gδ -topology: N (δG) = { ∞ T Un | Un ∈ N (G)}. n=1 Let L be an LCA group. Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.10/12 Gδ -topology in LCA Gδ -topology: N (δG) = { ∞ T Un | Un ∈ N (G)}. n=1 Let L be an LCA group. Σ(L̂) = {Σ ≤ L̂ | Σ is open and σ-compact}. Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.10/12 Gδ -topology in LCA Gδ -topology: N (δG) = { ∞ T Un | Un ∈ N (G)}. n=1 Let L be an LCA group. Σ(L̂) = {Σ ≤ L̂ | Σ is open and σ-compact}. {Σ⊥ | Σ ∈ Σ(L̂)} is a base at 0 for δL. Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.10/12 Gδ -topology in LCA Gδ -topology: N (δG) = { ∞ T Un | Un ∈ N (G)}. n=1 Let L be an LCA group. Σ(L̂) = {Σ ≤ L̂ | Σ is open and σ-compact}. {Σ⊥ | Σ ∈ Σ(L̂)} is a base at 0 for δL. a ∈ L is in the Gδ -closure of X ⊆ L ⇔ for each Σ ∈ Σ(L̂), there is x ∈ X such that x − a ∈ Σ⊥ . Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.10/12 Gδ -topology in LCA Gδ -topology: N (δG) = { ∞ T Un | Un ∈ N (G)}. n=1 Let L be an LCA group. Σ(L̂) = {Σ ≤ L̂ | Σ is open and σ-compact}. {Σ⊥ | Σ ∈ Σ(L̂)} is a base at 0 for δL. a ∈ L is in the Gδ -closure of X ⊆ L ⇔ for each Σ ∈ Σ(L̂), there is x ∈ X such that x|Σ = a|Σ . Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.10/12 Gδ -topology in LCA Gδ -topology: N (δG) = { ∞ T Un | Un ∈ N (G)}. n=1 Let L be an LCA group. Σ(L̂) = {Σ ≤ L̂ | Σ is open and σ-compact}. {Σ⊥ | Σ ∈ Σ(L̂)} is a base at 0 for δL. clδL X = {a ∈ L | ∀Σ ∈ Σ(L̂)∃x ∈ X, x|Σ = a|Σ }. Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.10/12 Local pseudocompactness G is locally pseudocompact if N (G) contains a pseudocompact set. Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.11/12 Local pseudocompactness G is locally pseudocompact if N (G) contains a pseudocompact set. (Comfort and Trigos-Arrieta, 1995) The following are equivalent: G is locally pseudocompact; β Ḡ = βG (Stone-Čech-compactification); υ Ḡ = υG (Hewitt-realcompactification); Gδ -dense in Ḡ and Ḡ is locally compact. Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.11/12 Local pseudocompactness G is locally pseudocompact if N (G) contains a pseudocompact set. (Comfort and Trigos-Arrieta, 1995) The following are equivalent: G is locally pseudocompact; Gδ -dense in Ḡ and Ḡ is locally compact. Let G be an LPA group. b̄ clδḠ G = {a ∈ Ḡ | ∀Σ ∈ Σ(G)∃g ∈ G, g|Σ = a|Σ }. Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.11/12 Local pseudocompactness (Comfort and Trigos-Arrieta, 1995) The following are equivalent: G is locally pseudocompact; Gδ -dense in Ḡ and Ḡ is locally compact. Let G be an LPA group. b̄ clδḠ G = {a ∈ Ḡ | ∀Σ ∈ Σ(G)∃g ∈ G, g|Σ = a|Σ }. ⇓ b̄ a is τ -continuous}. clδḠ G = {a ∈ Ḡ | ∀Σ ∈ Σ(G), G|Σ Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.11/12 Local pseudocompactness (Comfort and Trigos-Arrieta, 1995) The following are equivalent: G is locally pseudocompact; Gδ -dense in Ḡ and Ḡ is locally compact. Let G be an LPA group. b̄ a is τ -continuous}. clδḠ G = {a ∈ Ḡ | ∀Σ ∈ Σ(G), G|Σ ⇓ G is locally pseudocompact ⇔ ∀Σ, τG|Σ = τḠ|Σ . Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.11/12 Local pseudocompactness Let G be an LPA group. G is locally pseudocompact ⇔ ∀Σ, τG|Σ = τḠ|Σ . ⇓ (GL, 2006) Let G be a locally precompact abelian group. The following are equivalent: G is locally pseudocompact; τG|Σ is the Bohr-topology on Σ for every b̄ Σ ∈ Σ(G). Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.11/12 Local pseudocompactness Let G be an LPA group. G is locally pseudocompact ⇔ ∀Σ, τG|Σ = τḠ|Σ . (GL, 2006) Let G be a locally precompact abelian group. The following are equivalent: G is locally pseudocompact; τG|Σ is the Bohr-topology on Σ for every b̄ Σ ∈ Σ(G). Special case: G precompact (Hernández and Macario, 2003). Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.11/12 Realcompactness Index of precompactness ip(G) = smallest cardinal τ such that for every U ∈ N (G), there is S ⊆ G such that |S| ≤ τ and G = SU . Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.12/12 Realcompactness ip(G) = sup inf{|S| | G = SU }. U ∈N (G) (GL, 2006) Let G be a locally precompact group. Then G is realcompact if and only if ip(G) is not Ulam-measurable, and G is Gδ -closed in Ḡ. Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.12/12 Realcompactness ip(G) = sup inf{|S| | G = SU }. U ∈N (G) (GL, 2006) Let G be a locally precompact group. Then G is realcompact if and only if ip(G) is not Ulam-measurable, and G is Gδ -closed in Ḡ. κ is Ulam-measurable if there is a σ-additive measure µ : P (κ) → {0, 1} such that µ(κ) = 1; µ({λ}) = 0 for every λ ∈ κ. Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.12/12 Realcompactness ip(G) = sup inf{|S| | G = SU }. U ∈N (G) (GL, 2006) Let G be a locally precompact group. Then G is realcompact if and only if ip(G) is not Ulam-measurable, and G is Gδ -closed in Ḡ. Let G be an LPA group. b̄ a is τ -continuous}. clδḠ G = {a ∈ Ḡ | ∀Σ ∈ Σ(G), G|Σ Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.12/12 Realcompactness (GL, 2006) Let G be a locally precompact group. Then G is realcompact if and only if ip(G) is not Ulam-measurable, and G is Gδ -closed in Ḡ. Let G be an LPA group. b̄ a is τ -continuous}. clδḠ G = {a ∈ Ḡ | ∀Σ ∈ Σ(G), G|Σ ⇓ G = clδḠ G ⇔ if a ∈ Ḡ is τG|Σ -continuous for every b̄ then a ∈ G. Σ ∈ Σ(G), Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.12/12 Realcompactness (GL, 2006) Let G be a locally precompact group. Then G is realcompact if and only if ip(G) is not Ulam-measurable, and G is Gδ -closed in Ḡ. Let G be an LPA group. G = clδḠ G ⇔ if a ∈ Ḡ is τG|Σ -continuous for every b̄ then a is τ -continuous. Σ ∈ Σ(G), G Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.12/12 Realcompactness (GL, 2006) Let G be a locally precompact abelian group. Then G is realcompact if and only if ip(G) is not Ulam-measurable, and b̄ such that ψ is τ -continuevery character ψ of G |Σ G|Σ b̄ is τ -continuous. ous on every Σ ∈ Σ(G) G ⇑ Let G be an LPA group. G = clδḠ G ⇔ if a ∈ Ḡ is τG|Σ -continuous for every b̄ then a is τ -continuous. Σ ∈ Σ(G), G Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.12/12 Realcompactness (GL, 2006) Let G be a locally precompact abelian group. Then G is realcompact if and only if ip(G) is not Ulam-measurable, and b̄ such that ψ is τ -continuevery character ψ of G |Σ G|Σ b̄ is τ -continuous. ous on every Σ ∈ Σ(G) G Special cases: G LCA (Comfort, Hernández, Trigos-Arrieta, 1996); G precompact (Hernández and Macario, 2003). Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.12/12